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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='#definition'>Definition</a></li><li><a href='#properties'>Properties</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>An <strong>intersection</strong> is a <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a> of <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a>s or (more generally) <a class='existingWikiWord' href='/nlab/show/diff/subobject'>subobject</a><span> s.<ins class='diffins'> Dually,</ins><ins class='diffins'> a</ins></span><ins class='diffins'><strong>cointersection</strong></ins><ins class='diffins'> is a </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/union'>union</a></ins><ins class='diffins'>/</ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/join'>join</a></ins><ins class='diffins'> of cosubobjects.</ins></p> <del class='diffdel'><p>The dual notion is that of <a class='existingWikiWord' href='/nlab/show/diff/union'>union</a>/<a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>.</p></del><del class='diffdel'> </del><p>This includes the traditional <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set-theoretic</a> intersection of subsets of some ambient set, as well as intersections of completely arbitrary sets (which are subsets of the <a class='existingWikiWord' href='/nlab/show/diff/universe'>universe</a>) in <a class='existingWikiWord' href='/nlab/show/diff/material+set+theory'>material set theory</a>.</p> <p>In a <a class='existingWikiWord' href='/nlab/show/diff/finitely+complete+category'>finitely complete category</a>, the intersection of two <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a>s <math class='maruku-mathml' display='inline' id='mathml_5cb98cc7bb380e774561fedd9c4620a6ae7fe08d_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A\hookrightarrow X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_5cb98cc7bb380e774561fedd9c4620a6ae7fe08d_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>B\hookrightarrow X</annotation></semantics></math> can be computed by a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> of the <a class='existingWikiWord' href='/nlab/show/diff/cospan'>cospan</a> <math class='maruku-mathml' display='inline' id='mathml_5cb98cc7bb380e774561fedd9c4620a6ae7fe08d_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi><mo>←</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A\to X \leftarrow B</annotation></semantics></math><ins class='diffins'>. Dually, the cointersection of two </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphisms</a></ins><ins class='diffins'> is their </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a></ins>.</p> <p>The <strong>nullary intersection</strong> of the subsets of <math class='maruku-mathml' display='inline' id='mathml_5cb98cc7bb380e774561fedd9c4620a6ae7fe08d_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_5cb98cc7bb380e774561fedd9c4620a6ae7fe08d_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> itself. A <strong>binary intersection</strong> is the intersection of two sets, and a <strong>finitary intersection</strong> is the intersection of <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finitely</a> many sets. Finitary intersections may be built out of binary and nullary intersections.</p> <h2 id='properties'>Properties</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interactions+of+images+and+pre-images+with+unions+and+intersections'>interactions of images and pre-images with unions and intersections</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interactions+of+images+and+pre-images+with+unions+and+intersections'>the image of an intersection is contained in the intersection of the images</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pre-images+preserve+unions+and+intersections'>pre-images preserve intersections</a>,</p> </li> </ul> </li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/union'>union</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intersection+pairing'>intersection product</a>, <a class='existingWikiWord' href='/nlab/show/diff/intersection+theory'>intersection theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/brane+intersection'>brane intersection</a></p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on April 4, 2023 at 20:40:35. 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