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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="equality_and_equivalence">Equality and Equivalence</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a> (<a class="existingWikiWord" href="/nlab/show/definitional+equality">definitional</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional</a>, <a class="existingWikiWord" href="/nlab/show/computational+equality">computational</a>, <a class="existingWikiWord" href="/nlab/show/judgemental+equality">judgemental</a>, <a class="existingWikiWord" href="/nlab/show/extensional+equality">extensional</a>, <a class="existingWikiWord" href="/nlab/show/intensional+equality">intensional</a>, <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a>, <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a>, <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+equivalence">gauge equivalence</a></p> </li> <li> <p><strong>Examples.</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence+of+internal+categories">weak equivalence of internal categories</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/equation">equation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><strong>Examples.</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+equation">linear equation</a>, <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>, <a class="existingWikiWord" href="/nlab/show/ordinary+differential+equation">ordinary differential equation</a>, <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equation">Einstein equation</a>, <a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a>, <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+equation">Knizhnik-Zamolodchikov equation</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+equation">Maurer-Cartan equation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Arnold+equation">Euler-Arnold equation</a>, <a class="existingWikiWord" href="/nlab/show/Fuchsian+equation">Fuchsian equation</a>, <a class="existingWikiWord" href="/nlab/show/Fokker-Planck+equation">Fokker-Planck equation</a>, <a class="existingWikiWord" href="/nlab/show/Lax+equation">Lax equation</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/equality+and+equivalence+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="equivalence_of_categories">Equivalence of categories</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#Variants'>Variants</a></li> <ul> <li><a href='#Isomorphism'>Isomorphism</a></li> <li><a href='#StrongEquivalence'>Strong equivalence</a></li> <li><a href='#WeakEquivalence'>Weak equivalence</a></li> <li><a href='#Anaequivalence'>Anaequivalence</a></li> <li><a href='#fully_faithful_essentially_surjective_functors'>Fully faithful essentially surjective functors</a></li> <li><a href='#remarks'>Remarks</a></li> </ul> <li><a href='#adjoint_equivalence'>Adjoint equivalence</a></li> <ul> <li><a href='#an_example_of_a_nonadjoint_equivalence'>An example of a non-adjoint equivalence</a></li> </ul> <li><a href='#in_higher_categories'>In higher categories</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>equivalence of categories</em> is the correct <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> notion of “sameness” of <a class="existingWikiWord" href="/nlab/show/categories">categories</a>.</p> <p>Concretely, an equivalence between two categories is a pair of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between them which are <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> to each other <em>up to</em> <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of functors (<a class="existingWikiWord" href="/nlab/show/inverse+functors">inverse functors</a>).</p> <p>This is like an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, but weakened such as to accomodate for the fact that the correct ambient context for categories is not iself a 1-category, but is the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of all categories. Hence abstractly an equivalence of categories is just the special case of an <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+2-category">equivalence in a 2-category</a> specialized to <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>.</p> <p>If some foundational fine print is taken care of, then a functor exhibits an equivalence of categories precisely if it is both <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> and <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a>. This is true in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a> if the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> is assumed. It remains true non-classically, say for <a class="existingWikiWord" href="/nlab/show/internal+categories">internal categories</a>, if the concept of functor is suitably adapted (“<a class="existingWikiWord" href="/nlab/show/anafunctors">anafunctors</a>”), or the concept of essentially surjective is suitably adapted (“<a class="existingWikiWord" href="/nlab/show/split+essentially+surjective">split essentially surjective</a>”).</p> <p>From the point of view of <a class="existingWikiWord" href="/nlab/show/logic">logic</a> one may say that two <a class="existingWikiWord" href="/nlab/show/categories">categories</a> are <em>equivalent</em> if they have the same <a class="existingWikiWord" href="/nlab/show/properties">properties</a> — although this only applies (by definition) to properties that obey the <em><a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a></em>.</p> <p>Just as equivalence of categories is the generalization of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> from sets to categories, so the concept generalizes further to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categories</a> (see e.g. <em><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></em>, <em><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></em>) and ultimately to equivalence of <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>-<a class="existingWikiWord" href="/nlab/show/infinity-category">categories</a>.</p> <h2 id="definitions">Definitions</h2> <div class="num_defn" id="EquivalenceViaInverseFunctor"> <h6 id="definition">Definition</h6> <p>An <strong>equivalence</strong> between two <a class="existingWikiWord" href="/nlab/show/categories">categories</a> <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+2-category">equivalence in the 2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of all <a class="existingWikiWord" href="/nlab/show/categories">categories</a>, hence a pair of <a class="existingWikiWord" href="/nlab/show/inverse+functors">inverse functors</a>, hence it is</p> <ol> <li> <p>a pair of <a class="existingWikiWord" href="/nlab/show/functors">functors</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><munderover><mrow></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>F</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>G</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><mi>𝒟</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{\phantom{AA}F \phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}G\phantom{AA}}{\longleftarrow}} {} \mathcal{D}, </annotation></semantics></math></div></li> <li> <p><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><mi>F</mi><mo>∘</mo><mi>G</mi><mo>≅</mo><msub><mi>Id</mi> <mi>𝒟</mi></msub></mrow><annotation encoding="application/x-tex"> F \circ G \cong Id_{\mathcal{D}} </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><mi>G</mi><mo>∘</mo><mi>F</mi><mo>≅</mo><msub><mi>Id</mi> <mi>𝒞</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G \circ F \cong Id_{\mathcal{C}} \,. </annotation></semantics></math></div></li> </ol> <p>This is called an <strong><a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a></strong> if the natural isomorphisms above satisfy the <a class="existingWikiWord" href="/nlab/show/triangle+identities">triangle identities</a>, thus exhibiting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> as a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>.</p> <p>Two categories are called <strong>equivalent</strong> if there exists an equivalence between them.</p> </div> <div class="num_prop" id="ViaSplitEssentiallySurjectiveAndFullyFaithful"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \to \mathcal{D}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. Then the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is part of an equivalence of categories in the sense of def. <a class="maruku-ref" href="#EquivalenceViaInverseFunctor"></a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/split+essentially+surjective+functor">split essentially surjective functor</a> and</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a>.</p> </li> </ol> </li> </ol> </div> <p> <div class='proof'> <h6>Proof</h6> <p>This proof is from the <a class="existingWikiWord" href="/nlab/show/HoTT+book">HoTT book</a>, and is set in the context of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>. The categories here are what the HoTT book calls “precategories”, i.e. not <a class="existingWikiWord" href="/nlab/show/univalent+categories">univalent categories</a>:</p> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is an equivalence of categories with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">G,\eta,\epsilon</annotation></semantics></math> specified. Then we have the function</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>B</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mi>a</mi><mo>,</mo><mi>F</mi><mi>b</mi><mo stretchy="false">)</mo></mtd> <mtd><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></mtd></mtr> <mtr><mtd><mi>g</mi></mtd> <mtd><mo>↦</mo><msubsup><mi>η</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>a</mi></msub><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} hom_B(F a, F b) &\to hom_A(a,b), \\ g &\mapsto \eta_b^{-1}\circ G(g) \circ \eta_a. \end{aligned} </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:hom_A(a,b)</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>η</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>a</mi></msub><mo>=</mo><msubsup><mi>η</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>η</mi> <mi>b</mi></msub><mo>∘</mo><mi>f</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\eta_b^{-1} \circ G(F(f)) \circ \eta_a = \eta_b^{-1} \circ \eta_b \circ f = f</annotation></semantics></math></div> <p>while for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>hom</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mi>a</mi><mo>,</mo><mi>F</mi><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g: hom_B(F a, F b)</annotation></semantics></math> we have</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><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>η</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>η</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>ϵ</mi> <mrow><mi>F</mi><mi>b</mi></mrow></msub><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>g</mi><mo>∘</mo><msub><mi>ϵ</mi> <mrow><mi>F</mi><mi>a</mi></mrow></msub><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>g</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} F(\eta_b^{-1} \circ G(g) \circ \eta_a) &= F(\eta_b^{-1}) \circ F(G(g)) \circ F(\eta_a) \\ &= \epsilon_{F b} \circ F(G(g)) \circ F(\eta_a) \\ &= g \circ \epsilon_{F a} \circ F(\eta_a) \\ &= g \end{aligned} </annotation></semantics></math></div> <p>using <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>, and the triangle identities twice. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">F_{a,b}</annotation></semantics></math> is an equivalence, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a>. Finally, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b:B</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>b</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">G b : A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>b</mi></msub><mo>:</mo><mi>F</mi><mi>G</mi><mi>b</mi><mo>≅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\epsilon_b : F G b \cong b</annotation></semantics></math>.</p> <p>On the other hand, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a> and <a class="existingWikiWord" href="/nlab/show/split+essentially+surjective">split essentially surjective</a>. Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>:</mo><msub><mi>B</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_0:B_0\to A_0</annotation></semantics></math> by sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b:B</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math> given by the specified essential splitting, and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_b</annotation></semantics></math> for the likewise specified <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi><mi>b</mi><mo>≅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">F G b \cong b</annotation></semantics></math>.</p> <p>Now for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>hom</mi> <mi>B</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">g: hom_B(b,b')</annotation></semantics></math>, define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mi>b</mi><mo>,</mo><mi>G</mi><mi>b</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(g): hom_A(G b, G b')</annotation></semantics></math> to be the unique morphism such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mrow><mi>b</mi><mo>′</mo></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>g</mi><mo>∘</mo><msub><mi>ϵ</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">F(G(g))=(\epsilon_{b'})^{-1} \circ g \circ \epsilon_b</annotation></semantics></math> which exists since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a>. Finally for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math> define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>a</mi></msub><mo>:</mo><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>G</mi><mi>F</mi><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_a : hom_A(a,G F a)</annotation></semantics></math> to be the unique morphism such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><msub><mi>η</mi> <mi>a</mi></msub><mo>=</mo><msubsup><mi>ϵ</mi> <mrow><mi>F</mi><mi>a</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">F \eta_a = \epsilon^{-1}_{F a}</annotation></semantics></math>. It is easy to verify that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a functor and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>η</mi><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,\eta \epsilon)</annotation></semantics></math> exhibit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> as an equivalence of categories.</p> <p>We clearly recover the same function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>:</mo><msub><mi>B</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_0 : B_0 \to A_0</annotation></semantics></math>. For the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> on hom-sets, we must show that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>hom</mi> <mi>B</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">g:hom_B (b,b')</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(g)</annotation></semantics></math> is the necessarily unique morphism such taht <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mrow><mi>b</mi><mo>′</mo></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>g</mi><mo>∘</mo><msub><mi>ϵ</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">F(G(g))=(\epsilon_{b'})^{-1} \circ g \circ \epsilon_b</annotation></semantics></math>.</p> <p>But this holds by <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>. Then we show <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2) \to (1) \to (2)</annotation></semantics></math> gives the same data hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1)\simeq (2)</annotation></semantics></math>.</p> </div> </p> <h2 id="Variants">Variants</h2> <p>We discuss some possible variants of the definition of equivalence of categories, each of which comes naturally from a different view of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>.</p> <p>The first, <em>isomorphism</em>, comes from viewing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as a mere <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a>; it is too strong and is really only of interest for <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a>. The next, <em>strong equivalence</em>, comes from viewing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a>; it is the most common definition given and is correct if and only if the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds. The next definition, <em>weak equivalence</em>, comes from viewing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>; it is correct with or without choice and is about as simple to define as strong equivalence. The fourth, <em>anaequivalence</em>, comes from viewing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> that is not (without the axiom of choice) equivalent (as a bicategory!) to the strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category that defines strong equivalence; it is also always correct.</p> <p>It is also possible to define ‘category’ in such a way that only a correct definition can be stated, but here we use the usual algebraic definitions of category, <a class="existingWikiWord" href="/nlab/show/functor">functor</a>, and <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>.</p> <h3 id="Isomorphism">Isomorphism</h3> <p>Two <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a> <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> are <strong>isomorphic</strong> if there exist <a class="existingWikiWord" href="/nlab/show/strict+functors">strict functors</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>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G\colon D \to C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">F G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">G F</annotation></semantics></math> are each <a class="existingWikiWord" href="/nlab/show/equality">equal</a> to the appropriate <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a>. In this case, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is an <strong>isomorphism</strong> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is an isomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> to <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>) and call the 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>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F,G)</annotation></semantics></math> an <strong>isomorphism</strong> between <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is called the <strong>strict inverse</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the strict inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>).</p> <p>If you think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as the category of (strict) categories and functors, then this is the usual notion of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in a category. This is the most obvious notion of equivalence of categories and the first to be considered, but it is simply too strong for the purposes to which <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> is put. For example, here are some basic and important equivalences of categories that are not isomorphisms:</p> <ul> <li> <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 the category of <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be the category of sets and <a class="existingWikiWord" href="/nlab/show/partial+functions">partial functions</a>. The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F:C\to D</annotation></semantics></math> takes a pointed set to its subset of non-basepoint elements, and a pointed function to the induced partial function on these (which is defined on those elements not sent to the basepoint). See the section “The category of sets and partial functions” in <a class="existingWikiWord" href="/nlab/show/partial+function">partial function</a> for this equivalence.</p> </li> <li> <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 the category of finite-dimensional <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be the category whose objects are <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> and whose morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>→</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\to m</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m\times n</annotation></semantics></math> matrices of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> (which is equivalently the full subcategory 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">C</annotation></semantics></math> spanned by the specific vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>). Note in particular that here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, while <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> is not (though it is <a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small</a>, being equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>).</p> </li> </ul> <h3 id="StrongEquivalence">Strong equivalence</h3> <p>Two <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a> <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> are <strong>strongly equivalent</strong> if there exist strict functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G\colon D \to C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">F G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">G F</annotation></semantics></math> are each <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> (<a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> in the relevant <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>) to the appropriate <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a>. In this case, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a <strong>strong equivalence</strong> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a strong equivalence from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> to <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>). The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is called a <strong>weak inverse</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a weak inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>).</p> <p>Note that an isomorphism is precisely a strong equivalence in which the natural isomorphisms are <a class="existingWikiWord" href="/nlab/show/identity+natural+transformation">identity natural transformation</a>s.</p> <p>If you think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a> of (strict) categories, functors, and <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s, then this is the usual notion of <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-category">category</a>. This is the first ‘correct’ definition of equivalence to be considered and the one most often seen today; it is only correct using the axiom of choice.</p> <div class="query"> <p>If possible, use or modify the counterexample to isomorphism to show how choice follows if strong equivalence is assumed correct.</p> </div> <h3 id="WeakEquivalence">Weak equivalence</h3> <p>Two categories <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> are <strong>weakly equivalent</strong> if there exist a category <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 functors <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>D</mi></mrow><annotation encoding="application/x-tex">F\colon X \to D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G\colon X \to C</annotation></semantics></math> that are <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> and <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">fully faithful</a>. In this case, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a <strong>weak equivalence</strong> 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>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a weak equivalence 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>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>) and call the <a class="existingWikiWord" href="/nlab/show/span">span</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>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,F,G)</annotation></semantics></math> a <strong>weak equivalence</strong> between <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. (It is not entirely trivial to check that such spans can be composed, but they can be.)</p> <p>A strict functor with a weak inverse is necessarily essentially surjective and fully faithful; the converse is equivalent to the axiom of choice. Thus any strong equivalence becomes a weak equivalence in which <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 taken to be either <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (or even built symmetrically out 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">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> if you're so inclined); a weak equivalence becomes a strong equivalence using the axiom of choice to find weak inverses and composing across <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> <p>If you think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as the model category of categories and functors with the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure">canonical model structure</a>, then this is the usual notion of <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> in a model category.</p> <h3 id="Anaequivalence">Anaequivalence</h3> <p>Two categories <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> are <strong>anaequivalent</strong> if there exist <a class="existingWikiWord" href="/nlab/show/anafunctors">anafunctors</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>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G\colon D \to C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">F G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">G F</annotation></semantics></math> are each ananaturally isomorphic (isomorphic in the relevant <a class="existingWikiWord" href="/nlab/show/anafunctor+category">anafunctor category</a>) to the appropriate <a class="existingWikiWord" href="/nlab/show/identity+anafunctor">identity anafunctor</a>. In this case, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is an <strong>anaequivalence</strong> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is an anaequivalence from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> to <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>). The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is called an <strong>anainverse</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is an anainverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>). See also <em><a class="existingWikiWord" href="/nlab/show/weak+equivalence+of+internal+categories">weak equivalence of internal categories</a></em>.</p> <p>Any strict functor is an anafunctor, so any strong equivalence is an anaequivalence. Using the axiom of choice, any anafunctor is <a class="existingWikiWord" href="/nlab/show/ananatural+isomorphism">ananaturally isomorphic</a> to a strict functor, so any anaequivalence defines a strong equivalence. Using the definition of an anafunctor as an appropriate span of strict functors, a weak equivalence defines two anafunctors which form an anaequivalence; conversely, either anafunctor in an anaequivalence is (as a span) a weak equivalence.</p> <p>If you think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> of categories, anafunctors, and <a class="existingWikiWord" href="/nlab/show/ananatural+transformation">ananatural transformation</a>s, then this is the usual notion of <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category. It's fairly straightforward to turn any discussion of functors and strong equivalences in a context where the axiom of choice is assumed into a discussion of anafunctors and anaequivalences in a more general context.</p> <p>We can also regard the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> above as obtained from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Str</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Str Cat</annotation></semantics></math> of strict categories, strict functors, and natural transformations by formally inverting the weak equivalences as in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <h3 id="fully_faithful_essentially_surjective_functors">Fully faithful essentially surjective functors</h3> <p>Finally, there are <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a> and <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functors">essentially surjective functors</a>. However, while in general, these are not the same as equivalences in all mathematical foundations, they are the same under certain restrictions:</p> <div class="num_prop" id="ViaEssentiallySurjectiveAndFullyFaithful"> <h6 id="proposition_2">Proposition</h6> <p>Assume the ambient context is one of the following:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a> with the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> or <a class="existingWikiWord" href="/nlab/show/internal+category">internal</a> category theory with “functor” meaning <em><a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a></em>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher-level+foundations">higher-level foundations</a> with “category” meaning <a class="existingWikiWord" href="/nlab/show/univalent+category">univalent category</a>.</p> </li> </ul> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \to \mathcal{D}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. Then the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is part of an equivalence of categories in the sense of def. <a class="maruku-ref" href="#EquivalenceViaInverseFunctor"></a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective functor</a> and</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a>.</p> </li> </ol> </li> </ol> </div> <h3 id="remarks">Remarks</h3> <p>Note that <em>weak</em> inverses go with <em>strong</em> equivalences. The terminology isn't entirely inconsistent (weak inverses contrast with <em>strict</em> ones, while weak equivalences contrast with <em>strong</em> ones) but developed at different times. The prefix ‘ana‑’ developed last and is perfectly consistent.</p> <p>If you accept the axiom of choice, then you don't have to worry about the different kinds of equivalence (as long as you don't use isomorphism). This is not just a question of <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>, however, since the axiom of choice usually fails in <a class="existingWikiWord" href="/nlab/show/internalization">internal contexts</a>.</p> <p>It's also possible to use foundations (such as <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, some other forms of <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, or <a class="existingWikiWord" href="/nlab/show/FOLDS">FOLDS</a>) in which isomorphism and strong equivalence are impossible to state. In such a case, one usually drops the prefixes ‘weak’ and ‘ana‑’. In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Lab, we prefer to remain agnostic about foundations but usually drop these prefixes as well, leaving it up to the reader to insert them if necessary.</p> <h2 id="adjoint_equivalence">Adjoint equivalence</h2> <p>Any equivalence can be improved to an <strong><a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a></strong>: a strong equivalence or anaequivalence whose natural isomorphisms satisfy the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">triangle identities</a>. In that case, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is called <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <strong>adjoint inverse</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the adjoint inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>). When working in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, a good rule of thumb is that it is okay to consider either</p> <ul> <li>a functor with the <em>property</em> of being a <em>general</em> equivalence or</li> <li>a functor with the <em>structure</em> of being an <em>adjoint</em> equivalence (that is, a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and a pair of natural isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi><mo>≅</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">F G \cong 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≅</mo><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">1 \cong G F</annotation></semantics></math> satisfying the triangle identities),</li> </ul> <p>whereas considering</p> <ul> <li>a functor with the <em>structure</em> of being a <em>general</em> equivalence (that is, merely a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and a pair of natural isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi><mo>≅</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">F G \cong 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≅</mo><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">1 \cong G F</annotation></semantics></math>)</li> </ul> <p>is fraught with peril. For instance, an adjoint inverse is unique up to unique isomorphism (much as a strict inverse is unique up to equality), while a weak inverse or anainverse is not. Including adjoint equivalences is also the only way to make a higher-categorical structure completely <em><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic</a></em>.</p> <h3 id="an_example_of_a_nonadjoint_equivalence">An example of a non-adjoint equivalence</h3> <p>Identify a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>. One can check the following:</p> <ul> <li> <p>Any equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>H</mi><mo>⇆</mo><mi>H</mi><mo>:</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F : H \leftrightarrows H : G</annotation></semantics></math> of a group with itself comprises two automorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F, G</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">F G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">G F</annotation></semantics></math> are inner. The unit and counit are the group elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">g_{\rho}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GF</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mi>ρ</mi></msub><mi>k</mi><msubsup><mi>g</mi> <mi>ρ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">GF(k) = g_{\rho} k g_{\rho}^{-1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>σ</mi></msub></mrow><annotation encoding="application/x-tex">g_{\sigma}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FG</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>g</mi> <mi>σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mi>k</mi><msub><mi>g</mi> <mi>σ</mi></msub></mrow><annotation encoding="application/x-tex">FG(k) = g_{\sigma}^{-1} k g_{\sigma}</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">k \in H</annotation></semantics></math>.</p> </li> <li> <p>Any equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with itself where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> are themselves also inner is an adjoint equivalence.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> has trivial center, then any equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with itself is an adjoint equivalence.</p> </li> <li> <p>To obtain a non-adjoint equivalence, we therefore need a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with nontrivial center and nontrivial outer automorphisms, such that we can pick two whose products are inner.</p> </li> <li> <p>So take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">H = K</annotation></semantics></math> the Klein 4-group. This is a product of abelian groups, so abelian, so is its own center. In fact, it’s <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><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}</annotation></semantics></math>, so let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F = G</annotation></semantics></math> the automorphism which interchanges coordinates. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FG</mi><mo>=</mo><mi>GF</mi><mo>=</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">FG = GF = \operatorname{id}_{K}</annotation></semantics></math>, which is given by conjugation by any element.</p> </li> <li> <p>If this were adjoint, the triangle equality for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> will stipulate that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>ρ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>g</mi> <mi>σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">F(g_{\rho}) = g_{\sigma}^{-1}</annotation></semantics></math>. We can pick <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">g_{\rho}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>σ</mi></msub></mrow><annotation encoding="application/x-tex">g_{\sigma}</annotation></semantics></math> to break this. For example, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>ρ</mi></msub></mrow><annotation encoding="application/x-tex">g_{\rho}</annotation></semantics></math> be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>σ</mi></msub></mrow><annotation encoding="application/x-tex">g_{\sigma}</annotation></semantics></math> be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>.</p> </li> </ul> <p>This is a special case of the fact that, given a non-adjoint equivalence, you can always replace its unit with another unit (which determines the counit) to improve the equivalence to an <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>.</p> <h2 id="in_higher_categories">In higher categories</h2> <p>All of the above types of equivalence make sense for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/n-category">categories</a> and <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>-categories defined using an <a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a>; again, it is the weak notion that is usually wanted. When using a <a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a>, often isomorphism cannot even be written down, so equivalence comes more naturally.</p> <p>In particular, one expects (although a proof depends on the exact definition and hasn't always been done) that in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories, every <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> (in the sense of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category) will be essentially <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/k-surjective+functor">surjective</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\le k\le n+1</annotation></semantics></math>; this is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-version of “full, faithful, and essentially surjective.” The converse should be true assuming that</p> <ul> <li>we have an axiom of choice and use weak (pseudo) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/n-functor">functors</a>, or</li> <li>we use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<span class="newWikiWord">anafunctors<a href="/nlab/new/n-anafunctor">?</a></span> (which are automatically weak).</li> </ul> <p>If we use too strict a notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-functor, then we will not get the correct notion of equivalence; if we use weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-functors but not anafunctors, then we will get the correct notion of equivalence only if the axiom of choice holds, although again this can be corrected by moving to a span. Note that even strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories need weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-functors to get the correct notion of equivalence between them!</p> <p>For example, assuming choice, a <a class="existingWikiWord" href="/nlab/show/strict+2-functor">strict 2-functor</a> between strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-categories is an equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bicat</mi></mrow><annotation encoding="application/x-tex">Bicat</annotation></semantics></math> if and only if it is essentially (up to equivalence) surjective on objects, locally essentially surjective, and <a class="existingWikiWord" href="/nlab/show/locally+fully+faithful+2-functor">locally fully faithful</a>. However, its weak inverse may not be a <em>strict</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor, and even if it is, the equivalence transformations need not be strictly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-natural. Thus, it need not be an equivalence in the <a class="existingWikiWord" href="/nlab/show/strict+3-category">strict 3-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Str</mi><mn>2</mn><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Str 2 Cat</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-categories, strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functors, and strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-natural transformations, or even in the <span class="newWikiWord">semi-strict 3-category<a href="/nlab/new/semi-strict+3-category">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math> of strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-categories, strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functors, and pseudonatural transformations.</p> <p>As with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, we can recover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bicat</mi></mrow><annotation encoding="application/x-tex">Bicat</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a> sub<a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math> by formally inverting all such weak equivalences. Note that even with the axiom of choice, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bicat</mi></mrow><annotation encoding="application/x-tex">Bicat</annotation></semantics></math> is <em>not</em> equivalent (as a tricategory) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>, even though by the coherence theorem for tricategories it is equivalent to <em>some</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-category; see <a href="http://arxiv.org/abs/math/0612299">here</a>.</p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>basic properties of…</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/functors">functors</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/full+functor">full functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective-on-objects+functor">injective-on-objects functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+injective+functor">essentially injective functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+categories">embedding of categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective-on-objects+functor">surjective-on-objects functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bijective-on-objects+functor">bijective-on-objects functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dominant+functor">dominant functor</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/split+essentially+surjective+functor">split essentially surjective functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+surjective+and+full+functor">essentially surjective and full functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+functor">conservative functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+functor">localization functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+functor">pseudomonic functor</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/cocontinuous+functor">co</a>)<a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/final+functor">final functor</a>, <a class="existingWikiWord" href="/nlab/show/dense+functor">dense functor</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/2-functors">2-functors</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/full+and+faithful+2-functor">full and faithful 2-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+fully+faithful+2-functor">locally fully faithful 2-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functors">(∞,1)-functors</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+%28%E2%88%9E%2C1%29-functor">fully faithful (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+surjective+%28%E2%88%9E%2C1%29-functor">essentially surjective (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+final+functor">homotopy final functor</a>, <a class="existingWikiWord" href="/nlab/show/final+%28%E2%88%9E%2C1%29-functor">final (∞,1)-functor</a></p> </li> </ul> </li> </ul> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a>, <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence+of+toposes">homotopy equivalence of toposes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> </li> <li> <p><strong>equivalence of categories</strong>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence+of+internal+categories">weak equivalence of internal categories</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/equivalence+of+2-categories">equivalence of 2-categories</a>, <a class="existingWikiWord" href="/nlab/show/2-adjunction">2-adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 28, 2024 at 16:47:16. 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