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href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> </ul> <h2 id='sidebar_extensions'>Extensions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a><ins class='diffins'><ul><li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li></ul></ins></li><li><a href='#properties'>Properties</a><ul><li><a href='#representable_functors'>Representable functors</a></li><li><a href='#preservation_of_limits'>Preservation of limits</a></li><li><a href='#relation_to_profunctors'>Relation to profunctors</a></li></ul></li><li><a href='#examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>For <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/locally+small+category'>locally small category</a>, its <strong>hom-functor</strong> is the <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'> hom : C^{op} \times C \to Set </annotation></semantics></math></div> <p>from the <a class='existingWikiWord' href='/nlab/show/diff/product+category'>product category</a> of the category <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with its <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> to the category <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a>, which sends</p> <ul> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>∈</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>(c, c') \in C^{op} \times C</annotation></semantics></math>, i.e. a pair of objects in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, to the <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_C(c,c')</annotation></semantics></math> in Set, the set of <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>c</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>q : c \to c'</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>;</p> </li> <li> <p>a morphism <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow /></mover><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(c,c') \stackrel{}{\to} (d,d')</annotation></semantics></math>, i.e. a pair of morphisms</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>c</mi></mtd> <mtd><mi>c</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msup><mi>f</mi> <mi>op</mi></msup></mrow></mpadded></msup></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mi>d</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ c & c' \\ \downarrow^{\mathrlap{f^{op}}} & \downarrow^{\mathrlap{g}} \\ d & d' } </annotation></semantics></math></div> <p>in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_C(c,c') \to Hom_C(d,d')</annotation></semantics></math> that sends</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>q</mi><mo>:</mo><mi>c</mi><mover><mo>→</mo><mrow /></mover><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>↦</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thinmathspace' /><mrow><mo>(</mo><mi>g</mi><mo>∘</mo><mi>q</mi><mo>∘</mo><mi>f</mi><mspace width='thickmathspace' /><mo>:</mo><mspace width='thickmathspace' /><mrow><mtable><mtr><mtd><mi>c</mi></mtd> <mtd><mover><mo>→</mo><mi>q</mi></mover></mtd> <mtd><mi>c</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mi>f</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd /> <mtd><mi>d</mi><mo>′</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (q : c \stackrel{}{\to} c') \;\;\; \mapsto \;\;\, \left( g \circ q \circ f \; : \; \array{ c &\stackrel{q}{\to}& c' \\ \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ d && d' } \right) \,. </annotation></semantics></math></div></li> </ul> <div class='query'> <p>Note: when the symbol <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∘</mo></mrow><annotation encoding='application/x-tex'>\circ</annotation></semantics></math> is used, it denotes traditional right-to-left order of composition. For those who prefer the left-to-right order, the symbol <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>;</mo></mrow><annotation encoding='application/x-tex'>;</annotation></semantics></math> may be used in place of <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∘</mo></mrow><annotation encoding='application/x-tex'>\circ</annotation></semantics></math>. Further discussion of this should go to the nForum page <a href='https://nforum.ncatlab.org/discussion/7239/realigned-arrows-for-clarification-and-fixed-composition-order/'>here</a>.</p> </div> <p>More generally, for <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a> and <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched category</a>, its <em><a class='existingWikiWord' href='/nlab/show/diff/enriched+hom-functor'>enriched hom-functor</a></em> is the <a class='existingWikiWord' href='/nlab/show/diff/enriched+functor'>enriched functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><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>V</mi></mrow><annotation encoding='application/x-tex'> C(-,-) : C^{op} \times C \to V </annotation></semantics></math></div> <p>that sends objects <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c,c' \in C</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>C(c,c') \in V</annotation></semantics></math>.</p> <p>Some categories <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> are equipped with an operation that behaves like a hom-functor, but takes values in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> itself</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>C</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [-,-] : C^{op} \times C \to C \,. </annotation></semantics></math></div> <p>Such an operation is called an <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> functor, and categories carrying this are called <a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed categories</a>.</p> <ins class='diffins'><h3 id='in_homotopy_type_theory'>In homotopy type theory</h3></ins><ins class='diffins'> </ins><ins class='diffins'><p>Note: the <a class='existingWikiWord' href='/nlab/show/diff/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics'>HoTT book</a> calls a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> a “precategory” and a <a class='existingWikiWord' href='/nlab/show/diff/univalent+category'>univalent category</a> a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>For any <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, we have a hom-functor</p></ins><ins class='diffins'> </ins><ins class='diffins'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hom</mi> <mi>A</mi></msub><mo>:</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><mi>A</mi><mo>→</mo><mstyle mathvariant='italic'><mi>Set</mi></mstyle></mrow><annotation encoding='application/x-tex'>hom_A : A^{op} \times A \to \mathit{Set}</annotation></semantics></math></div></ins><ins class='diffins'> </ins><ins class='diffins'><p>It takes a pair <math class='maruku-mathml' display='inline' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>(</mo><msup><mi>A</mi> <mi>op</mi></msup><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub><mo>×</mo><msub><mi>A</mi> <mn>0</mn></msub><mo>≡</mo><msub><mi>A</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>(a,b):(A^{op})_0 \times A_0 \equiv A_0\times A_0</annotation></semantics></math> to the set <math class='maruku-mathml' display='inline' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>hom_A(a,b)</annotation></semantics></math>. For a morphism <math class='maruku-mathml' display='inline' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>:</mo><msub><mi>hom</mi> <mrow><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><mi>A</mi></mrow></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,f') : hom_{A^{op} \times A}((a,b),(a',b'))</annotation></semantics></math>, by definition we have <math class='maruku-mathml' display='inline' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f:hom_A(a',a)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>b</mi><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f': hom_A(b,b')</annotation></semantics></math>, so we can define</p></ins><ins class='diffins'> </ins><ins class='diffins'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_157b51e1b60eff4502133054bc616523a90650f6_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>hom</mi> <mi>A</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>≡</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>↦</mo><mi>f</mi><mo>′</mo><mi>g</mi><mi>f</mi><mo stretchy='false'>)</mo><mo>:</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(hom_A)_{(a,b),(a',b')}(f,f') \equiv (g\mapsto f' g f) : hom_A(a,b) \to hom_A(a',b')</annotation></semantics></math></div></ins><ins class='diffins'> </ins><h2 id='properties'>Properties</h2> <h3 id='representable_functors'>Representable functors</h3> <p>Given a hom-functor <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>hom:C^{op}\times C\to Set</annotation></semantics></math>, for any object <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> one obtains a functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>h</mi> <mi>c</mi></msup><mo>:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'> h^c: C \to Set </annotation></semantics></math></div> <p>given by <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>h</mi> <mi>c</mi></msup><mo>≔</mo><mi>hom</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>h^c\coloneqq hom(c,-)</annotation></semantics></math> and a functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>c</mi></msub><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'> h_c : C^{op} \to Set </annotation></semantics></math></div> <p>given by <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>c</mi></msub><mo>≔</mo><mi>hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>h_c\coloneqq hom(-,c)</annotation></semantics></math>, i.e. by fixing one of the arguments of <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>hom: C^{op} \times C \to Set</annotation></semantics></math> to be <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>.</p> <p>Formally this is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mover><mo>→</mo><mo>≃</mo></mover><mo>*</mo><mo>×</mo><mi>C</mi><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>Id</mi><mo stretchy='false'>)</mo></mrow></mover><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mover><mo>→</mo><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></mover><mi>Set</mi></mrow><annotation encoding='application/x-tex'> hom(c,-) : C \stackrel{\simeq}{\to} * \times C \stackrel{(c,Id)}{\to} C^{op} \times C \stackrel{hom(-,-)}{\to} Set </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mo>*</mo><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><mi>Id</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow></mover><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mover><mo>→</mo><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></mover><mi>Set</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> hom(-,c) : C \stackrel{\simeq}{\to} C^{op} \times * \stackrel{(Id,c)}{\to} C^{op} \times C \stackrel{hom(-,-)}{\to} Set \,. </annotation></semantics></math></div> <p>Functors of the form <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>C^{op} \to Set</annotation></semantics></math> are called <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaves</a> on <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and functors <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>naturally isomorphic</a> to <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>hom(-,c)</annotation></semantics></math> are called <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functors</a> or <strong>representable presheaves</strong> on <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <p>Functors of the form <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>C \to Set</annotation></semantics></math> are called <a class='existingWikiWord' href='/nlab/show/diff/copresheaf'>copresheaves</a> on <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and functors naturally isomorphic to <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>hom(c,-)</annotation></semantics></math> are called co<a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functors</a> or <strong>representable copresheaves</strong> on <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <h3 id='preservation_of_limits'>Preservation of limits</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/hom-functor+preserves+limits'>hom-functor preserves limits</a> in both arguments separately. This means:</p> <ul> <li> <p>for fixed object <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> the functor <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>hom(c,-) : C \to Set</annotation></semantics></math> sends limit <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagrams</a> in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> to limit diagrams in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math>;</p> </li> <li> <p>for fixed object <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c' \in C</annotation></semantics></math> the functor <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>hom(-,c') : C^{op} \to Set</annotation></semantics></math> sends limit diagrams in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> – which are <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagrams</a> in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>! – to limit diagrams in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math>.</p> </li> </ul> <p>For instance for</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>y</mi><msub><mo>×</mo> <mi>x</mi></msub><mi>z</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>y</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>z</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ y \times_x z &\to& y \\ \downarrow && \downarrow \\ z &\to& x } </annotation></semantics></math></div> <p>a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> in <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and for <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> any object, the induced diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo><mo>≃</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>y</mi><msub><mo>×</mo> <mi>x</mi></msub><mi>z</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd /> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ Hom_C(c,y) \times_{Hom_C(c,x)} Hom_C(c,z)\simeq & Hom_C(c,y \times_x z) &\to& Hom_C(c,y) \\ & \downarrow && \downarrow \\ & Hom_C(c,z) &\to& Hom_C(c,x) } </annotation></semantics></math></div> <p>in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> is again a pullback diagram. A moment of reflection shows that this statement is <em>equivalent</em> to the very definition of limit.</p> <h3 id='relation_to_profunctors'>Relation to profunctors</h3> <p>The hom-functor <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>hom : C^{op}\times C\to Set</annotation></semantics></math> is also the identity <a class='existingWikiWord' href='/nlab/show/diff/profunctor'>profunctor</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>C</mi></msub><mo>:</mo><mi>C</mi><mi>⇸</mi><mi>C</mi></mrow><annotation encoding='application/x-tex'>1_C: C &#8696; C</annotation></semantics></math>.</p> <p>One way to see this is to notice that its <a class='existingWikiWord' href='/nlab/show/diff/adjunct'>adjunct</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> C \to [C^{op}, Set] </annotation></semantics></math></div> <p>under the <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> in the 1-category <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a> is the functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mover><mo>→</mo><mi>id</mi></mover><mi>C</mi><mover><mo>→</mo><mi>j</mi></mover><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> C \stackrel{id}{\to} C \stackrel{j}{\to} [C^{op}, Set] \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a>. Profunctors <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>F</mi></mstyle><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\mathbf{F} : C^{op} \times C \to Set</annotation></semantics></math> whose hom-adjunct is of the form <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mover><mo>→</mo><mi>F</mi></mover><mi>C</mi><mover><mo>→</mo><mi>j</mi></mover><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>C \stackrel{F}{\to} C \stackrel{j}{\to} [C^{op}, Set]</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> an ordinary functor are those in the inclusion of these ordinary functors into profunctors. So the hom-functor is the image of the identity functor under this inclusion.</p> <h2 id='examples'>Examples</h2> <p>…</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a>, <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+hom-functor'>enriched hom-functor</a>, <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+hom-functor'>derived hom-functor</a></p> </li> </ul> <table><thead><tr><th /><th><a class='existingWikiWord' href='/nlab/show/diff/homotopy+%28as+an+operation%29'>homotopy</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a></th></tr></thead><tbody><tr><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S^n,-]</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[-,A]</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/functor'>covariant</a> <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/contravariant+functor'>contravariant</a> <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Tor'>Tor</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom space</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/derived+tensor+product'>derived tensor product</a> <math class='maruku-mathml' display='inline' id='mathml_3b8505472343a022e00fd5166c882eacdfdb2b50_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding='application/x-tex'>(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>, <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>enriched hom</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a></li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 7, 2022 at 15:38:27. 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