https://en.formulasearchengine.com/api.php?action=feedcontributions&user=75.98.19.140&feedformat=atomformulasearchengine - User contributions [en]2021-12-03T04:44:50ZUser contributionsMediaWiki 1.37.0-alphahttps://en.formulasearchengine.com/index.php?title=Quotient_category&diff=246615Quotient category2011-09-03T00:03:46Z<p>75.98.19.140: Give the correct page name</p>
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<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 />
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==Definition==<br />
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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''<sub>''X'',''Y''</sub> on Hom(''X'',''Y''), such that the equivalence relations respect composition of morphisms. That is, if<br />
:<math>f_1,f_2 : X \to Y\,</math><br />
are related in Hom(''X'', ''Y'') and<br />
:<math>g_1,g_2 : Y \to Z\,</math><br />
are related in Hom(''Y'', ''Z'') then ''g''<sub>1</sub>''f''<sub>1</sub> and ''g''<sub>2</sub>''f''<sub>2</sub> are related in Hom(''X'', ''Z'').<br />
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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 />
:<math>\mathrm{Hom}_{\mathcal C/\mathcal R}(X,Y) = \mathrm{Hom}_{\mathcal C}(X,Y)/R_{X,Y}.</math><br />
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Composition of morphisms in ''C''/''R'' is [[well-defined]] since ''R'' is a congruence relation.<br />
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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 <math>\varinjlim \mathrm{Hom}_A(X', Y/Y')</math> where the limit is over subobjects <math>X' \subseteq X</math> and <math>Y' \subseteq Y</math> such that <math>X/X', Y' \in B</math>. Then ''A/B'' is an Abelian category, and there is a canonical functor <math>Q \colon A \to A/B</math>. This Abelian quotient satisfies the universal property that if ''C'' is any other Abelian category, and <math>F \colon A \to C</math> is an [[exact functor]] such that ''F(b)'' is a zero object of ''C'' for each <math>b \in B</math>, then there is a unique exact functor <math>\overline{F} \colon A/B \to C</math> such that <math>F = \overline{F} \circ Q</math>. (See [Gabriel].)<br />
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==Properties==<br />
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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 />
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==Examples==<br />
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* [[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 />
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==See also==<br />
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*[[Subobject]]<br />
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==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 />
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[[Category:Category theory]]</div>75.98.19.140