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class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' 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class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition_of_hocat'>Definition of Ho(Cat)</a></li><li><a href='#subcategories_of_hocat'>Subcategories of Ho(Cat)</a></li><li><a href='#hocatcategories'>Ho(Cat)-categories</a></li><li><a href='#OtherLimits'>Other limits and colimits</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='definition_of_hocat'>Definition of Ho(Cat)</h2> <p><strong>Ho(Cat)</strong> is a name for the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> of <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>. That is, <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> <ul> <li>whose objects are (<a class='existingWikiWord' href='/nlab/show/diff/small+category'>small</a>) categories, and</li> <li>whose morphisms are <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphism</a> classes of <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a>.</li> </ul> <p>This is an instance of a general construction which, given a <a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a>, or more generally an <a class='existingWikiWord' href='/nlab/show/diff/n-category'>n-category</a>, produces a <a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-category</a> with the same objects and whose morphisms are <a class='existingWikiWord' href='/nlab/show/diff/equivalence'>equivalence</a> classes of 1-morphisms in the original <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category. Sometimes this is called the 1-<a class='existingWikiWord' href='/nlab/show/diff/truncation'>truncation</a> and denoted <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\tau_1</annotation></semantics></math>.</p> <p>It can also be viewed as an instance of the homotopy category of a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> (or more generally a <a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a>). The category <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> as defined above is equivalent to the category obtained from <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> by forcing all <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalences of categories</a> to be isomorphisms (by <a class='existingWikiWord' href='/nlab/show/diff/localization'>localizing</a>). This is for the same reason that the category <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hTop</mi></mrow><annotation encoding='application/x-tex'>hTop</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> and <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a> classes of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous maps</a> is equivalent to the category obtained from <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> by inverting the homotopy equivalences (namely, the existence of <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder objects</a> and/or <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path objects</a>). Indeed, a <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a> for a category <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/product+category'>product category</a> <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>C \times I</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is the category with two objects 0 and 1 and an isomorphism <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>0 \to 1</annotation></semantics></math>. It is not difficult to see that an isomorphism of functors is the same as a <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a> of functors with the respect to the <a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure'>canonical model structure</a> on <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>.</p> <h2 id='subcategories_of_hocat'>Subcategories of Ho(Cat)</h2> <p>Some notable full subcategories of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> include</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Gpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Gpd)</annotation></semantics></math>, the homotopy category of the category <a class='existingWikiWord' href='/nlab/show/diff/Grpd'>Gpd</a> of <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoids</a>. Note that this is equivalent to the homotopy category of (unbased) <a class='existingWikiWord' href='/nlab/show/diff/homotopy+1-type'>homotopy 1-types</a>.</li> <li>The category whose objects are <a class='existingWikiWord' href='/nlab/show/diff/group'>groups</a> and whose morphisms are <a class='existingWikiWord' href='/nlab/show/diff/conjugacy+class'>conjugacy class</a>es of <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>group homomorphism</a>s. This can be identified with the full subcategory of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Gpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Gpd)</annotation></semantics></math> whose objects are the connected groupoids. This category sometimes arises in the study of <a class='existingWikiWord' href='/nlab/show/diff/gerbe'>gerbes</a>.</li> </ul> <h2 id='hocatcategories'>Ho(Cat)-categories</h2> <p>Like the homotopy category of any model category, <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a> and <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproducts</a>, and is in particular a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a>. Therefore, we can talk about categories <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched over</a> <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>. Such a “<math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category” consists of</p> <ul> <li>a collection of objects <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>x,y,z</annotation></semantics></math></li> <li>for each pair of objects, a category <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(x,y)</annotation></semantics></math></li> <li>for each object <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, an objects <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>id</mi> <mi>x</mi></msub><mo>∈</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>id_x\in C(x,x)</annotation></semantics></math></li> <li>for each triple of objects, a functor <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(y,z)\times C(x,y)\to C(x,z)</annotation></semantics></math></li> </ul> <p>such that the usual associativity and unit diagrams for an enriched category commute up to isomorphism. The difference between a <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category and a <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a> is that in a <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category, <em>no coherence axioms</em> are required of the <a class='existingWikiWord' href='/nlab/show/diff/associator'>associator</a> and <a class='existingWikiWord' href='/nlab/show/diff/unitor'>unitor</a> isomorphisms; they are merely required to <em>exist</em>. Thus a <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category can be thought of as an “incoherent bicategory.” In particular, any bicategory has an underlying <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category.</p> <p>Although <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-categories are not very useful, there are some interesting things that can be said about them. For instance:</p> <ul> <li>Any <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category which is equivalent, as a <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category, to a bicategory, is itself in fact a bicategory.</li> <li>Any <a class='existingWikiWord' href='/nlab/show/diff/2-functor'>2-functor</a> between bicategories which induces an equivalence of underlying <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-categories is in fact itself an equivalence of bicategories (or “biequivalence”).</li> </ul> <p>An example of a <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>-category that does not come from any bicategory is sketched in <a href='https://mathoverflow.net/a/346613'>this MathOverflow answer</a>.</p> <h2 id='OtherLimits'>Other limits and colimits</h2> <p>Although <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> has products and coproducts, like most homotopy categories it is not well-endowed with other <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a>. The following concrete example shows that it (and also <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Gpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Gpd)</annotation></semantics></math>) fails to have <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullbacks</a>.</p> <p>Consider the <a class='existingWikiWord' href='/nlab/show/diff/cospan'>cospan</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mi>j</mi></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mtd> <mtd><munder><mo>→</mo><mi>i</mi></munder></mtd> <mtd><msub><mi>S</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{&& \mathbb{Z}/3\\ && \downarrow^j\\ \mathbb{Z}/2 & \underset{i}{\to} & S_3} </annotation></semantics></math></div> <p>where the two arrows are inclusions of subgroups. That is, we choose a 2-cycle and a 3-cycle in <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>S_3</annotation></semantics></math>, say <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>a=(1,2)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>=</mo><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>b=(1,2,3)</annotation></semantics></math>, and identify <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/3</annotation></semantics></math> with the subgroups generated by <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> respectively. Regard these groups as connected groupoids and thus as objects of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>, and suppose that this cospan had a pullback</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mtd></mtr> <mtr><mtd><msup><mo /><mi>g</mi></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mi>j</mi></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mtd> <mtd><munder><mo>→</mo><mi>i</mi></munder></mtd> <mtd><msub><mi>S</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{P & \overset{f}{\to} & \mathbb{Z}/3\\ ^g \downarrow && \downarrow^j\\ \mathbb{Z}/2 & \underset{i}{\to} & S_3} </annotation></semantics></math></div> <p>in <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Gpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Gpd)</annotation></semantics></math>.</p> <p>Note that for any category <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, the set <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)(1,C)</annotation></semantics></math> is the set of isomorphism classes of objects in <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> (where <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a>). Therefore, any pullbacks that exist in <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math> must induce pullbacks of sets of isomorphism classes of objects, and so <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> must also have only one isomorphism class of objects; i.e. it must be a <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a>, regarded as a one-object category. We choose monoid homomorphisms <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>P\to \mathbb{Z}/2</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>P\to \mathbb{Z}/3</annotation></semantics></math> representing <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>, respectively. We also choose a natural isomorphism <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo lspace='verythinmathspace'>:</mo><mi>j</mi><mi>f</mi><mo>≅</mo><mi>i</mi><mi>g</mi></mrow><annotation encoding='application/x-tex'>\sigma\colon j f \cong i g</annotation></semantics></math>, which consists of an element <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>\sigma\in S_3</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>σ</mi><mo>⋅</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>⋅</mo><msup><mi>σ</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>j(f(c)) = \sigma \cdot i(g(c)) \cdot \sigma^{-1}</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding='application/x-tex'>c\in P</annotation></semantics></math>.</p> <p>Now let <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mtd></mtr> <mtr><mtd><msup><mo /><mi>k</mi></msup><mo stretchy='false'>↓</mo></mtd> <mtd><mo>≅</mo></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mi>j</mi></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mtd> <mtd><munder><mo>→</mo><mi>i</mi></munder></mtd> <mtd><msub><mi>S</mi> <mn>3</mn></msub><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{Q & \overset{h}{\to} & \mathbb{Z}/3\\ ^k \downarrow & \cong & \downarrow^j\\ \mathbb{Z}/2 & \underset{i}{\to} & S_3.} </annotation></semantics></math></div> <p>Then the objects of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> are the elements of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>S_3</annotation></semantics></math>, and the morphisms from <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> consist of pairs <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mo>×</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>(u,v)\in \mathbb{Z}/2 \times \mathbb{Z}/3</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo>⋅</mo><mi>j</mi><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i(u)\cdot x = y \cdot j(v)</annotation></semantics></math>. Since the square defining <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> commutes in <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Cat)</annotation></semantics></math>, there must be a functor <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℓ</mi><mo lspace='verythinmathspace'>:</mo><mi>Q</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding='application/x-tex'>\ell\colon Q\to P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mi>ℓ</mi><mo>≅</mo><mi>h</mi></mrow><annotation encoding='application/x-tex'>f\ell\cong h</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mi>ℓ</mi><mo>≅</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>g\ell\cong k</annotation></semantics></math>.</p> <p>Now every element of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/3</annotation></semantics></math> is the image of some morphism of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi></mrow><annotation encoding='application/x-tex'>Q</annotation></semantics></math> under <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi></mrow><annotation encoding='application/x-tex'>h</annotation></semantics></math>, respectively. For instance, <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>a\in \mathbb{Z}/2</annotation></semantics></math> is the image of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mn>1</mn><mo>→</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>(a,1)\colon 1 \to a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>b\in \mathbb{Z}/3</annotation></semantics></math> is the image of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>(1,b)\colon b\to 1</annotation></semantics></math>. Therefore, since <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi></mrow><annotation encoding='application/x-tex'>h</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> factor through <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> up to isomorphism, <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> must be surjective as monoid homomorphisms.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>c_1</annotation></semantics></math> be such that <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f(c_1)=b</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g(c_1)</annotation></semantics></math> is not the identity, let <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>=</mo><msub><mi>c</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>c=c_1</annotation></semantics></math>. Otherwise <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>g(c_1)=1</annotation></semantics></math> and there is some <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>c_2</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>g(c_2)=a</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(c_2)</annotation></semantics></math> is not the identity, then let <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>=</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>c=c_2</annotation></semantics></math>. Otherwise <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>f(c_2)=1</annotation></semantics></math> and let <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>=</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>c = c_1\cdot c_2</annotation></semantics></math>. In either case, neither <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(c)</annotation></semantics></math> nor <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g(c)</annotation></semantics></math> is the identity. Therefore, neither <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>j(f(c))</annotation></semantics></math> nor <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i(g(c))</annotation></semantics></math> is the identity, and moreover <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>j(f(c))</annotation></semantics></math> is a 3-cycle and <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i(g(c))</annotation></semantics></math> is a 2-cycle in <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>S_3</annotation></semantics></math>. But the element <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> conjugates <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i(g(c))</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>j(f(c))</annotation></semantics></math>, a contradiction.</p> <p>Since all the categories involved were groupoids (except possibly <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>), the same argument shows that <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>Gpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(Gpd)</annotation></semantics></math> doesn’t have pullbacks. Moreover, basically the same argument, regarding groupoids as connected 1-types, shows that the homotopy category of topological spaces doesn’t have pullbacks either (in this case the final contradiction is derived from <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>P</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(P)</annotation></semantics></math> instead of <math class='maruku-mathml' display='inline' id='mathml_996fa4a55327a482eeffc2f5d868eb3ab464fdbd_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> itself).</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure'>canonical model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CombModCat'>Ho(CombModCat)</a></p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><p><strong><math class='maruku-mathml' display='inline' id='mathml_3d1567b8fb32288c3e83136857084b9847ce1814_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1,r+1)</annotation></semantics></math>-categories of <a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-categories</a></strong></p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pos'>Pos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Rel'>Rel</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grpd'>Grpd</a>, <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Cat%29'>Ho(Cat)</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/AccCat'>AccCat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PrCat'>PrCat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lex'>LexCat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MonCat'>MonCat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+V-enriched+categories'>VCat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CatAdj'>CatAdj</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Prof'>Prof</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Operad'>Operad</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2Cat'>2Cat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ModCat'>ModCat</a>, <a class='existingWikiWord' href='/nlab/show/diff/CombModCat'>CombModCat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Cat'>(∞,1)Cat</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pr%28infinity%2C1%29Cat'>Pr(∞,1)Cat</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29Operad'>(∞,1)Operad</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29Cat'>(∞,n)Cat</a></p> </li> </ul></ins> <p><div class='property'> category: <a class='category_link' href='/nlab/list/category'>category</a></div></p> </div> <div class="revisedby"> <p> Last revised on September 29, 2023 at 17:00:51. 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