Lie groupoid - Revision history

en>TakuyaMurata: /* Morita Morphisms and Smooth Stacks */

2014-04-21T21:10:24Z

Morita Morphisms and Smooth Stacks

← Older revision		Revision as of 23:10, 21 April 2014
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	~~{{For\|subcategories in wikipedia\|WP:Subcategories}}~~		Games can give you a universe of experience, relax and exhilaration. It's possible to learn, get a common-sense of success or usually enjoy beating down the bad-guy. No challenege show up form of video egaming you are into, are definitely the helpful tips in this particular post to give one self more fun whenever your site play your next sport Website.<br><br>In the instance as a parent you're concerned with movie online game content, control what downloadable mods are put previously sport. These down-loadable mods are usually developed by players, perhaps not often the gaming businesses, therefore there is no ranking system. What you thought was a reasonably un-risky game can move a lot worse using any of these mods.<br><br>Final results displayed in the chart are too apparent for you to become ignored. Even a child could work out how the nationwide debt has persistently relied upon clash of clans identify tool no survey to a certain extent, but now more that ever. A large number of analysts fear a subsequent depression.<br><br>There are no aftermaths in the least to assist you attacking other players or losing, so just attack and savor it. Win or lose, anybody may lose the many troops you have in only the attack since they are only beneficial on one mission, nevertheless, you can steal more guides with the enemy hamlet than it cost to make the troops. And you just formulate more troops within your barracks. If you loved this post and you would certainly like to receive more [http://Details.com details] concerning [http://prometeu.net clash of clans cheats android] kindly browse through the page. It''s per good idea to invest in them queued up previous to you decide to treat and that means the customer are rebuilding your soldiers through the battle.<br><br>View your child's xbox game enjoying. Video and also are now rated primarily like films and the can help. This enables you to maintain your an eye on unquestionably the information your kids often is exposed to. Conditional upon your child's age, continue to keep the boy clear of video video game that happen to be very meant for people individuals who are more fully learned than him.<br><br>You will notice for yourself that all of our Money Compromise of Clans i fairly effective, fully invisible by the team boss of the game, remarkably absolutely no price!<br><br>And all sorts of our options are considered and approved from the most effective virus recognition software and as a result anti-virus in the sell to ensure a security-level as large as you can, in argument you fear for protection of your computer perhaps cellular device, no concerns. In case you nevertheless have some sort of doubts, take a brows through the movie and you'll notice it operates and everyone 100% secure! It will only take a few moments of your time!

	~~In [[mathematics]],~~ a ~~'''subcategory'''~~ of ~~a [[category (mathematics)\|category]]~~ '~~'C'' is~~ a ~~category ''S'' whose objects~~ are ~~objects in ''C'' and whose [[morphism]]s~~ are ~~morphisms~~ in '~~'C''~~ with the ~~same identities and composition of morphisms~~. ~~Intuitively,~~ a ~~subcategory~~ of ~~''C'' is~~ a ~~category obtained from ''C'' by "removing" some~~ of ~~its objects and arrows.~~

	~~==Formal definition==~~
	~~Let ''C'' be~~ a ~~category~~. A ~~'''subcategory''' ''S''~~ of ~~''C'' is given by~~
	*a ~~subcollection of objects of ''C'', denoted ob(''S''),~~
	*a subcollection of morphisms of ''C'', denoted hom(''S'').
	~~such that~~
	*for every ''X'' in ob(''S''), the identity morphism id<~~sub~~>~~''X''~~<~~/sub~~> ~~is in hom(''S''),~~
	*for every morphism ''f'' : ''X'' → ''Y'' in ~~hom(''S''), both~~ the ~~source ''X'' and the target ''Y'' are in ob(''S'')~~,
	*for every pair of morphisms ''f'' and ~~''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever~~ it ~~is defined~~.

	~~These conditions ensure that ''S'' is a category in its own right: the collection of objects is ob(''S'')~~, the ~~collection of morphisms is hom(''S''), and~~ the ~~identities and composition~~ are ~~as in ''C''. There is an obvious [[Full and faithful functors\|faithful functor]] ''I'' : ''S'' → ''C''~~, ~~called~~ the ~~'''inclusion functor''' which takes objects and morphisms~~ to ~~themselves~~.

	~~Let ''S'' be a subcategory of a category ''C''~~. ~~We say that ''S'' is a '''full subcategory of''' ''C'' if for each pair of objects ''X''~~ and ~~''Y'' of ''S''~~
	:~~<math>\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y)~~.</~~math>~~
	~~A full subcategory is one that includes ''all'' morphisms between objects of ''S''~~. ~~For any collection~~ of ~~objects ''A'' in ''C'', there is a unique full subcategory of ''C'' whose objects are those in ''A''.~~

	~~==Embeddings==~~
	~~Given a subcategory ''S'' of ''C''~~ the ~~inclusion [[functor]] ''I'' : ''S'' → ''C'' is both [[faithful functor\|faithful]] and [[Injective function\|injective]] on objects~~. It ~~is [[full functor\|full]] if and only if ''S'' is a full subcategory.~~

	~~Some authors define an '''embedding'~~'' to ~~be a [[full and faithful functor]]. Such a functor is necessarily injective on objects~~ up-to~~-isomorphism. For instance, the [[Yoneda embedding]] is an embedding in this sense.~~

	~~Some authors define an '''embedding'''~~ to ~~be a full~~ and ~~faithful functor~~ that ~~is injective on objects (strictly)~~.<~~ref~~>~~{{cite web\|author=van Oosten\|title=Basic category theory\|url=http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf}}~~<~~/ref~~>

	~~Other authors define a functor to be an~~ '~~''embedding''' if it is~~
	~~faithful~~ and
	~~injective on objects~~.
	~~Equivalently, ''F'' is~~ an ~~embedding if it is injective~~ on ~~morphisms. A functor ''F'' is then called a '''full embedding''' if it is a full functor and an embedding.~~

	~~For any (full) embedding ''F'' : ''B'' → ''C''~~ the ~~image of ''F''~~ is ~~a (full) subcategory ''S'' of ''C'' and ''F'' induces an [[isomorphism of categories]] between ''B'' and ''S''~~. If '~~'F'' is not strictly injective on objects~~, ~~the image of ''F'' is [[equivalence of categories\|equivalent]]~~ to ~~''B''.~~

	~~In some categories, one can also speak of morphisms of~~ the ~~category being [[embedding#Category theory\|embedding]]s.~~

	~~==Types of subcategories==~~
	~~A subcategory ''S''~~ of ~~''C'' is said to be '''[[isomorphism-closed]]''' or '''replete''' if every [[isomorphism]] ''k'' : ''X'' → ''Y'' in ''C'' such~~ that ~~''Y'' is in ''S'' also belongs to ''S''. A isomorphism-closed full subcategory is said~~ to be ~~'''strictly full'''~~.

	~~A subcategory of ''C'' is '''wide''' or '''lluf''' (a term first posed by P. Freyd~~<~~ref~~>{{cite book \|last= Freyd\|first= Peter\|authorlink=Peter J. Freyd \|year= 1991\|month= \|pages=95–104 \|chapter= Algebraically complete categories\|series=Lecture Notes in Mathematics \|volume= 1488\|publisher=Springer\|title=Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990)\|doi=10.1007/BFb0084215}}<~~/ref~~>~~) if it contains~~ all the ~~objects~~ of ~~''C''. A lluf subcategory is typically not full:~~ the ~~only full lluf subcategory of a category is that category itself.~~

	~~A '''Serre subcategory''' is a non-empty full subcategory ''S'' of an [[abelian category]] ''C'' such that for all short [[exact sequence]]s~~

	~~:<math>0\to M'\to M\to M''\to 0</math>~~

	~~in ''C''~~, ~~''M'' belongs to ''S'' if and only if both~~ <~~math~~>M'<~~/math~~> and ~~<math>M''</math> do. This notion arises~~ from ~~[[Localization of~~ a ~~category#Serre's C~~-~~theory\|Serre's C~~-~~theory]]~~.

	~~== See also ==~~
	*[[Reflective subcategory]]
	*[[Exact category]], a ~~full subcategory closed under extensions.~~

	~~==References==~~
	~~<references />~~

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

en>Mark viking: Added wl

2014-01-22T20:54:26Z

Added wl

← Older revision		Revision as of 22:54, 22 January 2014
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	~~Kelvin Wilkerson~~ is ~~how he~~'s ~~called but it isn~~'t the ~~most masucline name out there~~. ~~Years ago she moved to Kansas~~. As a ~~girl what I prefer~~ is ~~playing dominoes having said~~ that ~~i don~~'~~t retain~~ the ~~time not too long ago~~. ~~The job~~ I'~~ve been occupying~~ a ~~long time~~ is ~~an auditing officer~~ and I'~~m doing very good financially~~. Go to ~~my website~~ to ~~find out more:~~ http://~~rigobertos1~~.~~bravejournal~~.~~com~~/~~entry~~/~~139899~~<br><br>~~Also visit my page~~ [~~http~~://~~rigobertos1~~.~~bravejournal~~.~~com~~/~~entry/139899 resinas~~]		{{For\|subcategories in wikipedia\|WP:Subcategories}}

			In [[mathematics]], a '''subcategory''' of a [[category (mathematics)\|category]] ''C'' is a category ''S'' whose objects are objects in ''C'' and whose [[morphism]]s are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows.

			==Formal definition==
			Let ''C'' be a category. A '''subcategory''' ''S'' of ''C'' is given by
			*a subcollection of objects of ''C'', denoted ob(''S''),
			*a subcollection of morphisms of ''C'', denoted hom(''S'').
			such that
			*for every ''X'' in ob(''S''), the identity morphism id<sub>''X''</sub> is in hom(''S''),
			*for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''),
			*for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined.

			These conditions ensure that ''S'' is a category in its own right: the collection of objects is ob(''S''), the collection of morphisms is hom(''S''), and the identities and composition are as in ''C''. There is an obvious [[Full and faithful functors\|faithful functor]] ''I'' : ''S'' → ''C'', called the '''inclusion functor''' which takes objects and morphisms to themselves.

			Let ''S'' be a subcategory of a category ''C''. We say that ''S'' is a '''full subcategory of''' ''C'' if for each pair of objects ''X'' and ''Y'' of ''S''
			:<math>\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y).</math>
			A full subcategory is one that includes ''all'' morphisms between objects of ''S''. For any collection of objects ''A'' in ''C'', there is a unique full subcategory of ''C'' whose objects are those in ''A''.

			==Embeddings==
			Given a subcategory ''S'' of ''C'' the inclusion [[functor]] ''I'' : ''S'' → ''C'' is both [[faithful functor\|faithful]] and [[Injective function\|injective]] on objects. It is [[full functor\|full]] if and only if ''S'' is a full subcategory.

			Some authors define an '''embedding''' to be a [[full and faithful functor]]. Such a functor is necessarily injective on objects up-to-isomorphism. For instance, the [[Yoneda embedding]] is an embedding in this sense.

			Some authors define an '''embedding''' to be a full and faithful functor that is injective on objects (strictly).<ref>{{cite web\|author=van Oosten\|title=Basic category theory\|url=http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf}}</ref>

			Other authors define a functor to be an '''embedding''' if it is
			faithful and
			injective on objects.
			Equivalently, ''F'' is an embedding if it is injective on morphisms. A functor ''F'' is then called a '''full embedding''' if it is a full functor and an embedding.

			For any (full) embedding ''F'' : ''B'' → ''C'' the image of ''F'' is a (full) subcategory ''S'' of ''C'' and ''F'' induces an [[isomorphism of categories]] between ''B'' and ''S''. If ''F'' is not strictly injective on objects, the image of ''F'' is [[equivalence of categories\|equivalent]] to ''B''.

			In some categories, one can also speak of morphisms of the category being [[embedding#Category theory\|embedding]]s.

			==Types of subcategories==
			A subcategory ''S'' of ''C'' is said to be '''[[isomorphism-closed]]''' or '''replete''' if every [[isomorphism]] ''k'' : ''X'' → ''Y'' in ''C'' such that ''Y'' is in ''S'' also belongs to ''S''. A isomorphism-closed full subcategory is said to be '''strictly full'''.

			A subcategory of ''C'' is '''wide''' or '''lluf''' (a term first posed by P. Freyd<ref>{{cite book \|last= Freyd\|first= Peter\|authorlink=Peter J. Freyd \|year= 1991\|month= \|pages=95–104 \|chapter= Algebraically complete categories\|series=Lecture Notes in Mathematics \|volume= 1488\|publisher=Springer\|title=Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990)\|doi=10.1007/BFb0084215}}</ref>) if it contains all the objects of ''C''. A lluf subcategory is typically not full: the only full lluf subcategory of a category is that category itself.

			A '''Serre subcategory''' is a non-empty full subcategory ''S'' of an [[abelian category]] ''C'' such that for all short [[exact sequence]]s

			:<math>0\to M'\to M\to M''\to 0</math>

			in ''C'', ''M'' belongs to ''S'' if and only if both <math>M'</math> and <math>M''</math> do. This notion arises from [[Localization of a category#Serre's C-theory\|Serre's C-theory]].

			== See also ==
			*[[Reflective subcategory]]
			*[[Exact category]], a full subcategory closed under extensions.

			==References==
			<references />

			[[Category:Category theory]]
			[[Category:Hierarchy]]

128.176.180.12: /* External links */

2011-09-14T12:10:45Z

External links

New page

Kelvin Wilkerson is how he's called but it isn't the most masucline name out there. Years ago she moved to Kansas. As a girl what I prefer is playing dominoes having said that i don't retain the time not too long ago. The job I've been occupying a long time is an auditing officer and I'm doing very good financially. Go to my website to find out more: http://rigobertos1.bravejournal.com/entry/139899<br><br>Also visit my page [http://rigobertos1.bravejournal.com/entry/139899 resinas]