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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="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="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'> References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>isomorphism</em> generalizes the concept of <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> from the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> to general <a class="existingWikiWord" href="/nlab/show/categories">categories</a>.</p> <p>An <em>isomorphism</em> is an invertible <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>, hence a morphism with an <a class="existingWikiWord" href="/nlab/show/inverse+morphism">inverse morphism</a>.</p> <p>Two <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> are said to be <em>isomorphic</em> if there exists an isomorphism between them. This means that they “are the same for all practical purposes” as long as one does not violate the <a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a>.</p> <p>But beware that two objects may be isomorphic by more than one isomorphism. In particular a single object may be isomorphic to itself by nontrivial isomorphisms other than the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a>. Frequently the particular choice of isomorphism matters.</p> <p>Every isomorphism is in particular an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> and a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, but the converse need not hold.</p> <p>Common jargon includes “is a mono” or “is monic” for “is a monomorphism”, and “is an epi” or “is epic” for “is an epimorphism”, and “is an iso” for “is an isomorphism”.</p> <p><a class="existingWikiWord" href="/nlab/show/Henri+Poincar%C3%A9">Henri Poincaré</a> wrote in <a href="#Poincaré1908">Poincaré 1908</a> about isomorphisms:</p> <blockquote> <p>“It is scarcely credible, as Mach said, how much a well-chosen word can economize thought. I do not know whether or not I have said somewhere that mathematics is the art of giving the same name to different things. We must so understand it. It is appropriate that things different in substance, but alike in form, should be put into the same mold, so to speak. When our language is well chosen, it is astonishing to see how all the demonstrations made upon some known fact immediately become applicable to many new facts. Nothing has to be changed, not even the words, since the names are the same in the new cases. There is an example, which comes at once to my mind; it is quaternions, upon which, however, I will not dwell.</p> <p>A word well chosen very often causes the disappearance of exceptions to rules as announced in their former forms; it was for this purpose that the terms ‘negative quantities’, ‘imaginary quantities’, ‘infinite points’, have been invented. And let us not forget that these exceptions are pernicious, for they conceal laws. Very well then, one of those marks by which we recognize the pregnancy of a result is in that it permits a happy innovation in our language. The mere fact is oftentimes without interest; it has been noted many times, but has rendered no service to science; it becomes of value only on that day when some happily advised thinker perceives a relationship, which he indicates and symbolizes by a word.</p> <p>The physicists also do it just the same way. They invented the term ‘energy’, a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form.</p> <p>Among the words which have had this happy result, I will mention the ‘group’ and the ‘invariant’. They make us perceive the gist of many mathematical demonstrations; they make us realize how often mathematicians of the past must have run across groups without recognizing them and how, believing these groups to be isolated things, they have found them to be in close relationship without knowing why. Today we would say that they were looking right in the face of isomorphic groups. We feel now that in a group the substance interests us but very little; it is the form alone which matters, and so, when we once know well a single group, then we know through it all the isomorphic groups; thanks to the words ‘groups’ and ‘isomorphism’, which sum in a few syllables this subtle law and make it at once familiar to us all, we take our step at once and in so doing economize all effort of thought.“</p> </blockquote> <h2 id="definitions">Definitions</h2> <p>An <strong>isomorphism</strong>, or <strong>iso</strong> for short, is an invertible <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>, i.e. a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> with a 2-sided <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a>.</p> <p>A morphism could be called <strong>isic</strong> (following the more common ‘monic’ and ‘epic’) if it is an isomorphism, but it's more common to simply call it <em>invertible</em>. Two <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are <strong>isomorphic</strong> if there exists an isomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> (or equivalently, from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> to <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>). An <strong><a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a></strong> is an isomorphism from one object to itself.</p> <h2 id="properties">Properties</h2> <p>It is immediate that isomorphisms satisfy the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> property. But they also satisfy <a class="existingWikiWord" href="/nlab/show/two-out-of-six+property">two-out-of-six property</a> satisfied by the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> in any <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>.</p> <p>Note that the <a class="existingWikiWord" href="/nlab/show/inverse+morphism">inverse morphism</a> of an isomorphism is an isomorphism, as is any <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> or <a class="existingWikiWord" href="/nlab/show/composite">composite</a> of isomorphisms. Thus, being isomorphic is an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on objects. The equivalence classes form the <a class="existingWikiWord" href="/nlab/show/fundamental+0-groupoid">fundamental 0-groupoid</a> of the category in question.</p> <p>Every isomorphism is both a <a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a> (and thus about any other kind of <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) and a <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a> (and thus about any other kind of <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>). In a <a class="existingWikiWord" href="/nlab/show/balanced+category">balanced category</a>, every morphism that is both a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> and an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> is invertible, but this does not hold in general. However, any monic <a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epimorphism</a> (or dually, any epic <a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular monomorphism</a>) must be an isomorphism.</p> <p>A <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> is precisely a <a class="existingWikiWord" href="/nlab/show/category">category</a> in which every morphism is an isomorphism. More generally, the <a class="existingWikiWord" href="/nlab/show/core">core</a> of any category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> consisting of all objects but only the isomorphisms; it is a kind of underlying groupoid 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>. In a similar way, the automorphisms of any given object <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> form a <a class="existingWikiWord" href="/nlab/show/group">group</a>, the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/n-category">higher categories</a>, isomorphisms generalise to <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>s, which we expect to have only <a class="existingWikiWord" href="/nlab/show/weak+inverse">weak inverse</a>s.</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, for every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <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> the <a class="existingWikiWord" href="/nlab/show/type">type</a> “f is an isomorphism” is a <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>. Therefore, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a,b:A</annotation></semantics></math> the type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \cong b</annotation></semantics></math> is a set. <div class='proof'> <h6>Proof</h6> <p>Suppose given <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>A</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g:hom_A(b,a)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><msub><mn>1</mn> <mi>a</mi></msub><mo>=</mo><mi>g</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\eta:1_a = g \circ f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>=</mo><msub><mn>1</mn> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon : f \circ g = 1_b</annotation></semantics></math>, and similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo>,</mo><mi>η</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g',\eta'</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\epsilon'</annotation></semantics></math>. We must show <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><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>η</mi><mo>′</mo><mo>,</mo><mi>ϵ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,\eta,\epsilon)=(g',\eta', \epsilon')</annotation></semantics></math>. But since all hom-sets are sets, their identity types are mere propositions, so it suffices to show <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mi>g</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g=g'</annotation></semantics></math>. For this we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo>=</mo><msub><mn>1</mn> <mi>a</mi></msub><mo>∘</mo><mi>g</mi><mo>′</mo><mo>=</mo><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>g</mi><mo>′</mo><mo>=</mo><mi>g</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo>∘</mo><msub><mn>1</mn> <mi>b</mi></msub><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex"> g' = 1_{a} \circ g'= (g \circ f) \circ g' = g \circ (f \circ g') = g \circ 1_{b}= g</annotation></semantics></math></p> </div> </p> <h2 id="examples">Examples</h2> <ul> <li> <p>A <strong><a class="existingWikiWord" href="/nlab/show/bijection">bijection</a></strong> is an isomorphism in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> </li> <li> <p>A <strong><a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></strong> is an isomorphism in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> </li> <li> <p>A <strong><a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a></strong> is an isomorphism in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>.</p> </li> <li> <p>Every morphism in a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> is an isomorphism. By definition of groupoid.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a></p> </li> <li> <p><strong>isomorphism</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+class">isomorphism class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exceptional+isomorphism">exceptional isomorphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a></p> </li> <li> <p><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/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/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/EI-category">EI-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a></p> </li> </ul> <h2 id="references"> References</h2> <ul> <li id="Poincaré1908"> <p><a class="existingWikiWord" href="/nlab/show/Henri+Poincar%C3%A9">Henri Poincaré</a>, <em>The future of mathematics</em>, Revue generale des Sciences pures et appliquees, Paris, 19th year, No. 23, December, 1908 (<a href="https://sites.math.rutgers.edu/courses/535/535-f08/Poincare.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 1.9 in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em> Vol. 1: <em>Basic Category Theory</em>, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994) [<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 17, 2024 at 13:37:54. 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