https://en.formulasearchengine.com/api.php?action=feedcontributions&user=72.229.39.72&feedformat=atom formulasearchengine - User contributions [en] 2021-12-03T03:24:00Z User contributions MediaWiki 1.37.0-alpha https://en.formulasearchengine.com/index.php?title=Quotient_category&diff=12794 Quotient category 2013-03-12T09:13:45Z <p>72.229.39.72: </p> <hr /> <div>In [[mathematics]], a '''quotient category''' is a [[category (mathematics)|category]] obtained from another one by identifying sets of [[morphism]]s. The notion is similar to that of a [[quotient group]] or [[quotient space]], but in the categorical setting.<br /> <br /> ==Definition==<br /> <br /> Let ''C'' be a category. A ''[[congruence relation]]'' ''R'' on ''C'' is given by: for each pair of objects ''X'', ''Y'' in ''C'', an [[equivalence relation]] ''R''&lt;sub&gt;''X'',''Y''&lt;/sub&gt; on Hom(''X'',''Y''), such that the equivalence relations respect composition of morphisms. That is, if<br /> :&lt;math&gt;f_1,f_2 : X \to Y\,&lt;/math&gt;<br /> are related in Hom(''X'', ''Y'') and<br /> :&lt;math&gt;g_1,g_2 : Y \to Z\,&lt;/math&gt;<br /> are related in Hom(''Y'', ''Z'') then ''g''&lt;sub&gt;1&lt;/sub&gt;''f''&lt;sub&gt;1&lt;/sub&gt;, ''g''&lt;sub&gt;1&lt;/sub&gt;''f''&lt;sub&gt;2&lt;/sub&gt;, ''g''&lt;sub&gt;2&lt;/sub&gt;''f''&lt;sub&gt;1&lt;/sub&gt; and ''g''&lt;sub&gt;2&lt;/sub&gt;''f''&lt;sub&gt;2&lt;/sub&gt; are related in Hom(''X'', ''Z'').<br /> <br /> Given a congruence relation ''R'' on ''C'' we can define the '''quotient category''' ''C''/''R'' as the category whose objects are those of ''C'' and whose morphisms are [[equivalence class]]es of morphisms in ''C''. That is,<br /> :&lt;math&gt;\mathrm{Hom}_{\mathcal C/\mathcal R}(X,Y) = \mathrm{Hom}_{\mathcal C}(X,Y)/R_{X,Y}.&lt;/math&gt;<br /> <br /> Composition of morphisms in ''C''/''R'' is [[well-defined]] since ''R'' is a congruence relation.<br /> <br /> There is also a notion of taking the quotient of an [[Abelian category]] ''A'' by a [[Serre subcategory]] ''B''. This is done as follows. The objects of ''A/B'' are the objects of ''A''. Given two objects ''X'' and ''Y'' of ''A'', we define the set of morphisms from ''X'' to ''Y'' in ''A/B'' to be &lt;math&gt;\varinjlim \mathrm{Hom}_A(X', Y/Y')&lt;/math&gt; where the limit is over subobjects &lt;math&gt;X' \subseteq X&lt;/math&gt; and &lt;math&gt;Y' \subseteq Y&lt;/math&gt; such that &lt;math&gt;X/X', Y' \in B&lt;/math&gt;. Then ''A/B'' is an Abelian category, and there is a canonical functor &lt;math&gt;Q \colon A \to A/B&lt;/math&gt;. This Abelian quotient satisfies the universal property that if ''C'' is any other Abelian category, and &lt;math&gt;F \colon A \to C&lt;/math&gt; is an [[exact functor]] such that ''F(b)'' is a zero object of ''C'' for each &lt;math&gt;b \in B&lt;/math&gt;, then there is a unique exact functor &lt;math&gt;\overline{F} \colon A/B \to C&lt;/math&gt; such that &lt;math&gt;F = \overline{F} \circ Q&lt;/math&gt;. (See [Gabriel].)<br /> <br /> ==Properties==<br /> <br /> There is a natural quotient [[functor]] from ''C'' to ''C''/''R'' which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a [[full functor]]).<br /> <br /> ==Examples==<br /> <br /> * [[Monoid]]s and [[group (mathematics)|group]] may be regarded as categories with one object. In this case the quotient category coincides with the notion of a [[quotient monoid]] or a [[quotient group]].<br /> * The [[homotopy category of topological spaces]] '''hTop''' is a quotient category of '''Top''', the [[category of topological spaces]]. The equivalence classes of morphisms are [[homotopy class]]es of continuous maps.<br /> <br /> ==See also==<br /> <br /> *[[Subobject]]<br /> <br /> ==References==<br /> * Gabriel, Pierre, ''Des categories abeliennes'', Bull. Soc. Math. France '''90''' (1962), 323-448.<br /> * [[Saunders Mac Lane|Mac Lane]], Saunders (1998) ''[[Categories for the Working Mathematician]]''. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.<br /> <br /> [[Category:Category theory]]</div> 72.229.39.72