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limits and colimits by example (changes) in nLab
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| </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/11423/#Item_5" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #49 to #50: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='limits_and_colimits'>Limits and colimits</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits and colimits</a></strong></p> <h2 id='1categorical'>1-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit and colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limits+and+colimits+by+example'>limits and colimits by example</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commutativity of limits and colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/directed+colimit'>directed colimit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/sequential+colimit'>sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sifted+colimit'>sifted colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+limit'>connected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/wide+pullback'>wide pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserved limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>created limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a>, <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a>, <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>, <a class='existingWikiWord' href='/nlab/show/diff/cobase+change'>cobase change</a>, <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>, <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+product'>direct product</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+extension'>Yoneda extension</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end and coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibered+limit'>fibered limit</a></p> </li> </ul> <h2 id='2categorical'>2-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isoinserter'>isoinserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PIE-limit'>PIE-limit</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> </ul> <h2 id='1categorical_2'>(∞,1)-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id='modelcategorical'>Model-categorical</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+product'>homotopy product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equalizer'>homotopy equalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+totalization'>homotopy totalization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy end</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coproduct'>homotopy coproduct</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coequalizer'>homotopy coequalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pushout'>homotopy pushout</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+realization'>homotopy realization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/infinity-limits+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <p>This entry lists and discusses examples and special types of the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal constructions</a> called <em><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a></em>.</p> <p>It starts with very elementary and simple examples and eventually passes to more sophisticated ones.</p> <p>For examples of the other kinds of <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal constructions</a> see</p> <ul> <li> <p><em><a class='existingWikiWord' href='/nlab/show/diff/examples+of+adjoint+functors'>examples of adjoint functors</a></em></p> </li> <li> <p><em><a class='existingWikiWord' href='/nlab/show/diff/examples+of+Kan+extensions'>examples of Kan extensions</a></em></p> </li> </ul> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#limcoliminset'>Limits and colimits of sets</a><ul><li><a href='#limits'>Limits</a><ul><li><a href='#terminal_object'>Terminal object</a></li><li><a href='#product'>Product</a></li><li><a href='#equalizer'>Equalizer</a></li><li><a href='#pullback'>Pullback</a></li><li><a href='#general_limits'>General limits</a></li></ul></li><li><a href='#colimits'>Colimits</a><ul><li><a href='#initial_object'>Initial object</a></li><li><a href='#coproduct'>Coproduct</a></li><li><a href='#coequalizer'>Coequalizer</a></li><li><a href='#pushout'>Pushout</a></li><li><a href='#general_colimits'>General colimits</a></li></ul></li></ul></li><li><a href='#OfTopologicalSpaces'>Limits and colimits of topological spaces</a></li><li><a href='#limits_and_colimits_in_a_preordered_set'>Limits and colimits in a preordered set</a></li><li><a href='#limits_and_colimits_in_functor_categories'>Limits and colimits in functor categories</a><del class='diffdel'><ul><li><a href='#ColimitOfARepresentablefunctor'>Colimit of a representable functor</a></li></ul></del></li><ins class='diffins'><li><a href='#ColimitOfARepresentablefunctor'>Colimit of a representable functor</a></li></ins><li><a href='#examplesoflimits'>Examples of limits</a><ul><li><a href='#limitssimplediagrams'>Simple diagrams</a></li><li><a href='#filteredlimits'>Filtered limits</a></li><li><a href='#limitsintermsofotherops'>In terms of other operations</a></li><li><a href='#limitsinpresheafcat'>Limits in presheaf categories</a></li><li><a href='#limitsinundercat'>Limits in under-categories</a></li></ul></li><li><a href='#further_resources'>Further resources</a></li></ul></div> <h2 id='limcoliminset'>Limits and colimits of sets</h2> <p>In the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a>, the concepts of <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a> and <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a> reduce to the familiar operations of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a> of sets;</li> <li><a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of sets;</li> <li><a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a> defined by <a class='existingWikiWord' href='/nlab/show/diff/equation'>equations</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/quotient+set'>quotient sets</a> of <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relations</a>.</li> </ul> <h3 id='limits'>Limits</h3> <h4 id='terminal_object'>Terminal object</h4> <p>The <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> is the limit of the empty functor <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>∅</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>F: \emptyset \to Set</annotation></semantics></math>. So a <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> is a set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> such that there is a unique function from any set to <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. This is given by any <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a> set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>a</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{a\}</annotation></semantics></math>, where the unique function <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>→</mo><mo stretchy='false'>{</mo><mi>a</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>Y \to \{a\}</annotation></semantics></math> from any set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is the function that sends every element in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>.</p> <h4 id='product'>Product</h4> <p>Given two sets <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A, B</annotation></semantics></math>, the categorical <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> is the limit of the diagram (with no non-trivial maps)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ A & B }. </annotation></semantics></math></div> <p>This is given by the usual product of sets, which can be constructed as the set of <a class='existingWikiWord' href='/nlab/show/diff/Kuratowski+pairs'>Kuratowski pairs</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>=</mo><mo stretchy='false'>{</mo><mo stretchy='false'>{</mo><mo stretchy='false'>{</mo><mi>a</mi><mo stretchy='false'>}</mo><mo>,</mo><mo stretchy='false'>{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>}</mo><mo stretchy='false'>}</mo><mo>:</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>B</mi><mo stretchy='false'>}</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> A \times B = \{ \{\{a\}, \{a, b\}\}: a \in A, b \in B\}. </annotation></semantics></math></div> <p>We tend to write <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a, b)</annotation></semantics></math> instead of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mo stretchy='false'>{</mo><mi>a</mi><mo stretchy='false'>}</mo><mo>,</mo><mo stretchy='false'>{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>}</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\{a\}, \{a, b\}\}</annotation></semantics></math>.</p> <p>The projection maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo>:</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\pi_1: A \times B \to A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo>:</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>\pi_2: A \times B \to B</annotation></semantics></math> are given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>a</mi><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>b</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1(a, b) = a, \pi_2(a, b) = b. </annotation></semantics></math></div> <p>To see this satisfies the universal property of products, given any pair of maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>f: X \to A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>g: X \to B</annotation></semantics></math>, we obtain a map <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>(f, g): X \to A \times B</annotation></semantics></math> given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> (f, g)(x) = (f(x), g(x)). </annotation></semantics></math></div> <p>More generally, given a (possibility infinite) collection of sets <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>A</mi> <mi>α</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>α</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{A_\alpha\}_{\alpha \in I}</annotation></semantics></math>, the product of the discrete diagram consisting of these sets is the usual product <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>α</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>A</mi> <mi>α</mi></msub></mrow><annotation encoding='application/x-tex'>\prod_{\alpha \in I} A_\alpha</annotation></semantics></math>. This can be constructed as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>α</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>A</mi> <mi>α</mi></msub><mo>=</mo><mrow><mo>{</mo><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>α</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>A</mi> <mi>α</mi></msub><mo lspace='mediummathspace' rspace='mediummathspace'>∣</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>A</mi> <mi>α</mi></msub><mspace width='thickmathspace' /><mtext> for all </mtext><mspace width='thickmathspace' /><mi>α</mi><mo>∈</mo><mi>I</mi><mo>}</mo></mrow><mo>.</mo></mrow><annotation encoding='application/x-tex'> \prod_{\alpha \in I} A_\alpha = \left\{f: I \to \bigcup_{\alpha \in I} A_\alpha \mid f(\alpha) \in A_\alpha\;\text{ for all }\;\alpha \in I\right\}. </annotation></semantics></math></div> <h4 id='equalizer'>Equalizer</h4> <p>Given a pair of functions <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f, g: X \to Y</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a> is the limit of the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><munder><mover><mo>⇉</mo><mi>f</mi></mover><mi>g</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X & \underset{g}{\overset{f}{\rightrightarrows}} & Y }. </annotation></semantics></math></div> <p>The limit is given by a map <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>e: A \to X</annotation></semantics></math> such that given any <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>a: B \to X</annotation></semantics></math>, it factors through <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi></mrow><annotation encoding='application/x-tex'>e</annotation></semantics></math> if and only if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>a</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>f \circ a = g \circ a</annotation></semantics></math>. In other words, <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> factors through <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi></mrow><annotation encoding='application/x-tex'>e</annotation></semantics></math> if and only if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>im</mo><mi>a</mi><mo>⊆</mo><mo stretchy='false'>{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>:</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\im a \subseteq \{x \in X: f(x) = g(x)\}</annotation></semantics></math>. Thus the limit of the diagram is given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy='false'>{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>:</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo><mo>,</mo></mrow><annotation encoding='application/x-tex'> A = \{x \in X: f(x) = g(x)\}, </annotation></semantics></math></div> <p>and the map <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>e: A \to X</annotation></semantics></math> is given by the inclusion.</p> <h4 id='pullback'>Pullback</h4> <p>Given two maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>f: A \to C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>g: B \to C</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> is the limit of the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>→</mo><mi>g</mi></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ & & A\\ & & \downarrow^\mathrlap{f}\\ B & \underset{g}{\to} & C }. </annotation></semantics></math></div> <p>This limit is given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>:</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo><mo>,</mo></mrow><annotation encoding='application/x-tex'> \{(a, b) \in A \times B: f(a) = g(b)\}, </annotation></semantics></math></div> <p>with the maps to <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> given by the projections.</p> <p>While the definition of a pullback is symmetric in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>, it is usually convenient to think of this as pulling back <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> (or the other way round). This has more natural interpretations in certain special cases.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>g: B \to C</annotation></semantics></math> is the inclusion of a subset (ie. is a <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a>), then the pullback of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo>:</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>B</mi><mo stretchy='false'>}</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> \{a \in A: f(a) \in B\}. </annotation></semantics></math></div> <p>So this is given by restricting <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> to the elements that are mapped into <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>.</p> <p>Further, if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>f: A \to C</annotation></semantics></math> is <em>also</em> the inclusion fo a subset, so that <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> are both subobjects of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, then the above formula tells us that the pullback is simply the intersection of the two subsets.</p> <p>Alternatively, we can view the map <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>f: A \to C</annotation></semantics></math> as a collection of sets indexed by elements of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, where the set indexed by <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> is given by <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mi>c</mi></msub><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A_c = f^{-1}(c)</annotation></semantics></math>. Under this interpretation, pulling <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> back along <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> gives a collection of sets indexed by elements of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, where the set indexed by <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>b \in B</annotation></semantics></math> is given b <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>A_{g(b)}</annotation></semantics></math>.</p> <h4 id='general_limits'>General limits</h4> <p>Given a general <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>-valued functor <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>F : D \to Set</annotation></semantics></math>, if the limit <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>lim F</annotation></semantics></math> exists, then by definition, for any set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, a function <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>f: A \to lim F</annotation></semantics></math> is equivalent to a compatible family of maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>d</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f_d: A \to F(d)</annotation></semantics></math> for each <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d \in Obj(D)</annotation></semantics></math>.</p> <p>In particular, since an element of a set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> bijects with maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>1 \to X</annotation></semantics></math> from the singleton <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>=</mo><mo stretchy='false'>{</mo><mi>∅</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>1 = \{\emptyset\}</annotation></semantics></math>, we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi><mo>≅</mo><mi>Set</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy='false'>)</mo><mo>≅</mo><mo stretchy='false'>[</mo><mi>D</mi><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><msub><mi>const</mi> <mn>1</mn></msub><mo>,</mo><mi>F</mi><mo stretchy='false'>)</mo><mo>,</mo></mrow><annotation encoding='application/x-tex'> lim F \cong Set (1, lim F) \cong [D, Set](const_1, F), </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>const</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>const_1</annotation></semantics></math> is the functor that constantly takes the value <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>. Thus the limit is given by the set of <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a> from <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>const</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>const_1</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>.</p> <p>More concretely, a compatible family of maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>1 \to F(d)</annotation></semantics></math> is given by an element <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>s</mi> <mi>d</mi></msub><mo>∈</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>s_d \in F(d)</annotation></semantics></math> for each <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d \in Obj(d)</annotation></semantics></math>, satisfying the appropriate compatibility conditions. Thus, the limit can be realized as a subset of the product <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow></msub><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\prod_{d \in Obj(d)} F(d)</annotation></semantics></math> of all objects:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi><mo>=</mo><mrow><mo>{</mo><mo stretchy='false'>(</mo><msub><mi>s</mi> <mi>d</mi></msub><msub><mo stretchy='false'>)</mo> <mi>d</mi></msub><mo>∈</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>d</mi></munder><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><mo>∀</mo><mo stretchy='false'>(</mo><mi>d</mi><mover><mo>→</mo><mi>f</mi></mover><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>s</mi> <mi>d</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>s</mi> <mrow><mi>d</mi><mo>′</mo></mrow></msub><mo>}</mo></mrow><mo>.</mo></mrow><annotation encoding='application/x-tex'> lim F = \left\{ (s_d)_d \in \prod_d F(d) | \forall (d \stackrel{f}{\to} d') : F(f)(s_{d}) = s_{d'} \right\}. </annotation></semantics></math></div> <h3 id='colimits'>Colimits</h3> <h4 id='initial_object'>Initial object</h4> <p>The <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> is a set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> such that there is a unique map from <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to any other set. This is given by the empty set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\emptyset</annotation></semantics></math>.</p> <h4 id='coproduct'>Coproduct</h4> <p>(…)</p> <h4 id='coequalizer'>Coequalizer</h4> <p>(…)</p> <h4 id='pushout'>Pushout</h4> <p>(…)</p> <h4 id='general_colimits'>General colimits</h4> <p>The colimit over a <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>-valued functor <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>F : D \to Set</annotation></semantics></math> is a quotient set of the disjoint union <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></msub><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\coprod_{d \in Obj(D)} F(d)</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>colim</mi><mi>F</mi><mo>≃</mo><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></munder><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><msub><mo stretchy='false'>/</mo> <mo>∼</mo></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> colim F \simeq \left(\coprod_{d\in D} F(d)\right)/_\sim \,, </annotation></semantics></math></div> <p>where the equivalence relation <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math> is that which is <em>generated</em> by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>∈</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∼</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>′</mo><mo>∈</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='1em' /><mi>if</mi><mspace width='1em' /><mo stretchy='false'>(</mo><mo>∃</mo><mo stretchy='false'>(</mo><mi>f</mi><mo>:</mo><mi>d</mi><mo>→</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mspace width='1em' /><mi>with</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>x</mi><mo>′</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> ((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists (f : d \to d') \quad with F(f)(x) = x') \,. </annotation></semantics></math></div> <p>If <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/filtered+category'>filtered category</a> then the resulting equivalence relation can be described as follows:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>∈</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∼</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>′</mo><mo>∈</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='1em' /><mi>iff</mi><mspace width='1em' /><mo stretchy='false'>(</mo><mo>∃</mo><mi>d</mi><mo>″</mo><mo>,</mo><mo stretchy='false'>(</mo><mi>f</mi><mo>:</mo><mi>d</mi><mo>→</mo><mi>d</mi><mo>″</mo><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>:</mo><mi>d</mi><mo>′</mo><mo>→</mo><mi>d</mi><mo>″</mo><mo stretchy='false'>)</mo><mspace width='1em' /><mi>with</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> ((x \in F(d)) \sim (x' \in F(d')))\quad iff \quad (\exists d'', (f : d \to d''), (g: d' \to d'') \quad with F(f)(x) = F(g)(x')) \,. </annotation></semantics></math></div> <p>(If <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is not filtered, then this description doesn’t yield an equivalence relation.)</p> <h2 id='OfTopologicalSpaces'>Limits and colimits of topological spaces</h2> <p>We discuss limits and colimits in the category <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>.</p> <p><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <p><strong>examples of <a href='Top#UniversalConstructions'>universal constructions of topological spaces</a>:</strong></p> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{AAAA}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a></th><th><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{AAAA}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union topological space</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> fiber space <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> </tbody></table> <p><math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <div class='num_defn' id='InitialAndFinalTopologies'> <h6 id='definition'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>∈</mo><mi>Top</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/class'>class</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, and let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S \in Set</annotation></semantics></math> be a bare <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a>. Then</p> <ul> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{S \stackrel{f_i}{\to} S_i \}_{i \in I}</annotation></semantics></math> a set of <a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> out of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, the <em><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a></em> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{initial}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/extremum'>minimum</a> collection of <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> such that all <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>.</p> </li> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>S</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><mi>S</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{S_i \stackrel{f_i}{\to} S\}_{i \in I}</annotation></semantics></math> a set of <a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> into <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, the <em><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></em> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{final}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/extremum'>maximum</a> collection of <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> such that all <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>.</p> </li> </ul> </div> <div class='num_example' id='TopologicalSubspace'> <h6 id='example'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a single topological space, and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>S</mi><mo>↪</mo><mi>U</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\iota_S \colon S \hookrightarrow U(X)</annotation></semantics></math> a subset of its underlying set, then the initial topology <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>intial</mi></msub><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{intial}(\iota_S)</annotation></semantics></math>, def. <a class='maruku-ref' href='#InitialAndFinalTopologies'>1</a>, is the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, making</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X </annotation></semantics></math></div> <p>a <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> inclusion.</p> </div> <div class='num_example' id='QuotientTopology'> <h6 id='example_2'>Example</h6> <p>Conversely, for <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>S</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>U</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⟶</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>p_S \colon U(X) \longrightarrow S</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphism</a>, then the final topology <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><msub><mi>p</mi> <mi>S</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{final}(p_S)</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is the <em><a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a></em>.</p> </div> <div class='num_prop' id='DescriptionOfLimitsAndColimitsInTop'> <h6 id='proposition'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small category</a> and let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>⟶</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>X_\bullet \colon I \longrightarrow Top</annotation></semantics></math> be an <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> (a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> from <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>), with components denoted <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X_i = (S_i, \tau_i)</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S_i \in Set</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\tau_i</annotation></semantics></math> a topology on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>S_i</annotation></semantics></math>. Then:</p> <ol> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>X_\bullet</annotation></semantics></math> exists and is given by <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> topological space whose underlying set is <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> limit in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of the underlying sets in the diagram, and whose topology is the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, def. <a class='maruku-ref' href='#InitialAndFinalTopologies'>1</a>, for the functions <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>p_i</annotation></semantics></math> which are the limiting <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> components:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mrow><msub><mi>p</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd /> <mtd><munder><mo>⟶</mo><mrow /></munder></mtd> <mtd /> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,. </annotation></semantics></math></div> <p>Hence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) </annotation></semantics></math></div></li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>X_\bullet</annotation></semantics></math> exists and is the topological space whose underlying set is the colimit in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of the underlying diagram of sets, and whose topology is the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a>, def. <a class='maruku-ref' href='#InitialAndFinalTopologies'>1</a> for the component maps <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\iota_i</annotation></semantics></math> of the colimiting <a class='existingWikiWord' href='/nlab/show/diff/cocone'>cocone</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd /> <mtd><msub><mo>↙</mo> <mpadded width='0'><mrow><msub><mi>ι</mi> <mi>j</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,. </annotation></semantics></math></div> <p>Hence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>ι</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right) </annotation></semantics></math></div></li> </ol> </div> <p>(e.g. <a href='#Bourbaki71'>Bourbaki 71, section I.4</a>)</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The required <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)</annotation></semantics></math> is immediate: for</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mi>i</mi></msub></mtd> <mtd /> <mtd><munder><mo>⟶</mo><mrow /></munder></mtd> <mtd /> <mtd><msub><mi>X</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_i } </annotation></semantics></math></div> <p>any <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> over the diagram, then by construction there is a unique function of underlying sets <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⟶</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i</annotation></semantics></math> making the required diagrams commute, and so all that is required is that this unique function is always <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>. But this is precisely what the initial topology ensures.</p> <p>The case of the colimit is <a class='existingWikiWord' href='/nlab/show/diff/duality'>formally dual</a>.</p> </div> <div class='num_example' id='PointTopologicalSpaceAsEmptyLimit'> <h6 id='example_3'>Example</h6> <p>The limit over the empty diagram in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math> with its unique topology.</p> </div> <div class='num_example' id='DisjointUnionOfTopologicalSpacesIsCoproduct'> <h6 id='example_4'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\sqcup} X_i \in Top</annotation></semantics></math> is their <em><a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a></em>.</p> </div> <p>In particular:</p> <div class='num_example' id='DiscreteTopologicalSpaceAsCoproduct'> <h6 id='example_5'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S \in Set</annotation></semantics></math>, the <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-indexed <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> of the point, <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mo>*</mo></mrow><annotation encoding='application/x-tex'>\underset{s \in S}{\coprod}\ast </annotation></semantics></math>, is the set <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> itself equipped with the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a>, hence is the <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>discrete topological space</a> on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>.</p> </div> <div class='num_example' id='ProductTopologicalSpace'> <h6 id='example_6'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\prod} X_i \in Top</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> of the underlying sets equipped with the <em><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a></em>, also called the <em><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>Tychonoff product</a></em>.</p> <p>In the case that <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite set</a>, such as for binary product spaces <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math>, then a <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>sub-basis</a> for the product topology is given by the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian products</a> of the open subsets of (a basis for) each factor space.</p> </div> <div class='num_example' id='EqualizerInTop'> <h6 id='example_7'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a> of two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> is the equalizer of the underlying functions of sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>eq</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>↪</mo><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub></mrow><annotation encoding='application/x-tex'> eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y </annotation></semantics></math></div> <p>(hence the largets subset of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>S_X</annotation></semantics></math> on which both functions coincide) and equipped with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, example <a class='maruku-ref' href='#TopologicalSubspace'>1</a>.</p> </div> <div class='num_example' id='CoequalizerInTop'> <h6 id='example_8'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a> of two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> is the coequalizer of the underlying functions of sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub><mo>⟶</mo><mi>coeq</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g) </annotation></semantics></math></div> <p>(hence the <a class='existingWikiWord' href='/nlab/show/diff/quotient+set'>quotient set</a> by the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> generated by <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∼</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(x) \sim g(x)</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>) and equipped with the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a>, example <a class='maruku-ref' href='#QuotientTopology'>2</a>.</p> </div> <div class='num_example' id='PushoutInTop'> <h6 id='example_9'>Example</h6> <p>For</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X } </annotation></semantics></math></div> <p>two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> out of the same <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a>, then the <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> under this diagram is also called the <em><a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a></em>, denoted</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mspace width='thinmathspace' /><mo>.</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,. </annotation></semantics></math></div> <p>(Here <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow><annotation encoding='application/x-tex'>g_\ast f</annotation></semantics></math> is also called the pushout of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>, or the <em><a class='existingWikiWord' href='/nlab/show/diff/base+change'>cobase change</a></em> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>.) If <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is an inclusion, one also write <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \cup_f Y</annotation></semantics></math> and calls this the <em><a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>attaching space</a></em>.</p> <div style='float: left; margin: 0 10px 10px 0;'><img src='http://ncatlab.org/nlab/files/AttachingSpace.jpg' width='450' /></div> <p>By example <a class='maruku-ref' href='#CoequalizerInTop'>8</a> the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>/<a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>attaching space</a> is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mo>≃</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'> X \sqcup_A Y \simeq (X\sqcup Y)/\sim </annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> subject to the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> which identifies a point in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with a point in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> if they have the same pre-image in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <p>(graphics from <a href='#AguilarGitlerPrieto02'>Aguilar-Gitler-Prieto 02</a>)</p> </div> <div class='num_example' id='TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself'> <h6 id='example_10'>Example</h6> <p>As an important special case of example <a class='maruku-ref' href='#PushoutInTop'>9</a>, let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace='verythinmathspace'>:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> i_n \colon S^{n-1}\longrightarrow D^n </annotation></semantics></math></div> <p>be the canonical inclusion of the standard <a class='existingWikiWord' href='/nlab/show/diff/sphere'>(n-1)-sphere</a> as the <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> of the standard <a class='existingWikiWord' href='/nlab/show/diff/ball'>n-disk</a> (both regarded as <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> with their <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> as subspaces of the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math>).</p> <div style='float: left; margin: 0 10px 10px 0;'> <img src='http://ncatlab.org/nlab/files/GluingHemispheres.jpg' width='400' /></div> <p>Then the colimit in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> under the diagram, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>i_n</annotation></semantics></math> along itself,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>{</mo><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>⟵</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>}</mo></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,, </annotation></semantics></math></div> <p>is the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. </annotation></semantics></math></div> <p>(graphics from Ueno-Shiga-Morita 95)</p> </div> <p>W</p> <h2 id='limits_and_colimits_in_a_preordered_set'>Limits and colimits in a preordered set</h2> <p>(..)</p> <h2 id='limits_and_colimits_in_functor_categories'>Limits and colimits in functor categories</h2> <del class='diffmod'><h3 id='ColimitOfARepresentablefunctor'>Colimit of a representable functor</h3></del><ins class='diffmod'><p>The main point is that <em><a class='existingWikiWord' href='/nlab/show/diff/limits+of+presheaves+are+computed+objectwise'>limits of functors are computed objectwise</a></em>. See there for more</p></ins> <del class='diffmod'><p>The <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> over a <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a> (with values in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>) is the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a>, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>.</p></del><ins class='diffmod'><h2 id='ColimitOfARepresentablefunctor'>Colimit of a representable functor</h2></ins> <ins class='diffins'><p>The <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> of a <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a> (with values in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>) is the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a>, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>.</p></ins><ins class='diffins'> </ins><p>One can readily see this from a <a class='existingWikiWord' href='/nlab/show/diff/universal+element'>universal element</a>-style argument, by direct inspection of <a class='existingWikiWord' href='/nlab/show/diff/cone'>cocones</a>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi><mo stretchy='false'>(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mi>D</mi><mover><mo>→</mo><mi>f</mi></mover><mi>C</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow></mover></mtd> <mtd /> <mtd><mi>𝒞</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>↘</mo></mtd> <mtd /> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mrow><mo>{</mo><mi>C</mi><mover><mo>→</mo><mi>id</mi></mover><mi>C</mi><mo>}</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \mathcal{C}(D,C) && \overset{ (D \overset{f}{\to}C)^\ast }{\longleftarrow} && \mathcal{C}(C,C) \\ & \searrow && \swarrow \\ && \left\{ C \overset{id}{\to}C \right\} } </annotation></semantics></math></div> <p>However, this style of reasoning does not easily generalize to <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a>. The following gives a more abstract argument that is short and generalizes. We state it in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> just for definiteness of notation:</p> <div class='num_prop' id='HodgeStarFollowedByHodgeStar'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a> over a <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a> is <a class='existingWikiWord' href='/nlab/show/diff/n-truncated+object+of+an+%28infinity%2C1%29-category'>contractible</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/essentially+small+%28infinity%2C1%29-category'>small (∞,1)-category</a> and consider an <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞-groupoid</a>-valued <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>→</mo><msub><mi>Groupoids</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>F \colon \mathcal{C}^{op} \to Groupoids_\infty</annotation></semantics></math> which is <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable</a>, i.e. in the <a class='existingWikiWord' href='/nlab/show/diff/essential+image'>image</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>≃</mo><mi>y</mi><mi>C</mi></mrow><annotation encoding='application/x-tex'>F \simeq y C</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>y</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>Func</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>Groupoids</mi> <mn>∞</mn></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>y</mi><mi>C</mi><mo>≔</mo><mi>𝒞</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \mathcal{C} & \overset{ \phantom{AA} y \phantom{AA} }{\longrightarrow} & Func \big( \mathcal{C}^{op}, Groupoids_\infty \big) \\ C &\mapsto& y C \coloneqq \mathcal{C}(-,C) } </annotation></semantics></math></div> <p>Then the <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-colimit</a> over this <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a> is contractible, i.e. is the point, the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object</a> in <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Groupoids</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi><munder><mo>⟶</mo><mi>𝒞</mi></munder></munder><mo stretchy='false'>(</mo><mi>y</mi><mi>C</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mo>*</mo></mrow><annotation encoding='application/x-tex'> \underset{ \underset{\mathcal{C}}{\longrightarrow} }{\lim} (y C) \;\simeq\; \ast </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>The terminal <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math> is characterized by the fact that for each <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi>Groupoids</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>S \in Groupoids_\infty</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Groupoids</mi> <mn>∞</mn></msub><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>Groupoids_\infty(\ast, S) \simeq S</annotation></semantics></math>. Therefore it is sufficient to show that <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi><mo>⟶</mo></munder><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>y</mi><mi>C</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>\underset{\longrightarrow}{\lim}\big(y C\big)</annotation></semantics></math> has the same property:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>Groupoids</mi> <mn>∞</mn></msub><mrow><mo>(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>y</mi><mi>C</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>,</mo><mi>S</mi><mo>)</mo></mrow></mtd> <mtd><mo>≃</mo><mspace width='thickmathspace' /><mi>Func</mi><mo stretchy='false'>(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mspace width='thinmathspace' /><msub><mi>Groupoids</mi> <mn>∞</mn></msub><mo stretchy='false'>)</mo><mrow><mo>(</mo><mi>y</mi><mi>C</mi><mo>,</mo><mi>const</mi><mi>S</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>const</mi><mi>S</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><mspace width='thickmathspace' /><mi>S</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} Groupoids_\infty \left( \underset{\longrightarrow}{\lim} \big( y C \big) , S \right) & \simeq\; Func( \mathcal{C}^{op} ,\, Groupoids_\infty ) \left( y C , const S \right) \\ & \simeq\; (const S)(C) \\ & \simeq\; S \end{aligned} </annotation></semantics></math></div> <p>Here the first step is the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Func</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>Groupoids</mi> <mn>∞</mn></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>⟵</mo><mi>const</mi></munder><mover><mo>⟶</mo><munder><mi>lim</mi><mo>⟶</mo></munder></mover></munderover><msub><mo lspace='0em' rspace='thinmathspace'>Groupoids</mo> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'> Func \big( \mathcal{C}^{op}, Groupoids_\infty \big) \underoverset { \underset{const}{\longleftarrow} } { \overset{ \underset{\longrightarrow}{\lim} }{\longrightarrow} } {\phantom{AA}\bot\phantom{AA}} \Groupoids_\infty </annotation></semantics></math></div> <p>and the second step is the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a></p> </div> <h2 id='examplesoflimits'>Examples of limits</h2> <p>In the following examples, <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small category</a>, <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is any category and the limit is taken over a functor <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F : D^{op} \to C</annotation></semantics></math>.</p> <h3 id='limitssimplediagrams'>Simple diagrams</h3> <ul> <li> <p>the limit of the empty diagram <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>D = \emptyset</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is, if it exists <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>;</p> </li> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/discrete+category'>discrete category</a>, i.e. a category with only identity morphisms, then a diagram <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F : D \to C</annotation></semantics></math> is just a collection <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>c_i</annotation></semantics></math> of objects of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. Its limit is the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>i</mi></msub><msub><mi>c</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\prod_i c_i</annotation></semantics></math> of these.</p> </li> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>=</mo><mo stretchy='false'>{</mo><mi>a</mi><mover><mo>→</mo><mo>→</mo></mover><mi>b</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>D = \{a \stackrel{\to}{\to} b\}</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>lim F</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a> of the two morphisms <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(b) \to F(a)</annotation></semantics></math>.</p> </li> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> (so that <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>D^{op}</annotation></semantics></math>), then the limit of any <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F : D^{op} \to C</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(I)</annotation></semantics></math>.</p> </li> </ul> <h3 id='filteredlimits'>Filtered limits</h3> <ul> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a>, then the limit over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>D^{op}</annotation></semantics></math> is the supremum over the <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(d)</annotation></semantics></math> with respect to <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>≤</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>⇔</mo><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mover><mo>←</mo><mrow><mi>F</mi><mo stretchy='false'>(</mo><mo>≤</mo><mo stretchy='false'>)</mo></mrow></mover><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(F(d) \leq F(d')) \Leftrightarrow (F(d) \stackrel{F(\leq)}{\leftarrow} F(d'))</annotation></semantics></math>;</p> </li> <li> <p>the generalization of this is where the term “limit” for categorical limit (probably) originates from: for <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/filtered+category'>filtered category</a>, hence <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>D^{op}</annotation></semantics></math> a cofiltered category, one may think of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>d</mi><mover><mo>→</mo><mi>f</mi></mover><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>↦</mo><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mover><mo>←</mo><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mover><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(d \stackrel{f}{\to} d') \mapsto (F(d) \stackrel{F(f)}{\leftarrow} F(d')</annotation></semantics></math> as witnessing that <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(d')</annotation></semantics></math> is “larger than” <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(d)</annotation></semantics></math> in some sense, and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>lim F</annotation></semantics></math> is then the “largest” of all these objects, the limiting object. This interpretation is perhaps more evident for filtered <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a>, where the codomain category <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is thought of as being the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo><msup><mi>E</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C = E^{op}</annotation></semantics></math>. See the motivation at <a class='existingWikiWord' href='/nlab/show/diff/ind-object'>ind-object</a>.</p> </li> </ul> <h3 id='limitsintermsofotherops'>In terms of other operations</h3> <p>If products and equalizers exist in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, then limit of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F : D^{op} \to C</annotation></semantics></math> can be exhibited as a <a class='existingWikiWord' href='/nlab/show/diff/subobject'>subobject</a> of the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> of the <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(d)</annotation></semantics></math>, namely the <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a> of</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></munder><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><mo stretchy='false'>⟨</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>∘</mo><msub><mi>p</mi> <mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>⟩</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></msub></mrow></mover><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></munder><mi>F</mi><mo stretchy='false'>(</mo><mi>s</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \prod_{d \in Obj(D)} F(d) \stackrel{\langle F(f) \circ p_{F(t(f))} \rangle_{f \in Mor(D)} }{\to} \prod_{f \in Mor(D)} F(s(f)) </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></munder><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><mo stretchy='false'>⟨</mo><msub><mi>p</mi> <mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>s</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>⟩</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></msub></mrow></mover><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow></munder><mi>F</mi><mo stretchy='false'>(</mo><mi>s</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \prod_{d \in Obj(D)} F(d) \stackrel{\langle p_{F(s(f))} \rangle_{f \in Mor(D)} }{\to} \prod_{f \in Mor(D)} F(s(f)) \,. </annotation></semantics></math></div> <p>See the explicit formula for the limit in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> in terms of a subset of a product set.</p> <p>In particular therefore, a category has all limits already if it has all products and equalizers.</p> <h3 id='limitsinpresheafcat'>Limits in presheaf categories</h3> <p>Consider limits of functors <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F : D^{op} \to PSh(C)</annotation></semantics></math> with values in the <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a> over a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <div class='num_prop' id='LimitsOfPresheaves'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/limits+of+presheaves+are+computed+objectwise'>limits of presheaves are computed objectwise</a>)</strong></p> <p>Limits of presheaves are computed objectwise:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi><mo>:</mo><mi>c</mi><mo>↦</mo><mi>lim</mi><mi>F</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> lim F : c \mapsto lim F(-)(c) </annotation></semantics></math></div> <p>Here on the right the limit is over the functor <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>F(-)(c) : D^{op} \to Set</annotation></semantics></math>.</p> <p>Similarly for colimits</p> </div> <p>Similarly <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a>s of presheaves are computed objectwise.</p> <div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Y : C \to PSh(C)</annotation></semantics></math> commutes with small limits:</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F : D^{op} \to C</annotation></semantics></math>, then we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo stretchy='false'>(</mo><mi>lim</mi><mi>F</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>lim</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>∘</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Y(lim F) \simeq lim (Y\circ F) </annotation></semantics></math></div> <p>if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>lim F</annotation></semantics></math> exists.</p> </div> <p><strong>Warning</strong> The Yoneda embedding does <em>not</em> in general preserve colimits.</p> <h3 id='limitsinundercat'>Limits in under-categories</h3> <p>Limits in <a class='existingWikiWord' href='/nlab/show/diff/under+category'>under categories</a> are a special case of limits in <a class='existingWikiWord' href='/nlab/show/diff/comma+category'>comma categories</a>. These are explained elsewhere. It may still be useful to spell out some details for the special case of under-categories. This is what the following does.</p> <div class='num_prop'> <h6 id='proposition_5'>Proposition</h6> <p>Limits in an <a class='existingWikiWord' href='/nlab/show/diff/under+category'>under category</a> are computed as limits in the underlying category.</p> <p>Precisely: let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>t \in C</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a>, and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>t/C</annotation></semantics></math> the corresponding <a class='existingWikiWord' href='/nlab/show/diff/under+category'>under category</a>, and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>p : t/C \to C</annotation></semantics></math> the obvious projection.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F : D \to t/C</annotation></semantics></math> be any <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a>. Then, if it exists, the <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>p \circ F</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is the image under <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> of the limit over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mi>lim</mi><mi>F</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>lim</mi><mo stretchy='false'>(</mo><mi>p</mi><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> p(\lim F) \simeq \lim (p F) </annotation></semantics></math></div> <p>and <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>\lim F</annotation></semantics></math> is uniquely characterized by <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mo stretchy='false'>(</mo><mi>p</mi><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\lim (p F)</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>Over a morphism <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>:</mo><mi>d</mi><mo>→</mo><mi>d</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\gamma : d \to d'</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> the limiting cone over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>p F</annotation></semantics></math> (which exists by assumption) looks like</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>lim</mi><mi>p</mi><mi>F</mi></mtd></mtr> <mtr><mtd /> <mtd><mo>↙</mo></mtd> <mtd /> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd /> <mtd><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') } </annotation></semantics></math></div> <p>By the universal property of the limit this has a unique lift to a cone in the <a class='existingWikiWord' href='/nlab/show/diff/under+category'>under category</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>t/C</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>t</mi></mtd></mtr> <mtr><mtd /> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>lim</mi><mi>p</mi><mi>F</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd /> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd /> <mtd><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') } </annotation></semantics></math></div> <p>It therefore remains to show that this is indeed a limiting cone over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>. Again, this is immediate from the universal property of the limit in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. For let <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding='application/x-tex'>t \to Q</annotation></semantics></math> be another cone over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>t/C</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> is another cone over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>p F</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and we get in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a universal morphism <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mo>→</mo><mi>lim</mi><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>Q \to \lim p F</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>t</mi></mtd></mtr> <mtr><mtd /> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>Q</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>lim</mi><mi>p</mi><mi>F</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd /> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd /> <mtd><mi>p</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') } </annotation></semantics></math></div> <p>A glance at the diagram above shows that the composite <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>→</mo><mi>Q</mi><mo>→</mo><mi>lim</mi><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>t \to Q \to \lim p F</annotation></semantics></math> constitutes a morphism of cones in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> into the limiting cone over <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>p F</annotation></semantics></math>. Hence it must equal our morphism <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>→</mo><mi>lim</mi><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>t \to \lim p F</annotation></semantics></math>, by the universal property of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>\lim p F</annotation></semantics></math>, and hence the above diagram does commute as indicated.</p> <p>This shows that the morphism <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mo>→</mo><mi>lim</mi><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>Q \to \lim p F</annotation></semantics></math> which was the unique one giving a cone morphism on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> does lift to a cone morphism in <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>t/C</annotation></semantics></math>, which is then necessarily unique, too. This demonstrates the required universal property of <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>→</mo><mi>lim</mi><mi>p</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>t \to \lim p F</annotation></semantics></math> and thus identifies it with <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding='application/x-tex'>\lim F</annotation></semantics></math>.</p> </div> <ul> <li>Remark: One often says “<math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U: A \to C</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic</a> (i.e., has a left adjoint <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> such that the canonical <a class='existingWikiWord' href='/nlab/show/diff/comparison+functor'>comparison functor</a> <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>→</mo><mo stretchy='false'>(</mo><mi>U</mi><mi>F</mi><mo stretchy='false'>)</mo><mtext>-</mtext><mi>Alg</mi></mrow><annotation encoding='application/x-tex'>A \to (U F)\text{-}Alg</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence</a>), then <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> both reflects and preserves limits. In the present case, the projection <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><mo>=</mo><mi>t</mi><mo stretchy='false'>/</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>p: A = t/C \to C</annotation></semantics></math> is monadic, is essentially the category of algebras for the monad <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_306' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>t</mi><mo>+</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>T(-) = t + (-)</annotation></semantics></math>, at least if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_307' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> admits binary coproducts. (Added later: the proof is even simpler: if <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_308' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U: A \to C</annotation></semantics></math> is the underlying functor for the category of algebras of an <em>endofunctor</em> on <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> (as opposed to algebras of a monad), then <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> reflects and preserves limits; then apply this to the endofunctor <math class='maruku-mathml' display='inline' id='mathml_3501aa82402e9016b16014fe5ed583b44b8abf90_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> above.)</li> </ul> <h2 id='further_resources'>Further resources</h2> <p>Pedagogical vidoes that explain limits and colimits are at</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/The+Catsters'>The Catsters</a>, <a href='http://www.youtube.com/watch?v=g47V6qxKQNU'>General Limits and Colimits</a></li> </ul> </div> <div class="revisedby"> <p> Last revised on April 20, 2023 at 17:58:37. See the <a href="/nlab/history/limits+and+colimits+by+example" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/limits+and+colimits+by+example" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11423/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/limits+and+colimits+by+example/49" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/limits+and+colimits+by+example" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/limits+and+colimits+by+example" accesskey="S" class="navlink" id="history" rel="nofollow">History (49 revisions)</a> <a href="/nlab/show/limits+and+colimits+by+example/cite" style="color: black">Cite</a> <a href="/nlab/print/limits+and+colimits+by+example" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/limits+and+colimits+by+example" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>