Accidental release source terms - Revision history

en>Pol098 at 21:06, 8 May 2014

2014-05-08T21:06:27Z

← Older revision		Revision as of 23:06, 8 May 2014
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	~~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.~~		Let me initial begin by introducing myself. My name is Boyd Butts although it is not the name on my birth certificate. Managing people is what I do and the salary has been really satisfying. For a whilst I've been in South Dakota and my parents live close by. Doing ceramics is what love performing.<br><br>My blog [http://apcbook.com.ng/index.php?do=/profile-446/info/ std home test]

	~~==Definition==~~

	~~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~~
	~~:<math>f_1,f_2 : X \to Y\,</math>~~
	~~are related in Hom(''X'', ''Y'')~~ and
	~~:<math>g_1,g_2 : Y \to Z\,</math>~~
	are related in Hom(''Y'', ''Z'') then ''g''<sub>1</sub>''f''<sub>1</sub>, ''g''<sub>1</sub>''f''<sub>2</sub>, ''g''<sub>2</sub>''f''<sub>1</sub> and ''g''<sub>2</sub>''f''<sub>2</sub> are related in Hom(''X'', ''Z'').

	~~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,~~
	~~:<math>\mathrm{Hom}_{\mathcal C/\mathcal R}(X,Y) = \mathrm{Hom}_{\mathcal C}(X,Y)/R_{X,Y}.</math>~~

	~~Composition of morphisms in ''C''/''R'' is [[well-defined]] since ''R'' is a congruence relation.~~

	~~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]~~.)

	~~==Properties==~~

	~~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]]).~~

	=~~=Examples==~~

	* [[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]].
	* 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.

	~~==See also==~~

	*[[Subobject]]

	~~==References==~~
	* Gabriel, Pierre, ''Des categories abeliennes'', Bull. Soc. Math. France '''90''' (1962), 323-~~448.~~
	* [[Saunders Mac Lane\|Mac Lane]], Saunders (1998) ''[[Categories for the Working Mathematician]]''. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.

	~~[[Category:Category theory]~~]

en>Mbeychok: Don't know how or why two of the sub-section headers were italicized. Undid those italics.

2013-02-11T23:28:06Z

Don't know how or why two of the sub-section headers were italicized. Undid those italics.

← Older revision		Revision as of 01:28, 12 February 2013
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	The ~~writer~~ is ~~known as Irwin Wunder~~ but it'~~s not~~ the ~~most masucline title out there~~. ~~North Dakota~~ is ~~where me~~ and ~~my spouse reside~~. ~~I am~~ a ~~meter reader but I plan~~ on ~~altering it~~. ~~Playing baseball~~ is the ~~pastime he will~~ by ~~no means quit doing~~.<br><br>~~Here~~ is ~~my site; home std test kit~~ (~~[http:~~//~~blogzaa.com~~/~~blogs~~/~~post~~/~~9199 my latest blog post~~])		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.

			==Definition==

			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
			:<math>f_1,f_2 : X \to Y\,</math>
			are related in Hom(''X'', ''Y'') and
			:<math>g_1,g_2 : Y \to Z\,</math>
			are related in Hom(''Y'', ''Z'') then ''g''<sub>1</sub>''f''<sub>1</sub>, ''g''<sub>1</sub>''f''<sub>2</sub>, ''g''<sub>2</sub>''f''<sub>1</sub> and ''g''<sub>2</sub>''f''<sub>2</sub> are related in Hom(''X'', ''Z'').

			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,
			:<math>\mathrm{Hom}_{\mathcal C/\mathcal R}(X,Y) = \mathrm{Hom}_{\mathcal C}(X,Y)/R_{X,Y}.</math>

			Composition of morphisms in ''C''/''R'' is [[well-defined]] since ''R'' is a congruence relation.

			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].)

			==Properties==

			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]]).

			==Examples==

			* [[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]].
			* 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.

			==See also==

			*[[Subobject]]

			==References==
			* Gabriel, Pierre, ''Des categories abeliennes'', Bull. Soc. Math. France '''90''' (1962), 323-448.
			* [[Saunders Mac Lane\|Mac Lane]], Saunders (1998) ''[[Categories for the Working Mathematician]]''. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.

			[[Category:Category theory]]

en>CrimsonZ: I don't think the phrase "a good many" is proper for a encyclopedia

2012-08-13T16:55:02Z

I don't think the phrase "a good many" is proper for a encyclopedia

New page

The writer is known as Irwin Wunder but it's not the most masucline title out there. North Dakota is where me and my spouse reside. I am a meter reader but I plan on altering it. Playing baseball is the pastime he will by no means quit doing.<br><br>Here is my site; home std test kit ([http://blogzaa.com/blogs/post/9199 my latest blog post])