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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/318/#Item_14" 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 #28 to #29: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <blockquote> <p>This entry is mostly about cones in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>. For more geometric cones see at <em><a class='existingWikiWord' href='/nlab/show/diff/cone+%28Riemannian+geometry%29'>cone (Riemannian geometry)</a></em>.</p> </blockquote> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'>Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></strong></p> <h2 id='sidebar_concepts'>Concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </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</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> </ul> <h2 id='sidebar_extensions'>Extensions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></div> <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+limit'>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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#in_homotopy_theory'>In homotopy theory</a></li><li><a href='#as_a_monad'>As a monad</a></li><li><a href='#in_category_theory'>In category theory</a></li><li><a href='#ConesOverADiagram'>Cones over a diagram</a></li><li><a href='#over_a_diagram_in_an_category'>Over a diagram in an <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category</a></li></ul></li><li><a href='#see_also'>See also</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>In <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, the <em>cone</em> of a <a class='existingWikiWord' href='/nlab/show/diff/space'>space</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the space obtained by taking the <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>-shaped <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X \times I</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> may be an <a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a>, and squashing one end down to a point. The eponymous example is where <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is the standard interval <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math>. Then the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X \times I</annotation></semantics></math> really is a <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, and the cone of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is likewise a cone.</p> <p>This notion also makes sense when <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, if <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is taken to be the <a class='existingWikiWord' href='/nlab/show/diff/interval+category'>interval category</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{ 0 \to 1 \}</annotation></semantics></math>, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/ordinal+number'>ordinal</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mn>2</mn></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{2}</annotation></semantics></math>. Note that since the interval category is directed, this gives two different kinds of cone, depending on which end we squash down to a point.</p> <p>Another, perhaps more common, meaning of ‘cone’ in category theory is that of a <em>cone over (or under) a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a></em>. This is just a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> over the cone category, as above. Explicitly, a cone over <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F\colon J \to C</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> equipped with a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> from <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> to each vertex of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_20' 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 every <em>new</em> triangle arising in this way <a class='existingWikiWord' href='/nlab/show/diff/commutative+triangle'>commutes</a>. A cone which is <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal</a> is a <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>, the word <em><a class='existingWikiWord' href='/nlab/show/diff/cocone'>cocone</a></em> is sometimes used for the case when we squash the other end of the interval; thus <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> is equipped with a morphism <em>to</em> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> <em>from</em> each vertex of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> (but <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> itself still belongs to <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>). A cocone in this sense which is universal is a <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a>. However, one should beware that in homotopy theory, the word <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>cocone</a> is used for a different dualization.</p> <p>This definition generalizes to <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a>. In particular in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> a cone over an <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a> is essentially a cone in the sense of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>.</p> <h2 id='definition'>Definition</h2> <h3 id='in_homotopy_theory'>In homotopy theory</h3> <p>If <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a space, then the cone of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+pushout'>homotopy pushout</a> of the identity on <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> along the unique map to the point:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X & \to & X \\ \downarrow & & \downarrow \\ * & \to & cone(X) }\,. </annotation></semantics></math></div> <p>This homotopy pushout can be computed as the ordinary <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>X</mi><mo>×</mo><mi>I</mi><msub><mo>⨿</mo> <mi>X</mi></msub><mo>*</mo></mrow><annotation encoding='application/x-tex'>cone(X) := X\times I \amalg_X *</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>I</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X &\stackrel{d_1}{\to} & X \times I \\ \downarrow && \downarrow \\ * &\to& cone(X) } \,. </annotation></semantics></math></div> <p>If <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>, then the cone of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/join+of+simplicial+sets'>join</a> of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with the point.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a> (q.v.) of a morphism <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> is then the pushout along <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> of the inclusion <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>cone</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \to cone(X)</annotation></semantics></math>.</p> <h3 id='as_a_monad'>As a monad</h3> <p>In contexts where <a class='existingWikiWord' href='/nlab/show/diff/interval'>intervals</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> can be treated as <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> objects, the cone construction as quotient of a cylinder with one end identified with a point,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>I</mi><mo>×</mo><mi>X</mi><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mn>0</mn><mo>×</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>∼</mo><mi>p</mi><mo>,</mo></mrow><annotation encoding='application/x-tex'>C(X) = I \times X/(0 \times X) \sim p,</annotation></semantics></math></div> <p>carries a structure of <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. In such cases, the monoid has a multiplicative <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> and an absorbing element <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>, where multiplication by <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> is the constant map at <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>. In that case, a <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-algebra consists of an object <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> together with</p> <ul> <li> <p>An <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> of the monoid, <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>:</mo><mi>I</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>a: I \times X \to X</annotation></semantics></math>.</p> </li> <li> <p>A constant or basepoint <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo lspace='verythinmathspace'>:</mo><mn>1</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x_0 \colon 1 \to X</annotation></semantics></math></p> </li> </ul> <p>such that <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>a(0, x) = x_0</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>. This equation can be expressed in any category <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math> with finite products and a suitable interval object <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> as monoid (for example, <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>I = [0, 1]</annotation></semantics></math> is a monoid under real multiplication, or under <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>min</mi></mrow><annotation encoding='application/x-tex'>min</annotation></semantics></math> as multiplication). Under some reasonable assumptions (e.g., if the <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math> has quotients, and these are preserved by the functor <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>×</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo></mrow><annotation encoding='application/x-tex'>I \times -</annotation></semantics></math>), the category of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-algebras will be monadic over <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math> and the free <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-algebra on <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> will be <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(X)</annotation></semantics></math> as described above. The category of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-algebras will also be monadic over the category of pointed <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math>-objects, <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>↓</mo><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>1 \downarrow \mathbf{C}</annotation></semantics></math>.</p> <p>These observations apply for example to <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>, and also to <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> where the <a class='existingWikiWord' href='/nlab/show/diff/interval+category'>interval category</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mn>2</mn></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{2}</annotation></semantics></math> is a monoid in <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> under the <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>min</mi></mrow><annotation encoding='application/x-tex'>min</annotation></semantics></math> operation (see below).</p> <p>If in addition the underlying category <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed</a>, or more generally if <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/exponential+object'>exponentiable</a>, the monad <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> on pointed <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>C</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{C}</annotation></semantics></math>-objects also has a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> which can be regarded as a <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a> construction <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>, where we have a pullback</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msup><mi>X</mi> <mi>I</mi></msup></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>eval</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ P(X) & \to & 1 \\ \downarrow & & \downarrow \\ X^I & \stackrel{eval_0}{\to} & X. } </annotation></semantics></math></div> <p>For general abstract reasons, the right adjoint <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> carries a comonad structure whereby <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-algebras are equivalent to <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>-coalgebras. Considered over the category of simplicial sets, this is closely connected to <a class='existingWikiWord' href='/nlab/show/diff/decalage'>decalage</a>.</p> <h3 id='in_category_theory'>In category theory</h3> <p>If <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a category, then the cone of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_82' 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 <span class='newWikiWord'>cocomma category<a href='/nlab/new/cocomma+category'>?</a></span> of the identity on <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and the unique map to the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>⇒</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>cone</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ C & \to & C \\ \downarrow & \Rightarrow & \downarrow \\ * & \to & cone(C) }\,. </annotation></semantics></math></div> <p>Again, this may be computed as a pushout:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>C</mi><mo>×</mo><mstyle mathvariant='bold'><mn>2</mn></mstyle></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>cone</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ C &\stackrel{d_1}{\to} & C \times \mathbf{2} \\ \downarrow && \downarrow \\ * &\to& cone(C) } \,. </annotation></semantics></math></div> <p>The cone of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> may equivalently be thought of, or defined, as the result of adjoining a new <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> to <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <h3 id='ConesOverADiagram'>Cones over a diagram</h3> <p>A cone <em>in</em> a category <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is given by a category <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> together with a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cone</mi><mo stretchy='false'>(</mo><mi>J</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>cone(J) \to C</annotation></semantics></math>. By the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the cocomma category, to give such a functor is to give an object <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, a functor <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>F \colon J \to C</annotation></semantics></math>, and a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mo>:</mo><mi>Δ</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>T: \Delta(c) \to F</annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>\Delta(c):J\to C</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/diff/constant+functor'>constant functor</a> at the object <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>. Such a transformation is called a <em>cone over the diagram</em> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_97' 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>In other words, a cone consists of morphisms (called the <strong>components</strong> of the cone)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>j</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>c</mi><mo>⟶</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> T_j \;\colon\; c \longrightarrow F(j) \,, </annotation></semantics></math></div> <p>one for each object <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math>, which are compatible with all the morphisms <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(f): F(j) \to F(k)</annotation></semantics></math> of the diagram, in the sense that each diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mrow /></mtd> <mtd><mrow /></mtd> <mtd><mi>c</mi></mtd> <mtd><mrow /></mtd> <mtd><mrow /></mtd></mtr> <mtr><mtd><mrow /></mtd> <mtd><mpadded lspace='-100%width' width='0'><mstyle scriptlevel='1'><mrow><msub><mi>T</mi> <mi>j</mi></msub></mrow></mstyle></mpadded><mo>↙</mo></mtd> <mtd><mrow /></mtd> <mtd><mo>↘</mo><mpadded width='0'><mstyle scriptlevel='1'><mrow><msub><mi>T</mi> <mi>k</mi></msub></mrow></mstyle></mpadded></mtd> <mtd><mrow /></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo></mtd> <mtd><mrow /></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mrow /></mtd> <mtd><mi>F</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ {}&{}&c&{}&{} \\ {}& \mathllap{\scriptsize{T_j}}\swarrow &{}& \searrow\mathrlap{\scriptsize{T_k}} &{} \\ F(j) &{}& \stackrel{F(f)}{\longrightarrow} &{}& F(k) \\ } </annotation></semantics></math></div> <p>commutes.</p> <p>It’s called a <em>cone</em> because one pictures <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> as sitting at the vertex, and the diagram itself as forming the base of the cone.</p> <p>A <em><a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a></em><span> <del class='diffmod'> if</del><ins class='diffmod'> of</ins> such cones is a</span><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo lspace='verythinmathspace'>:</mo><mi>Δ</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Δ</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha\colon \Delta(c)\to\Delta(c')</annotation></semantics></math> such that the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>Δ</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd> <mtd><mrow /></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mrow /></mtd> <mtd><mi>Δ</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mrow /></mtd> <mtd><mpadded lspace='-100%width' width='0'><mstyle scriptlevel='1'><mi>T</mi></mstyle></mpadded><mo>↘</mo></mtd> <mtd><mrow /></mtd> <mtd><mo>↙</mo><mpadded width='0'><mstyle scriptlevel='1'><mrow><mi>T</mi><mo>′</mo></mrow></mstyle></mpadded></mtd> <mtd><mrow /></mtd></mtr> <mtr><mtd><mrow /></mtd> <mtd><mrow /></mtd> <mtd><mi>F</mi></mtd> <mtd><mrow /></mtd> <mtd><mrow /></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \Delta(c) &{}&\overset{\alpha}{\longrightarrow} &{}& \Delta(c') \\ {}& \mathllap{\scriptsize{T}}\searrow &{}& \swarrow\mathrlap{\scriptsize{T'}} &{} \\ {}&{}&F&{}&{} } </annotation></semantics></math></div> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commutes</a>. Note that naturality of any such <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> implies that for all <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>i,j\in J</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>α</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>\alpha_i=\alpha_j</annotation></semantics></math>, so that <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>=</mo><mi>Δ</mi><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha=\Delta(\phi)</annotation></semantics></math> for some <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mi>c</mi><mo>→</mo><mi>c</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\phi \colon c \to c'</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. The single component <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> itself is often referred to as the cone morphism.</p> <p>An equivalent definition of a cone morphism <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>T</mi><mo>→</mo><mi>T</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\phi : T \to T'</annotation></semantics></math> says that all component diagrams</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>c</mi></mtd> <mtd><mrow /></mtd> <mtd><mover><mo>⟶</mo><mi>ϕ</mi></mover></mtd> <mtd><mrow /></mtd> <mtd><mi>c</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mrow /></mtd> <mtd><mpadded lspace='-100%width' width='0'><mstyle scriptlevel='1'><mrow><msub><mi>T</mi> <mi>j</mi></msub></mrow></mstyle></mpadded><mo>↘</mo></mtd> <mtd><mrow /></mtd> <mtd><mo>↙</mo><mpadded width='0'><mstyle scriptlevel='1'><mrow><mi>T</mi><msub><mo>′</mo> <mi>j</mi></msub></mrow></mstyle></mpadded></mtd> <mtd><mrow /></mtd></mtr> <mtr><mtd><mrow /></mtd> <mtd><mrow /></mtd> <mtd><mi>F</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo></mtd> <mtd><mrow /></mtd> <mtd><mrow /></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ c &{}& \overset{\phi}{\longrightarrow} &{}& c' \\ {}& \mathllap{\scriptsize{T_j}}\searrow &{}& \swarrow\mathrlap{\scriptsize{T'_j}} &{} \\ {}&{}&F(j)&{}&{} } </annotation></semantics></math></div> <p>commute.</p> <p>Cones and their morhisms over a given <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> clearly form a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>. The <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in this category, if it exists, is the <em><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a></em> of the diagram (see <a href='limit#InTermsOfUniversalCones'>there</a>).</p> <p>A <em><a class='existingWikiWord' href='/nlab/show/diff/cocone'>cocone</a></em> in <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is precisely a cone in the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>.</p> <h3 id='over_a_diagram_in_an_category'>Over a diagram in an <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_119' 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> a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a>, i.e. an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a>, the <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category of <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-cones over <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>over quasi-category</a> denoted <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy='false'>/</mo><mi>F</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>C_{/F}</annotation></semantics></math>. Its objects are cones over <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>. Its <a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a>s are <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-homotopies between cones. The <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-categorical limit</a> over <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is, if it exists, the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in <math class='maruku-mathml' display='inline' id='mathml_1632f3692572cae1650c1de76fb21007ac6f9bed_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy='false'>/</mo><mi>F</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>C_{/F}</annotation></semantics></math>.</p> <h2 id='see_also'>See also</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cone+type'>cone type</a></li> </ul> <p>These are shaped like the homotopy-theoretic cone, so maybe there is a deeper relationship:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/positive+cone'>positive cone</a> (in an <a class='existingWikiWord' href='/nlab/show/diff/ordered+group'>ordered group</a>, such as an <a class='existingWikiWord' href='/nlab/show/diff/operator+algebra'>operator algebra</a>),</li> <li><a class='existingWikiWord' href='/nlab/show/diff/future+cone'>future cone</a> (of an <a class='existingWikiWord' href='/nlab/show/diff/event'>event</a> in a <a class='existingWikiWord' href='/nlab/show/diff/smooth+Lorentzian+space'>Lorentzian manifold</a>, such as <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a>),</li> <li><a class='existingWikiWord' href='/nlab/show/diff/convex+cone'>convex cone</a> (in a <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a>).</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on April 25, 2024 at 20:46:18. 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