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width: 0.3em;"></span> <a href="/nlab/show/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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <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/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"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/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></div> <h4 id="mapping_space">Mapping space</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>/<a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong></p> <h3 id="general_abstract">General abstract</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>, <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>, <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>, <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>, <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a></p> </li> </ul> <h3 id="topology">Topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (<a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topology+of+mapping+spaces">topology of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/evaluation+fibration+of+mapping+spaces">evaluation fibration of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/free+loop+space+of+a+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/simplicial+mapping+complex">simplicial mapping complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+groupoid">inertia groupoid</a></p> </li> </ul> <h3 id="differential_topology">Differential topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+topology+of+mapping+spaces">differential topology of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/C-k+topology">C-k topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tangent+spaces+of+mapping+spaces">tangent spaces of mapping spaces</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/mapping+spectrum">mapping spectrum</a></li> </ul> <h3 id="geometric_homotopy_theory">Geometric homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+stack">inertia stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+loop+stack">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> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#EvaluationMap'>Evaluation map</a></li> <li><a href='#CompositionMap'>Composition map</a></li> </ul> <li><a href='#properties'>Properties</a></li> <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><a href='#Examples'>Examples</a></li> <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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/objects">objects</a>, the <em>internal hom</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/object">object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/morphisms">morphisms</a>” from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/internalization">internal version</a> of the ordinary <a class="existingWikiWord" href="/nlab/show/hom+set">hom set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/hom+object">hom object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/locally+small+category">locally small category</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/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/function+set">function set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/functions">functions</a> between two <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/natural+bijection">natural bijection</a> with the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cartesian+product">cartesian product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. That is: for each set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/right+adjoint">right adjoint</a>, given by the construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cartesian+monoidal+category">cartesian monoidal category</a> or in fact in any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/tensor+product">tensor product</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/closed+monoidal+category">closed</a></em> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>. Explicity, the condition is that there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>(<a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/natural+isomorphism">natural</a> in all three <a class="existingWikiWord" href="/nlab/show/variables">variables</a>. (The leftward map here is often called <strong><a class="existingWikiWord" href="/nlab/show/currying">currying</a></strong>, especially in a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (and more especially for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/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/generalized+elements">generalized elements</a> out of the <a class="existingWikiWord" href="/nlab/show/unit+object">tensor unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/cartesian+monoidal+category">cartesian monoidal category</a> is indeed the hom as seen in the <em><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></em> of that category (the <em><a class="existingWikiWord" href="/nlab/show/function+type">function type</a></em>).</p> <p>More generally, one can consider objects that satisfy some basic <a class="existingWikiWord" href="/nlab/show/universal+properties">universal properties</a> that an internal hom should satisfy even in the absence of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal structure</a>. If such objects exist one speaks therefore just of a <em><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></em>. Every <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a> may be seen as a category <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over itself. Accordingly, an internal hom is after all a special case of a <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>. An <strong>internal hom</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> is not <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (Def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>).</p> <div class="num_defn" id="EvalMap"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/objects">objects</a>, the <strong><a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/adjunct">adjunct</a> of the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/locally+cartesian+closed+category">locally cartesian closed category</a>, then in terms of the <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> is the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of the <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> which in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><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/function+application">function application</a></em> on the right.</p> </div> <h3 id="CompositionMap">Composition map</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (Def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>)</p> <div class="num_defn" id="CompositionMorphism"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/objects">objects</a>, the <strong><a class="existingWikiWord" href="/nlab/show/composition">composition</a> morphism</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/adjunct">adjunct</a> of the following composite of two <a class="existingWikiWord" href="/nlab/show/evaluation+maps">evaluation maps</a>, def. <a class="maruku-ref" href="#EvalMap"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (Def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>) happen to share all the key abstract properties of ordinary (“external”) <a class="existingWikiWord" href="/nlab/show/hom-functors">hom-functors</a>, even though this is not completely manifest from Def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>:</p> <div class="num_prop" id="InternalHomBifunctor"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <a class="existingWikiWord" href="/nlab/show/bifunctor">bifunctor</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/symmetric+monoidal+category">symmetric monoidal category</a> such that for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>, in that there is a unique functor out of the <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/opposite+category">opposite category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math>, but also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, and hence in particular for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/adjoint+functor">adjoint functor</a></em>.</p> </div> <p>In fact the 3-variable adjunction from Prop. <a class="maruku-ref" href="#InternalHomBifunctor"></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/closed+monoidal+category">closed monoidal category</a> (def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>) there are <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/natural+bijections">natural bijections</a> of Prop. <a class="maruku-ref" href="#InternalHomBifunctor"></a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>, there are composite <a class="existingWikiWord" href="/nlab/show/natural+bijections">natural bijections</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><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> <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> <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> <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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithfulness</a> of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> says that there is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/unitor">unitor</a> isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <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"></mspace><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/hom-functors">hom-functors</a> for <a class="existingWikiWord" href="/nlab/show/limits">limits</a> is inherited by <a class="existingWikiWord" href="/nlab/show/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/internal+hom-functor+preserves+limits">internal hom-functor preserves limits</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>-<a class="existingWikiWord" href="/nlab/show/bifunctor">bifunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>). Then this bifunctor preserves <a class="existingWikiWord" href="/nlab/show/limits">limits</a> in the second variable, and sends colimits in the first variable to limits:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/right+adjoint">right adjoint</a> by definition, and hence preserves limits by <em><a class="existingWikiWord" href="/nlab/show/adjoints+preserve+%28co-%29limits">adjoints preserve (co-)limits</a></em>.</p> <p>For the other case, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><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> <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> <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> <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> <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/natural+isomorphism">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/hom-functor">hom-functor</a> respects (co)limits as shown (see at <em><a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">hom-functor preserves limits</a></em>), and that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/adjoints+preserve+%28co-%29limits">adjoints preserve (co-)limits</a></em>).</p> <p>Hence by the <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithfulness</a> of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>, there is an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/categorical+semantics">categorical semantics</a> of what in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> are <a class="existingWikiWord" href="/nlab/show/function+types">function types</a></p> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/function+type">function type</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a></strong></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+formation">type formation</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo>⊢</mo><mspace width="mediummathspace"></mspace><mi>X</mi><mo lspace="verythinmathspace">:</mo><mi>Type</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo lspace="verythinmathspace">:</mo><mi>Type</mi></mrow><mrow><mo>⊢</mo><mspace width="thickmathspace"></mspace><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/term+introduction">term introduction</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><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/term+elimination">term elimination</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo>⊢</mo><mspace width="thickmathspace"></mspace><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><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi></mrow><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><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/computation+rule">computation rule</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> </div> <h3 id="induced_monad_state_monad">Induced monad (state monad)</h3> <p>For each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> the (internal hom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>)-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> induces a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/computer+science">computer science</a> this <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a> is called the <em><a class="existingWikiWord" href="/nlab/show/state+monad">state monad</a></em>.</p> <h3 id="StableSplitting">Stable splitting</h3> <p>In <a class="existingWikiWord" href="/nlab/show/topology">topology</a> the <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a>/<a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/mapping+spaces">mapping spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/CW-complexes">CW-complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/direct+sum">direct sum</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> in a useful way, related to the expression of the <a class="existingWikiWord" href="/nlab/show/Goodwillie+derivatives">Goodwillie derivatives</a> of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>, the internal hom is given by <a class="existingWikiWord" href="/nlab/show/function+sets">function sets</a>. This exists, by the discussion there, as soon as the <a class="existingWikiWord" href="/nlab/show/foundations">foundational</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> are strong enough, for instance as soon as there are <a class="existingWikiWord" href="/nlab/show/power+objects">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/sSet">sSet</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, the internal hom between two <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/n-simplex">n-simplex</a>. This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/function+complex">function complex</a></em> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/category+of+presheaves">category of presheaves</a> over the <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/sheaf+topos">sheaf topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a>. We discuss the internal hom of this regard as a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>/<a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+%28%E2%88%9E%2C1%29-category">cartesian monoidal (∞,1)-category</a>.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The sheaf topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cartesian+closed+category">cartesian closed category</a> / <a class="existingWikiWord" href="/nlab/show/cartesian+closed+%28%E2%88%9E%2C1%29-category">cartesian closed (∞,1)-category</a>. In fact it is a <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed category</a> / <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28%E2%88%9E%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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/objects">objects</a>, the internal hom-object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/sheaf">sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a> given by the assignment</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Yoneda+embedding">Yoneda embedding</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">∞-Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cartesian+product">cartesian product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/hom+set">hom set</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a> / <a class="existingWikiWord" href="/nlab/show/derived+hom-space">hom space</a>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/Yoneda+lemma">Yoneda lemma</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> we have <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/adjunction">adjunction</a> this is naturally equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/homotopy+type+theory">homotopy</a>-)<a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> of the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/categorical+semantics">categorical semantics</a> of the <a class="existingWikiWord" href="/nlab/show/function+type">function type</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/evaluation+map">evaluation map</a>, def. <a class="maruku-ref" href="#EvalMap"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/objects">objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/composition">composition</a> <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>, def. <a class="maruku-ref" href="#CompositionMorphism"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/presheaves">presheaves</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaves">(∞,1)-presheaves</a> whose component over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/sets">sets</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/diagonal">diagonal</a> morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> the morphism is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the double <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/cartesian+product">cartesian product</a> of two sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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. <a class="maruku-ref" href="#..."></a> 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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized family of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized family of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-family given by the parameter-wise composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/automorphism+group">automorphism group</a>/<a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> of an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/subobject">subobject</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></a> becomes invertible.</p> <p>The (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy</a>-)<a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> for this is given by the <a class="existingWikiWord" href="/nlab/show/type">type</a> of <a class="existingWikiWord" href="/nlab/show/equivalences+in+homotopy+type+theory">equivalences in homotopy type theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/locally+cartesian+closed+category">locally cartesian closed category</a>. This means that for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cartesian+closed+category">cartesian closed category</a>. The <a class="existingWikiWord" href="/nlab/show/product">product</a> in the slice is given by the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/base+change">base change</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/isomorphisms">isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/locally+cartesian+closed+category">locally cartesian closed category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/morphism">morphism</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/hom+sets">hom sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><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> <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> <mtd><mover><mo>→</mo><mrow></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> <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> <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/adjunction">adjunction</a> properties, where the morphism in the middle is the component of the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> functor, and where “Frob.Rec.” is <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a>.</p> <p>Since this is <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/unit+of+an+adjunction">adjunction (co)units</a> and the <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> of the internal hom. Its hom-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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"></a> on the right. The hom-adjunct of that in turn is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> (for simplicity), then the morphism of prop. <a class="maruku-ref" href="#MorphismFromDepProductOfFuncTypeToFuncTypeOfDepSum"></a> can be understood as follows: a <a class="existingWikiWord" href="/nlab/show/global+element">global element</a> of the <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/commuting+diagram">commuting diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></munder><mi>A</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></munder><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><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"></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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (and the <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>) or equivalently over the <a class="existingWikiWord" href="/nlab/show/dense+subsite">dense subsite</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> and <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between these.</p> <p>The <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/smooth+spaces">smooth spaces</a>/<a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/smooth+manifolds">smooth manifolds</a>, the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/mapping+space">mapping space</a> between them regarded as a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></em> for when this internal hom is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> again by a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/diffeomorphism+group">diffeomorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/diffeological+space">diffeological</a> group</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> is the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/graded+vector+spaces">graded vector spaces</a>. Thus, its objects are pairs of vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a>.</p> <p>We can equivalently regard a super vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/direct+sum">direct sum</a> vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ban</mi></mrow><annotation encoding="application/x-tex">Ban</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a> and <a class="existingWikiWord" href="/nlab/show/short+linear+operators">short linear operators</a>. The external hom consists of only the <em>short</em> linear maps (those bounded by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><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/isomorphism">isomorphism</a> of Banach spaces, as well as defining the <a class="existingWikiWord" href="/nlab/show/product">product</a> and <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> as the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> completed with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">p = \infty</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ban</mi></mrow><annotation encoding="application/x-tex">Ban</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+hom-functor">enriched hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+hom-functor">derived hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+type">function type</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/implication">implication</a>, <a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/power+object">power object</a>, <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+monad">function monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential law for spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a>, <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strong+adjoint+functor">strong adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+mapping+space">pointed mapping space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributions+are+the+smooth+linear+functionals">distributions are the smooth linear functionals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+mapping+space">Sullivan model of mapping space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/residual">residual</a></p> </li> </ul> <h2 id="references">References</h2> <p>See any reference on <a class="existingWikiWord" href="/nlab/show/closed+categories">closed categories</a> and <a class="existingWikiWord" href="/nlab/show/closed+monoidal+categories">closed monoidal categories</a>.</p> <p>Also for instance:</p> <ul> <li id="MacLaneMoerdijk"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 2, 2024 at 22:29:07. 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