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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/3372/#Item_28" 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 #77 to #78: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <blockquote> <p>This page discusses the general concept of mapping spaces and internal homs. For mapping spaces in topology, see at <em><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</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='mapping_space'>Mapping space</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>/<a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a></strong></p> <h3 id='general_abstract'>General abstract</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a>, <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a>, <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>, <a class='existingWikiWord' href='/nlab/show/diff/exponential+object'>exponential object</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom-space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/loop+space+object'>loop space object</a>, <a class='existingWikiWord' href='/nlab/show/diff/free+loop+space+object'>free loop space object</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+loop+space'>derived loop space</a></p> </li> </ul> <h3 id='topology'>Topology</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> (<a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topology+of+mapping+spaces'>topology of mapping spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/evaluation+fibration+of+mapping+spaces'>evaluation fibration of mapping spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/free+loop+space'>free loop space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/free+loop+space+of+classifying+space'>free loop space of a classifying space</a></li> </ul> </li> </ul> <h3 id='simplicial_homotopy_theory'>Simplicial homotopy theory</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/function+complex'>simplicial mapping complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inertia+orbifold'>inertia groupoid</a></p> </li> </ul> <h3 id='differential_topology'>Differential topology</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+topology+of+mapping+spaces'>differential topology of mapping spaces</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/C-infinity+topology'>C-k topology</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/tangent+spaces+of+mapping+spaces'>tangent spaces of mapping spaces</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+loop+space'>smooth loop space</a></p> </li> </ul> <h3 id='stable_homotopy_theory'>Stable homotopy theory</h3> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/function+spectrum'>mapping spectrum</a></li> </ul> <h3 id='geometric_homotopy_theory'>Geometric homotopy theory</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+stack'>mapping stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inertia+orbifold'>inertia stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/free+loop+orbifold'>free loop stack</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/mapping+space+-+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='#EvaluationMap'>Evaluation map</a></li><li><a href='#CompositionMap'>Composition map</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#BasicProperties'>Basic properties</a></li><li><a href='#relation_to_function_types'>Relation to function types</a></li><li><a href='#induced_monad_state_monad'>Induced monad (state monad)</a></li><li><a href='#StableSplitting'>Stable splitting</a></li></ul></li><li><a href='#Examples'>Examples</a><ul><li><a href='#in_sets'>In sets</a></li><li><a href='#in_simplicial_sets'>In simplicial sets</a></li><li><a href='#InASheafTopos'>In a sheaf topos or <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_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>-sheaf <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_2' 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>-topos</a></li><li><a href='#ExampleInSliceCategories'>In slice categories</a></li><li><a href='#for_smooth_spaces_and_smooth_groupoids'>For smooth spaces and smooth <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</a></li><li><a href='#for_chain_complexes'>For chain complexes</a></li><li><a href='#for_super_vector_spaces'>For super vector spaces</a></li><li><a href='#for_banach_spaces'>For Banach spaces</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> and <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X, Y \in \mathcal{C}</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a>, the <em>internal hom</em> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>[X,Y] \in \mathcal{C}</annotation></semantics></math> from <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is, if it exists, another <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> which behaves like the “object of <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a>” from <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. In other words it is, if it exists, an <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internal version</a> of the ordinary <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom set</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}(X, Y) \in Set</annotation></semantics></math> or more generally <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom object</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>𝒱</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}(X, Y) \in \mathcal{V}</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/locally+small+category'>locally small category</a> or <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding='application/x-tex'>\mathcal{V}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched category</a>.</p> <p>One way to make this precise starts by mimicking a property of the <a class='existingWikiWord' href='/nlab/show/diff/function+set'>function set</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>{</mo><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>[X,Y] = \{f : X \to Y\}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> between two <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_16' 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_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>: this set is characterized by the fact that for any other set <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, the functions <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>→</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>S \to [X,Y]</annotation></semantics></math> are in <a class='existingWikiWord' href='/nlab/show/diff/natural+bijection'>natural bijection</a> with the functions <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>S \times X \to Y</annotation></semantics></math> out of the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. That is: for each set <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>(-) \times X</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a>, given by the construction <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math>.</p> <p>One can verbalize this thus: <em>taking the cartesian product with the set <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></em> is left-adjoint to <em>taking the set of all functions with domain <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></em>.</p> <p>This, then, is, generally, the definition of <em>internal hom</em> in any <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a> or in fact in any <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math>: the <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math> to the given <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> functor <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊗</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>(-)\otimes X</annotation></semantics></math> for all objects <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. It may or may not exist. If it exists, one says that <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math> is a <em><a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed</a></em> <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a>. Explicity, the condition is that there is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>(<a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>𝒞</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \mathcal{C}(A, [X,Z]) \stackrel{\simeq}{\to} \mathcal{C}(A \otimes X, Z) </annotation></semantics></math></div> <p>which is <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural</a> in all three <a class='existingWikiWord' href='/nlab/show/diff/variable'>variables</a>. (The leftward map here is often called <strong><a class='existingWikiWord' href='/nlab/show/diff/currying'>currying</a></strong>, especially in a <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a> (and more especially for the <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>λ</mi></mrow><annotation encoding='application/x-tex'>\lambda</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/lambda-calculus'>calculus</a>).)</p> <p>In particular this implies that in a closed monoidal category the external hom is re-obtained from the internal hom as its set of <a class='existingWikiWord' href='/nlab/show/diff/generalized+element'>generalized elements</a> out of the <a class='existingWikiWord' href='/nlab/show/diff/unit+object'>tensor unit</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>I \in \mathcal{C}</annotation></semantics></math> in that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mfrac><mrow><mi>I</mi><mo>→</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo></mrow><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></mfrac></mrow><annotation encoding='application/x-tex'> \frac{I \to [X,Y]}{X \to Y} </annotation></semantics></math></div> <p>using that <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>X</mi><mo>≃</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>I \otimes X \simeq X</annotation></semantics></math> by definition of the tensor unit.</p> <p>Here “closed” in “<a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a>” is in the sense that forming “hom-sets” does not lead “out of the category”. In fact the internal hom of a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a> is indeed the hom as seen in the <em><a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a></em> of that category (the <em><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a></em>).</p> <p>More generally, one can consider objects that satisfy some basic <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal properties</a> that an internal hom should satisfy even in the absence of a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal structure</a>. If such objects exist one speaks therefore just of a <em><a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed category</a></em>. Every <a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed category</a> may be seen as a category <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched</a> over itself. Accordingly, an internal hom is after all a special case of a <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a>, for the special case of this enrichment over itself.</p> <h2 id='definition'>Definition</h2> <div class='num_defn' id='ClosedMonoidalCategory'> <h6 id='definition_2'>Definition</h6> <p><strong>(internal hom)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a>. An <strong>internal hom</strong> in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>:</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>×</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'> [-,-] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C} </annotation></semantics></math></div> <p>such that for every <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X \in \mathcal{C}</annotation></semantics></math> we have a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊗</mo><mi>X</mi><mo>⊣</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> ((-) \otimes X \dashv [X, -]) : \mathcal{C} \to \mathcal{C} \,. </annotation></semantics></math></div> <p>If this exists, <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math> is called a <em><a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a></em>.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>If the monoidal category <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> in Def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a> is not <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a>, there is instead a concept of left- and right-internal hom.</p> </div> <h3 id='EvaluationMap'>Evaluation map</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a> (Def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a>).</p> <div class='num_defn' id='EvalMap'> <h6 id='definition_3'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X,Y \in \mathcal{C}</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a>, the <strong><a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation map</a></strong></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>eval</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> eval_{X,Y} : [X,Y] \otimes X \to Y </annotation></semantics></math></div> <p>is the <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊗</mo><mi>X</mi><mo>⊣</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>((-)\otimes X \dashv [X,-])</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjunct</a> of the <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>id</mi> <mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo></mrow></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>id_{[X,Y]} : [X,Y] \to [X,Y]</annotation></semantics></math>.</p> </div> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is specifically a <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+category'>locally cartesian closed category</a>, then in terms of the <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> <a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal language</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation map</a> is the <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a> of the <a class='existingWikiWord' href='/nlab/show/diff/dependent+type'>dependent type</a> which in <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> <a class='existingWikiWord' href='/nlab/show/diff/syntax'>syntax</a> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_52' 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><mo>,</mo><mspace width='thickmathspace' /><mi>x</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mspace width='thickmathspace' /><mo>⊢</mo><mspace width='thickmathspace' /><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>Y</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> f \colon X \to Y,\; x \colon X \;\vdash\; f(x) \colon Y \,, </annotation></semantics></math></div> <p>with <em><a class='existingWikiWord' href='/nlab/show/diff/function+application'>function application</a></em> on the right.</p> </div> <h3 id='CompositionMap'>Composition map</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>×</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \times)</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a> (Def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a>)</p> <div class='num_defn' id='CompositionMorphism'> <h6 id='definition_4'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X, Y, Z \in \mathcal{C}</annotation></semantics></math> three <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a>, the <strong><a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> morphism</strong></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo>×</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> \circ_{X,Y,Z} : [Y, Z] \times [X, Y] \to [X, Z] </annotation></semantics></math></div> <p>is the <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi><mo>⊣</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>((-)\times X \dashv [X,-])</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjunct</a> of the following composite of two <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation maps</a>, def. <a class='maruku-ref' href='#EvalMap'>2</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo>×</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>×</mo><mi>X</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><msub><mi>id</mi> <mrow><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo></mrow></msub><mo>,</mo><msub><mi>eval</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo stretchy='false'>)</mo></mrow></mover><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo>×</mo><mi>Y</mi><mover><mo>→</mo><mrow><msub><mi>eval</mi> <mrow><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub></mrow></mover><mi>Z</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [Y, Z] \times ([X , Y] \times X) \stackrel{(id_{[Y,Z]}, eval_{X,Y})}{\to} [Y,Z] \times Y \stackrel{eval_{Y,Z}}{\to} Z \,. </annotation></semantics></math></div></div> <h2 id='properties'>Properties</h2> <h3 id='BasicProperties'>Basic properties</h3> <p>The internal homs in a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a> (Def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a>) happen to share all the key abstract properties of ordinary (“external”) <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functors</a>, even though this is not completely manifest from Def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a>:</p> <div class='num_prop' id='InternalHomBifunctor'> <h6 id='proposition'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> <a class='existingWikiWord' href='/nlab/show/diff/bifunctor'>bifunctor</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_58' 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/symmetric+monoidal+category'>symmetric monoidal category</a> such that for each object <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X \in \mathcal{C}</annotation></semantics></math> the functor <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>⊗</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \otimes (-)</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math>. Then this is already equivalent to Def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a>, in that there is a unique functor out of the <a class='existingWikiWord' href='/nlab/show/diff/product+category'>product category</a><span><ins class='diffins'> </ins> of</span><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> with its <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>×</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'> [-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>such that for each <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X \in \mathcal{C}</annotation></semantics></math> it coincides with the <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math> as a functor in the second variable, and such that there is a <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Hom</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>Hom</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Hom(X, [Y,Z]) \;\simeq\; Hom(X \otimes Y, Z) </annotation></semantics></math></div> <p>which is natural not only in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_67' 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_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi></mrow><annotation encoding='application/x-tex'>Z</annotation></semantics></math>, but also in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>We have a natural isomorphism for each fixed <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, and hence in particular for fixed <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> and fixed <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi></mrow><annotation encoding='application/x-tex'>Z</annotation></semantics></math>. With this the statement follows directly by <a href='https://ncatlab.org/nlab/show/adjoint%20functor#AdjointFunctorFromObjectwiseRepresentingObject'>this prop.</a> at <em><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></em>.</p> </div> <p>In fact the 3-variable adjunction from Prop. <a class='maruku-ref' href='#InternalHomBifunctor'>1</a> even holds internally:</p> <div class='num_prop' id='TensorHomAdjunctionIsoInternally'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(internal tensor/hom-adjunction)</strong></p> <p>In a <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a> (def. <a class='maruku-ref' href='#ClosedMonoidalCategory'>1</a>) there are <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphisms</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> [X \otimes Y, Z] \;\simeq\; [X, [Y,Z]] </annotation></semantics></math></div> <p>whose image under <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_{\mathcal{C}}(1,-)</annotation></semantics></math> are the defining <a class='existingWikiWord' href='/nlab/show/diff/natural+bijection'>natural bijections</a> of Prop. <a class='maruku-ref' href='#InternalHomBifunctor'>1</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>A \in \mathcal{C}</annotation></semantics></math> be any object. By applying the natural bijections from Prop. <a class='maruku-ref' href='#InternalHomBifunctor'>1</a>, there are composite <a class='existingWikiWord' href='/nlab/show/diff/natural+bijection'>natural bijections</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_76' 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>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>⊗</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} Hom_{\mathcal{C}}(A , [X \otimes Y, Z]) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, [Y,Z]) \\ & \simeq Hom_{\mathcal{C}}(A, [X, [Y,Z]]) \end{aligned} </annotation></semantics></math></div> <p>Since this holds for all <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/full+and+faithful+functor'>fully faithfulness</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a> says that there is an isomorphism <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo>≃</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[ X\otimes Y, Z ] \simeq [X, [Y,Z]]</annotation></semantics></math>. Moreover, by taking <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>A = 1</annotation></semantics></math> in the above and using the left <a class='existingWikiWord' href='/nlab/show/diff/unitor'>unitor</a> isomorphisms <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_80' 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>Y</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>A \otimes (X \otimes Y) \simeq X \otimes Y</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>≃</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A\otimes X \simeq X</annotation></semantics></math> we get a <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commuting diagram</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mo>≃</mo></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ Hom_{\mathcal{C}}(1, [X\otimes Y, Z ]) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1, [X, [Y,Z]]) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, [Y,Z]) } \,. </annotation></semantics></math></div></div> <p>Also the key respect of <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functors</a> for <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a> is inherited by <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>-functors:</p> <div class='num_prop' id='InternalHomPreservesLimits'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/hom-functor+preserves+limits'>internal hom-functor preserves limits</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_83' 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/symmetric+monoidal+category'>symmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a> with <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>-<a class='existingWikiWord' href='/nlab/show/diff/bifunctor'>bifunctor</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[-,-]</annotation></semantics></math> (Prop. <a class='maruku-ref' href='#InternalHomBifunctor'>1</a>). Then this bifunctor preserves <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a> in the second variable, and sends colimits in the first variable to limits:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> [X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)] </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mo stretchy='false'>[</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> [\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X] </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>X \in \mathcal{C}</annotation></semantics></math> any object, <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> by definition, and hence preserves limits by <em><a class='existingWikiWord' href='/nlab/show/diff/adjoints+preserve+%28co-%29limits'>adjoints preserve (co-)limits</a></em>.</p> <p>For the other case, let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>ℒ</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>Y \;\colon\; \mathcal{L} \to \mathcal{C}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, and let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_91' 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 \mathcal{C}</annotation></semantics></math> be any object. Then there are isomorphisms</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_92' 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>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mo stretchy='false'>[</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo>⊗</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mo stretchy='false'>(</mo><mi>C</mi><mo>⊗</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>C</mi><mo>⊗</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mo stretchy='false'>[</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} Hom_{\mathcal{C}}(C, [ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ] ) & \simeq Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) \\ & \simeq Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( C, [Y(j), X] ) \\ & \simeq Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] ) \end{aligned} </annotation></semantics></math></div> <p>which are <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural</a> in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_93' 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 \mathcal{C}</annotation></semantics></math>, where we used that the ordinary <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a> respects (co)limits as shown (see at <em><a class='existingWikiWord' href='/nlab/show/diff/hom-functor+preserves+limits'>hom-functor preserves limits</a></em>), and that the <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊗</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C \otimes (-)</annotation></semantics></math> preserves colimits (see at <em><a class='existingWikiWord' href='/nlab/show/diff/adjoints+preserve+%28co-%29limits'>adjoints preserve (co-)limits</a></em>).</p> <p>Hence by the <a class='existingWikiWord' href='/nlab/show/diff/full+and+faithful+functor'>fully faithfulness</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a>, there is an isomorphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>[</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo>]</mo></mrow><mover><mo>⟶</mo><mo>≃</mo></mover><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>j</mi><mo>∈</mo><mi>𝒥</mi></mrow></munder></munder><mo stretchy='false'>[</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right] \overset{\simeq}{\longrightarrow} \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] \,. </annotation></semantics></math></div></div> <h3 id='relation_to_function_types'>Relation to function types</h3> <p>The internal hom is the <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a> of what in <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> are <a class='existingWikiWord' href='/nlab/show/diff/function+type'>function types</a></p> <table><thead><tr><th /><th><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></th></tr></thead><tbody><tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/syntax'>syntax</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/semantics'>semantics</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>natural deduction</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a></strong></td><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a></strong></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/type+formation'>type formation</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mfrac><mrow><mo>⊢</mo><mspace width='mediummathspace' /><mi>X</mi><mo lspace='verythinmathspace'>:</mo><mi>Type</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>⊢</mo><mspace width='thickmathspace' /><mi>A</mi><mo lspace='verythinmathspace'>:</mo><mi>Type</mi></mrow><mrow><mo>⊢</mo><mspace width='thickmathspace' /><mrow><mo>(</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo>)</mo></mrow><mo lspace='verythinmathspace'>:</mo><mi>Type</mi></mrow></mfrac></mrow><annotation encoding='application/x-tex'>\frac{\vdash\: X \colon Type \;\;\;\;\; \vdash\; A\colon Type}{\vdash \; \left(X \to A\right) \colon Type}</annotation></semantics></math></td><td style='text-align: left;'><img src='' /></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>term introduction</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mfrac><mrow><mi>x</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mspace width='thickmathspace' /><mo>⊢</mo><mspace width='thickmathspace' /><mi>a</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>A</mi></mrow><mrow><mo>⊢</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>↦</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mrow><mo>(</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo>)</mo></mrow></mrow></mfrac></mrow><annotation encoding='application/x-tex'>\frac{x \colon X \;\vdash\; a(x) \colon A}{\vdash (x \mapsto a\left(x\right)) \colon \left(X \to A\right) }</annotation></semantics></math></td><td style='text-align: left;'><img src='' /></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/term+elimination'>term elimination</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mfrac><mrow><mo>⊢</mo><mspace width='thickmathspace' /><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mrow><mo>(</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo>)</mo></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>⊢</mo><mspace width='thickmathspace' /><mi>x</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi></mrow><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>⊢</mo><mspace width='thickmathspace' /><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding='application/x-tex'>\frac{\vdash\; f \colon \left(X \to A\right)\;\;\;\; \vdash \; x \colon X}{\;\;\;\vdash\; f(x) \colon A}</annotation></semantics></math></td><td style='text-align: left;'><img src='' /></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/conversion+rule'>computation rule</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>y</mi><mo>↦</mo><mi>a</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>a</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(y \mapsto a(y))(x) = a(x)</annotation></semantics></math></td><td style='text-align: left;'><img src='' /></td></tr> </tbody></table> <h3 id='induced_monad_state_monad'>Induced monad (state monad)</h3> <p>For each object <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> the (internal hom <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a>)-<a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> induces a <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>S</mi><mo>,</mo><mi>S</mi><mo>⊗</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S, S \otimes (-)]</annotation></semantics></math>. In <a class='existingWikiWord' href='/nlab/show/diff/computer+science'>computer science</a> this <a class='existingWikiWord' href='/nlab/show/diff/monad+%28in+computer+science%29'>monad (in computer science)</a> is called the <em><a class='existingWikiWord' href='/nlab/show/diff/state+monad'>state monad</a></em>.</p> <h3 id='StableSplitting'>Stable splitting</h3> <p>In <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> the <a class='existingWikiWord' href='/nlab/show/diff/stabilization'>stabilization</a>/<a class='existingWikiWord' href='/nlab/show/diff/suspension+spectrum'>suspension spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Sigma^\infty Maps(X,A)</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(X,A)</annotation></semantics></math> between suitable <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complexes</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>X, A</annotation></semantics></math> happens to decompose as a <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a> of <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectra</a> in a useful way, related to the expression of the <a class='existingWikiWord' href='/nlab/show/diff/Goodwillie+calculus'>Goodwillie derivatives</a> of the functor <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(X,-)</annotation></semantics></math>.</p> <p>For more on this see at <em><a class='existingWikiWord' href='/nlab/show/diff/stable+splitting+of+mapping+spaces'>stable splitting of mapping spaces</a></em>.</p> <h2 id='Examples'>Examples</h2> <h3 id='in_sets'>In sets</h3> <p>In the category <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a>, regarded as a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a>, the internal hom is given by <a class='existingWikiWord' href='/nlab/show/diff/function+set'>function sets</a>. This exists, by the discussion there, as soon as the <a class='existingWikiWord' href='/nlab/show/diff/foundation+of+mathematics'>foundational</a> <a class='existingWikiWord' href='/nlab/show/diff/axiom'>axioms</a> are strong enough, for instance as soon as there are <a class='existingWikiWord' href='/nlab/show/diff/power+object'>power objects</a>, which is the special case of a function set into the 2-element set.</p> <h3 id='in_simplicial_sets'>In simplicial sets</h3> <p>In the category <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a> of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a>, the internal hom between two <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_107' 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,Y</annotation></semantics></math> is given by the formula</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy='false'>]</mo> <mi>n</mi></msub><mo>=</mo><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> [X,Y]_n = Hom_{sSet}(X\times \Delta[n],Y) \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta[n]</annotation></semantics></math> is the simplicial <a class='existingWikiWord' href='/nlab/show/diff/simplex'>n-simplex</a>. This <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>[X,Y] \in sSet</annotation></semantics></math> is also called the <em><a class='existingWikiWord' href='/nlab/show/diff/function+complex'>function complex</a></em> between <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_111' 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_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>.</p> <p>Since <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi><mo>≃</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sSet \simeq PSh(\Delta)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a> over the <a class='existingWikiWord' href='/nlab/show/diff/simplex+category'>simplex category</a>, this is a special case of internal homs in sheaf toposes, discussed <a href='#InASheafTopos'>below</a>.</p> <h3 id='InASheafTopos'>In a sheaf topos or <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_114' 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>-sheaf <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_115' 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>-topos</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_116' 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/site'>site</a>. Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo>=</mo><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{H} = Sh(C)</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf topos</a> over <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> or in fact the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-sheaf (∞,1)-topos</a>. We discuss the internal hom of this regard as a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a>/<a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+%28infinity%2C1%29-category'>cartesian monoidal (∞,1)-category</a>.</p> <div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>The sheaf topos <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a> / <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+%28infinity%2C1%29-category'>cartesian closed (∞,1)-category</a>. In fact it is a <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+category'>locally cartesian closed category</a> / <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+%28infinity%2C1%29-category'>locally cartesian closed (∞,1)-category</a>.</p> </div> <p>Hence the internal hom exist.</p> <div class='num_prop' id='InternalHomInSheaves'> <h6 id='proposition_5'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X, Y \in \mathbf{H}</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a>, the internal hom-object</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'> [X,Y] \in \mathbf{H} </annotation></semantics></math></div> <p>is the <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a> given by the assignment</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>:</mo><mi>U</mi><mo>↦</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> [X,Y] : U \mapsto \mathbf{H}(U \times X, Y) \,, </annotation></semantics></math></div> <p>for all objects <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> which on the right we regard under the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a>/<a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>∞-Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi><mover><mo>↪</mo><mi>Yoneda</mi></mover><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>U \in C \stackrel{Yoneda}{\hookrightarrow} \mathbf{H}</annotation></semantics></math>.</p> <p>Here</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>×</mo><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>U \times X \in \mathbf{H}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{H}(-,-)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom set</a>-<a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>hom space</a>-<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>.</p> </li> </ul> </div> <p>See also at <em><a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+presheaves'>closed monoidal structure on presheaves</a></em>.</p> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a>/<a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a> we have <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural equivalences</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>≃</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> [X,Y](U) \simeq \mathbf{H}(U , [X,Y]) </annotation></semantics></math></div> <p>and by the defining <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi><mo>⊢</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>((-)\times X \vdash [X,-])</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> this is naturally equivalent to</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mo>≃</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdots \simeq \mathbf{H}(U \times X, Y) \,. </annotation></semantics></math></div></div> <div class='num_remark'> <h6 id='remark_3'>Remark</h6> <p>In the (<a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy</a>-)<a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> <a class='existingWikiWord' href='/nlab/show/diff/syntax'>syntax</a> of the <a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal language</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> the internal hom <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>[X, Y] \in \mathbf{H}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a> of the <a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊢</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>Type</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \vdash (X \to Y) : Type \,. </annotation></semantics></math></div></div> <div class='num_prop' id='EvaluationOfInternalFunctionsInSheafTopos'> <h6 id='proposition_6'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X, Y \in \mathbf{H}</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation map</a>, def. <a class='maruku-ref' href='#EvalMap'>2</a>,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>eval</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> eval_{X,Y} : [X,Y] \times X \to Y </annotation></semantics></math></div> <p>is the morphism of sheaves which over each <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> sends a morphism of sheaves <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>θ</mi><mo>:</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>Y</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\theta : \mathbf{H}(-,U) \times X(-) \to Y(-)</annotation></semantics></math> (which is the first component by prop. <a class='maruku-ref' href='#InternalHomInSheaves'>5</a>) and an <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x \in \mathbf{H}(U,X)</annotation></semantics></math> to</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>eval</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>(</mo><mi>θ</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>↦</mo><msub><mi>θ</mi> <mi>U</mi></msub><mo stretchy='false'>(</mo><msub><mi>id</mi> <mi>U</mi></msub><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>Y</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> eval_{X,Y}(U) : (\theta, x) \mapsto \theta_U(id_U, x) \in Y(U) \,. </annotation></semantics></math></div></div> <p>See (<a href='#MacLane-Moerdijk'>MacLane-Moerdijk, p. 46</a>).</p> <div class='num_prop' id='CompositionOfInternalFunctionsInSheafTopos'> <h6 id='proposition_7'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X, Y, Z \in \mathbf{H}</annotation></semantics></math> three <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>, the canonical <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a>, def. <a class='maruku-ref' href='#CompositionMorphism'>3</a>,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo>×</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> \circ_{X,Y,Z} : [Y, Z] \times [X, Y] \to [X, Z] </annotation></semantics></math></div> <p>is given by the morphism of <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaves</a>/<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-presheaf'>(∞,1)-presheaves</a> whose component over <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> is the morphism of <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoids</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>:</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>×</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \circ_{X,Y,Z}(U) : \mathbf{H}(U \times X, Y) \times \mathbf{H}(U \times Y, Z) \to \mathbf{H}(U \times X, Z) </annotation></semantics></math></div> <p>which sends a pair <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>:</mo><mi>U</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>U</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f : U \times X \to Y, g : U \times Y \to Z)</annotation></semantics></math> to the composite</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>U</mi><mo>×</mo><mi>X</mi><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><msub><mi>Δ</mi> <mi>U</mi></msub><mo>,</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo></mrow></mover><mi>U</mi><mo>×</mo><mi>U</mi><mo>×</mo><mi>X</mi><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><msub><mi>id</mi> <mi>U</mi></msub><mo>,</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mover><mi>U</mi><mo>×</mo><mi>Y</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \circ_{X,Y,Z}(U)(f,g) = U \times X \stackrel{(\Delta_U, id_X)}{\to} U \times U \times X \stackrel{(id_U, f)}{\to} U \times Y \stackrel{g}{\to} Z \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Δ</mi> <mi>U</mi></msub><mo>:</mo><mi>U</mi><mo>→</mo><mi>U</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>\Delta_U : U \to U \times U</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a> morphism on <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>By definition <a class='maruku-ref' href='#CompositionMorphism'>3</a> the morphism is the <a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjunct</a> of the double <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation map</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo>×</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>×</mo><mi>X</mi><mo>→</mo><mi>Z</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [Y,Z] \times [X,Y] \times X \to Z \,. </annotation></semantics></math></div> <p>Since the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a> of two sheaves <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>A, B \in \mathbf{H}</annotation></semantics></math> is computed objectwise</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>:</mo><mi>U</mi><mo>↦</mo><mi>A</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> A \times B : U \mapsto A(U) \times B(U) </annotation></semantics></math></div> <p>it follows that over each <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> this double evaluation map is the morphism of sets/<math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>×</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>X</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Z</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> [Y,Z](U) \times [X,Y](U) \times X(U) \to Z(U) </annotation></semantics></math></div> <p>hence by prop. <a class='maruku-ref' href='#InternalHomInSheaves'>5</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo><mo>×</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>×</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \mathbf{H}(U \times Y, Z) \times \mathbf{H}(U \times X, Y) \times \mathbf{H}(U,X) \to \mathbf{H}(U,Z) \,, </annotation></semantics></math></div> <p>where now by prop. \ref this is the external evaluation.</p> </div> <div class='num_remark'> <h6 id='remark_4'>Remark</h6> <p>Intuitively this says that the composite of a <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-parameterized family of maps <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy='false'>|</mo><mi>u</mi><mo>∈</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{f(u) : X \to Y| u \in U\}</annotation></semantics></math> with a <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-parameterized family of maps <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Z</mi><mo stretchy='false'>|</mo><mi>u</mi><mo>∈</mo><mi>U</mi></mrow></mrow><annotation encoding='application/x-tex'>{g(u) : Y \to Z| u \in U}</annotation></semantics></math> is the <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-family given by the parameter-wise composite <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><mi>u</mi><mo>∈</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{g(u)\circ f(u) | u \in U\}</annotation></semantics></math>.</p> </div> <div class='num_example' id='InternalAutomorphismGroup'> <h6 id='example'>Example</h6> <p>The internal <a class='existingWikiWord' href='/nlab/show/diff/automorphism'>automorphism group</a>/<a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-group'>automorphism ∞-group</a> of an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \in \mathbf{H}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/subobject'>subobject</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Aut</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>↪</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> \mathbf{Aut}(X) \hookrightarrow [X,X] </annotation></semantics></math></div> <p>of the internal hom which is maximal subject to the property that the composition of prop. <a class='maruku-ref' href='#CompositionOfInternalFunctionsInSheafTopos'>7</a> becomes invertible.</p> <p>The (<a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy</a>-)<a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> <a class='existingWikiWord' href='/nlab/show/diff/syntax'>syntax</a> for this is given by the <a class='existingWikiWord' href='/nlab/show/diff/type'>type</a> of <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+types'>equivalences in homotopy type theory</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊢</mo><mo stretchy='false'>(</mo><mi>X</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>Type</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \vdash (X \stackrel{\simeq}{\to} X) : Type \,. </annotation></semantics></math></div></div> <h3 id='ExampleInSliceCategories'>In slice categories</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+category'>locally cartesian closed category</a>. This means that for each object <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \in \mathbf{H}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice category</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathbf{H}_{/X}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a>. The <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> in the slice is given by the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a> over <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> computed in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>. Fairly detailed discussion of constructions of the internal hom in such slices <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathbf{H}_{/X}</annotation></semantics></math> is at <em><a href='locally%20cartesian%20closed%20category#EquivalentCharacterizations'>locally cartesian closed category – cartesian closure in terms of base change and dependent product</a></em>.</p> <p>We record some further properties</p> <div class='num_prop' id='InverseImageBaseChangeIsCartesianClosed'> <h6 id='proposition_8'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+category'>locally cartesian closed category</a> and <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_174' 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> any morphism in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/inverse+image'>inverse image</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> of the corresponding <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a> <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub><mover><mover><munder><mo>→</mo><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder></mrow></mover></mover><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \mathcal{C}_{/X} \stackrel{\overset{\sum_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{\prod_f}{\to}}} \mathcal{C}_{/Y} </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+functor'>cartesian closed functor</a>.</p> </div> <p>This is discussed in more detail at <em><a href='cartesian+closed+functor#Examples'>cartesian closed functor – Examples</a></em>.</p> <p>So for <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>A,B \in \mathcal{C}_{/Y}</annotation></semantics></math> we have <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphisms</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mrow><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo></mrow><mover><mo>→</mo><mo>≃</mo></mover><mrow><mo>[</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>A</mi><mo>,</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>B</mi><mo>]</mo></mrow></mrow><annotation encoding='application/x-tex'> f^* \left[A,B\right] \stackrel{\simeq}{\to} \left[f^* A , f^* B\right] </annotation></semantics></math></div> <p>between the image of the internal hom under <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> and the internal hom of the images of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_181' 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_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> separately.</p> <div class='num_prop' id='MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum'> <h6 id='proposition_9'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+category'>locally cartesian closed category</a>, <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_184' 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> any <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a>, and <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>A, B \in \mathbf{H}_{/X}</annotation></semantics></math> two objects in the slice over <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, there is a natural morphism (not in general an isomorphism)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mrow><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo></mrow><mo>→</mo><mrow><mo>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>A</mi><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo>]</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \prod_f \left[A,B \right] \to \left[ \sum_f A, \sum_f B\right] \,. </annotation></semantics></math></div></div> <p>Here are two ways to get this morphism:</p> <div class='proof'> <h6 id='proofconstruction_1'>Proof/Construction 1</h6> <p>For any object <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>U \in \mathbf{H}_{/Y}</annotation></semantics></math> we have a canonical morphism of <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom sets</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_189' 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><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub><mo stretchy='false'>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>U</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub><mo stretchy='false'>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>U</mi><mo>×</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mo>→</mo><mrow /></mover><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub><mo stretchy='false'>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mo stretchy='false'>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>U</mi><mo>×</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mo>≃</mo><mrow><mi>Frob</mi><mo>.</mo><mi>Rec</mi><mo>.</mo></mrow></mover><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>A</mi><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>Y</mi></mrow></msub><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mo stretchy='false'>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>A</mi><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} \mathbf{H}_{/Y}( U, \prod_f [A,B] ) & \simeq \mathbf{H}_{/X}( f^* U, [A,B] ) \\ & \simeq \mathbf{H}_{/X}(f^* U \times A, B) \\ & \stackrel{}{\to} \mathbf{H}_{/Y}( \sum_f( f^* U \times A ), \sum_f B ) \\ & \stackrel{Frob.Rec.}{\simeq} \mathbf{H}_{/Y}( U \times \sum_f A , \sum_f B ) \\ & \simeq \mathbf{H}_{/Y}(U, [\sum_f A , \sum_f B]) \end{aligned} </annotation></semantics></math></div> <p>where the first and the last steps use <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> properties, where the morphism in the middle is the component of the <a class='existingWikiWord' href='/nlab/show/diff/dependent+sum'>dependent sum</a> functor, and where “Frob.Rec.” is <a class='existingWikiWord' href='/nlab/show/diff/Frobenius+reciprocity'>Frobenius reciprocity</a>.</p> <p>Since this is <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural</a> in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a> implies the claimed morphism.</p> </div> <div class='proof'> <h6 id='proofconstruction_2'>Proof/Construction 2</h6> <p>There is the composite morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><msup><mi>f</mi> <mo>*</mo></msup><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>)</mo></mrow><mo>×</mo><mi>A</mi><mover><mo>→</mo><mrow><mi>counit</mi><mo>×</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow></mover><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>×</mo><mi>A</mi><mover><mo>→</mo><mi>eval</mi></mover><mi>B</mi><mover><mo>→</mo><mi>unit</mi></mover><msup><mi>f</mi> <mo>*</mo></msup><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi></mrow><annotation encoding='application/x-tex'> \left(f^\ast \prod_f [A, B]\right) \times A \stackrel{counit \times id_A}{\to} [A, B] \times A \stackrel{eval}{\to} B \stackrel{unit}{\to} f^\ast \sum_f B </annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>adjunction (co)units</a> and the <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation map</a> of the internal hom. Its hom-<a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjunct</a> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>f</mi> <mo>*</mo></msup><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>,</mo><msup><mi>f</mi> <mo>*</mo></msup><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>]</mo><mo>≅</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> A \to [f^\ast \prod_f [A, B], f^\ast \sum_f B] \cong f^\ast [\prod_f [A, B], \sum_f B] \,, </annotation></semantics></math></div> <p>using prop. <a class='maruku-ref' href='#InverseImageBaseChangeIsCartesianClosed'>8</a> on the right. The hom-adjunct of that in turn is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>A</mi><mo>→</mo><mo stretchy='false'>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\sum_f A \to [\prod_f [A, B], \sum_f B]</annotation></semantics></math></div> <p>and by symmetry the morphism that we are after:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></munder><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>A</mi><mo>,</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \prod_f [A, B] \to [\sum_f A, \sum_f B] \,. </annotation></semantics></math></div></div> <div class='num_remark' id='RememberingTopMorphismInHomInSlice'> <h6 id='remark_5'>Remark</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_195' 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 <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> (for simplicity), then the morphism of prop. <a class='maruku-ref' href='#MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum'>9</a> can be understood as follows: a <a class='existingWikiWord' href='/nlab/show/diff/global+element'>global element</a> of the <a class='existingWikiWord' href='/nlab/show/diff/dependent+product'>dependent product</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>f</mi></msub><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\prod_f [A,B]</annotation></semantics></math> is given by a <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commuting diagram</a> in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>A</mi></mtd> <mtd /> <mtd><mo>→</mo></mtd> <mtd /> <mtd><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>f</mi></munder><mi>B</mi></mtd></mtr> <mtr><mtd /> <mtd><mo>↘</mo></mtd> <mtd /> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \sum_f A &&\to&& \sum_f B \\ & \searrow && \swarrow \\ && X } \,. </annotation></semantics></math></div> <p>The map in prop. <a class='maruku-ref' href='#MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum'>9</a> picks out the top horizontal morphism in this diagram.</p> </div> <h3 id='for_smooth_spaces_and_smooth_groupoids'>For smooth spaces and smooth <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</h3> <p>Consider the <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C = </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/SmoothManifolds'>SmthMfd</a> of <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifolds</a> (and the <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <a class='existingWikiWord' href='/nlab/show/diff/coverage'>coverage</a>) or equivalently over the <a class='existingWikiWord' href='/nlab/show/diff/dense+sub-site'>dense subsite</a> <a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a> of <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian spaces</a> and <a class='existingWikiWord' href='/nlab/show/diff/smooth+map'>smooth functions</a> between these.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf topos</a>/<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-sheaf (∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo>=</mo><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{H} = Sh(C)</annotation></semantics></math> is that of <a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>smooth spaces</a>/<a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoids</a>. So the discussion of internal homs here is a special case of the above discussion <em><a href='#InASheafTopos'>In a sheaf topos</a></em>.</p> <div class='num_example'> <h6 id='example_2'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>SmthMfd</mi><mo>↪</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X , Y \in SmthMfd \hookrightarrow \mathbf{H}</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifolds</a>, the <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>[X,Y] \in \mathbf{H}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> between them regarded as a <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>.</p> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a></em> for when this internal hom is <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable</a> again by a <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>.</p> </div> <div class='num_example'> <h6 id='example_3'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmthMfd</mi><mo>↪</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>X \in SmthMfd \hookrightarrow \mathbf{H}</annotation></semantics></math> the internal automorphism group, example <a class='maruku-ref' href='#InternalAutomorphismGroup'>1</a>, of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_205' 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/diffeomorphism+group'>diffeomorphism group</a> of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, regarded as a <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological</a> group</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Aut</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mstyle mathvariant='bold'><mi>Diff</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbf{Aut}(X) = \mathbf{Diff}(X) \,. </annotation></semantics></math></div></div> <h3 id='for_chain_complexes'>For chain complexes</h3> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/internal+hom+of+chain+complexes'>internal hom of chain complexes</a></li> </ul> <h3 id='for_super_vector_spaces'>For super vector spaces</h3> <p>The category <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/super+vector+space'>super vector spaces</a> is the category of <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/graded+vector+space'>graded vector spaces</a>. Thus, its objects are pairs of vector spaces <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(V_+,V_-)</annotation></semantics></math>, with <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>V_+</annotation></semantics></math> called the <em>even</em> part and <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mo>−</mo></msub></mrow><annotation encoding='application/x-tex'>V_-</annotation></semantics></math> the <em>odd</em> part. The morphisms in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> are likewise pairs of linear maps, i.e. we define <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> to be <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vect</mi><mo>×</mo><mi>Vect</mi><mo>=</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>Vect \times Vect = Vect^{\mathbb{Z}/2}</annotation></semantics></math>, as usual for any sort of graded object. With this definition of the category <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math>, we capture the concepts of superalgebra and so on in succinct categorical terms.</p> <p>Because the morphisms in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> send even things to even things and odd things to odd things, they are sometimes called <em>even</em> linear maps, and one may write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>Even</mi><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> sVect(V, W) = Even Lin(V,W). </annotation></semantics></math></div> <p>Note that <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched</a> over <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vect</mi></mrow><annotation encoding='application/x-tex'>Vect</annotation></semantics></math>, i.e. these hom-sets are vector spaces.</p> <p>Occasionally, however, one does need to refer to the <em>odd</em> linear maps, which send even things to odd things and odd things to even things. That is, an odd linear map <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>V\to W</annotation></semantics></math> is a pair of linear maps <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>W</mi> <mo>−</mo></msub></mrow><annotation encoding='application/x-tex'>V_+ \to W_-</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mo>−</mo></msub><mo>→</mo><msub><mi>W</mi> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>V_-\to W_+</annotation></semantics></math>. The internal-hom in <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> allows us to capture these as well: it is the following super vector space:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>V</mi><mo>,</mo><mi>W</mi><msub><mo stretchy='false'>]</mo> <mo>+</mo></msub><mo>=</mo><mi>Even</mi><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mspace width='2em' /><mo stretchy='false'>[</mo><mi>V</mi><mo>,</mo><mi>W</mi><msub><mo stretchy='false'>]</mo> <mo>−</mo></msub><mo>=</mo><mi>Odd</mi><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> [V,W]_+ = Even Lin(V,W) \qquad [V,W]_- = Odd Lin(V,W). </annotation></semantics></math></div> <p>With this definition, <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sVect</mi></mrow><annotation encoding='application/x-tex'>sVect</annotation></semantics></math> becomes a <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a>.</p> <p>We can equivalently regard a super vector spaces <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(V_+,V_-)</annotation></semantics></math> as being the <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a> vector space <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mo>+</mo></msub><mo>⊕</mo><msub><mi>V</mi> <mo>−</mo></msub></mrow><annotation encoding='application/x-tex'>V_+ \oplus V_-</annotation></semantics></math> equipped with this direct sum decomposition. If we view the internal-hom <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[V,W]</annotation></semantics></math> in this way as well, then we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>]</mo><mo>=</mo><mi>Even</mi><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>⊕</mo><mi>Odd</mi><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> [V, W] = Even Lin(V,W) \oplus Odd Lin(V,W) = Lin(V,W). </annotation></semantics></math></div> <p>In other words, any linear map between these “summed” super vector spaces decomposes uniquely as the sum of an even linear map and an odd one.</p> <h3 id='for_banach_spaces'>For Banach spaces</h3> <p>A similar thing happens in the category <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ban</mi></mrow><annotation encoding='application/x-tex'>Ban</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach spaces</a> and <a class='existingWikiWord' href='/nlab/show/diff/short+linear+map'>short linear operators</a>. The external hom consists of only the <em>short</em> linear maps (those bounded by <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>):</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ban</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>{</mo><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><mrow><mo stretchy='false'>‖</mo><mi>f</mi><mo stretchy='false'>‖</mo></mrow><mo>≤</mo><mn>1</mn><mo stretchy='false'>}</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> Ban(V,W) = \{ f\colon Lin(V,W) \;|\; {\|f\|} \leq 1 \} .</annotation></semantics></math></div> <p>This definition of morphism recovers the most specific notion of <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of Banach spaces, as well as defining the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> and <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> as the <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a> completed with <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding='application/x-tex'>p = \infty</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>p = 1</annotation></semantics></math> respectively.</p> <p>But the internal hom is the Banach space of <em>all</em> bounded linear maps:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>{</mo><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>Lin</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><mrow><mo stretchy='false'>‖</mo><mi>f</mi><mo stretchy='false'>‖</mo></mrow><mo><</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> [V,W] = \{ f\colon Lin(V,W) \;|\; {\|f\|} \lt \infty \} .</annotation></semantics></math></div> <p>This is a Banach space and makes <math class='maruku-mathml' display='inline' id='mathml_ee6f3b3f02891105cc2dc6b89abaaaa41409c5dd_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ban</mi></mrow><annotation encoding='application/x-tex'>Ban</annotation></semantics></math> into a <a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed category</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a>, <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+hom-functor'>enriched hom-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+hom-functor'>derived hom-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a>, <a class='existingWikiWord' href='/nlab/show/diff/linear+implication'>linear implication</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/power+object'>power object</a>, <a class='existingWikiWord' href='/nlab/show/diff/exponential+object'>exponential object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/function+monad'>function monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/exponential+law+for+spaces'>exponential law for spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed category</a>, <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strong+adjoint+functor'>strong adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+stack'>mapping stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/space+of+sections'>space of sections</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+mapping+space'>pointed mapping space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/distributions+are+the+smooth+linear+functionals'>distributions are the smooth linear functionals</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/rational+model+of+mapping+space'>Sullivan model of mapping space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/residual'>residual</a></p> </li> </ul> <h2 id='references'>References</h2> <p>See any reference on <a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed categories</a> and <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal categories</a>.</p> <p>Also for instance:</p> <ul> <li id='MacLaneMoerdijk'><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ieke+Moerdijk'>Ieke Moerdijk</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Sheaves+in+Geometry+and+Logic'>Sheaves in Geometry and Logic</a></em></li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on October 2, 2024 at 22:29:07. 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