Inseparable differential equation - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-28T18:12:47Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

← Older revision		Revision as of 20:12, 28 March 2014
Line 1:		Line 1:
	~~This is a glossary of properties and concepts in [[category theory]] in [[mathematics]].~~		Greetings! I am Myrtle Shroyer. For a while I've been in South Dakota and my mothers and fathers reside close by. To do aerobics is a factor that I'm completely addicted to. Bookkeeping is my profession.<br><br>Feel free to visit my web blog ... [http://www.lankaclipstv.com/blog/165665 http://www.lankaclipstv.com/]

	~~==Categories==~~
	~~A [[category (mathematics)\|category]] '''A''' is said to be:~~
	* '''small''' if the class of all morphisms is a [[Set (mathematics)\|set]] (i.e.~~, not~~ a ~~[[proper class]]); otherwise~~ '~~''large'''.~~
	* '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set.
	* Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in ~~which case a '''quasicategory''' is a category whose objects~~ and ~~morphisms merely form a conglomerate.<ref>{{cite book \|last=Adámek \|first=Jiří \|coauthors=Herrlich, Horst,~~ and ~~Strecker, George E \|title=Abstract and Concrete Categories (The Joy of Cats) \|origyear=1990 \|url=http://katmat.math.uni-bremen~~.de/acc/ \|format=PDF \|year=2004 \|publisher= Wiley & Sons \|location=New York \|isbn=0-471-60922-6 \|page=40}}</ref> (NB other authors use the term "quasicategory" with a different meaning.<ref>{{cite journal\|doi=10.1016/S0022-4049(02)00135-4\|last=Joyal\|first=A.\|title=Quasi-categories and Kan complexes\|journal=Journal of Pure and Applied Algebra\|volume=175\|year=2002\|issue=1-3\|pages=207–222\|ref=harv}}</ref>)
	* '''isomorphic''' to a category '''B''' if there is ~~an isomorphism between them.~~
	* '''equivalent''' to a ~~category '''B''' if there is an [[equivalence of categories\|equivalence]] between them.~~
	* '''[[concrete category\|concrete]]''' if there is a faithful functor from '''A''' to ~~'''[[Category of sets\|Set]]'''; e.g., '''[[category of vector spaces\|Vec]]''', '''[[category of groups\|Grp]]''' and '''[[category of topological spaces\|Top]]'''.~~
	* '''[[discrete category\|discrete]]''' if each morphism is an identity morphism (of some object).
	* '''[[thin category\|thin]]''' category if there is at most one morphism between any pair of objects.
	* a '''subcategory''' of a category '''B''' if there is an inclusion functor given from '''A''' to '''B'''.
	* a '''full subcategory''' of a category '''B''' if the inclusion functor is full.
	* '''wellpowered''' if for each object ''A'' there is only a set of pairwise non-isomorphic [[subobject]]s.
	* '''[[complete category\|complete]]''' if all small limits exist.
	* '''[[Cartesian closed category\|cartesian closed]]''' if it has a terminal object and that any two objects have a product and exponential.
	* '''[[Abelian category\|abelian]]''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
	* '''[[Normal category\|normal]]''' if every monic is ~~normal~~.<~~ref~~>~~http://planetmath.org/encyclopedia/NormalCategory.html~~<~~/ref~~>
	* '''balanced''' if every bimorphism is an isomorphism.
	* <span id="linear category"></span>'''''R''-linear''' (''R'' is a [[commutative ring]]) if '''A''' is locally small, each hom set is an ''R''-module, and composition of morphisms is ''R''-bilinear. ~~The category '''A''' is also said to be '''over ''R'''''~~.
	* '''[~~[preadditive category\|preadditive]]''' if it is [[enriched category\|enriched]] over the [[monoidal category]] of [[abelian group]]s.~~

	~~==Morphisms==~~
	~~A [[morphism]] ''f'' in a category is called~~:
	* an '''[[epimorphism]]''' if <math>g=h</~~math> whenever <math>g\circ f=h\circ f<~~/~~math>~~. ~~In other words, ''f'' is the dual of a monomorphism~~.
	* an '''[[identity (mathematics)\|identity]]''' if ''f'' maps an object ''A'' to ''A'' and for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', <math>g\circ f=g</~~math> and <math>f\circ h=h<~~/~~math>.~~
	* an '''[[Inverse (mathematics)\|inverse]]''' to a morphism ''g'' if <math>g\circ f</math> is defined and is equal to the identity morphism on the codomain of ''g'', and <math>f\circ g</math> is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a '''left inverse''' to ''g'' if <math>f\circ g</math> is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.
	* an '''[[isomorphism]]''' if there exists an ''inverse'' of ''f''.
	* a '''[[monomorphism]]''' (also called '''monic''') if <math>g=h</math> whenever <math>f\circ g=f\circ h</math>; e.g., an [[Injective function\|injection]] in '''[[Category of sets\|Set]]'''. In other words, ''f'' is the dual of an epimorphism.
	* a '''[[section (category theory)\|retraction]]''' if it has a right inverse.
	* a '''[[section (category theory)\|coretraction]]''' if it has a left inverse.

	~~==Functors==~~
	~~A [[functor]] ''F'' is said to be:~~
	* a '''[[constant functor\|constant]]''' if ''F'' maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''.
	* '''faithful''' if ''F'' is injective when restricted to each [[hom-set]].
	* '''full''' if ''F'' is surjective when restricted to each hom-set.
	* '''isomorphism-dense''' (sometimes called '''essentially surjective''') if for every ''B'' there exists ''A'' such that ''F''(''A'') is isomorphic to ''B''.
	* an '''[[equivalence of categories\|equivalence]]''' if ''F'' is faithful, full and isomorphism-dense.
	* '''amnestic''' provided that if ''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.
	* '''reflect identities''' provided that if ''F''(''k'') is an identity then ''k'' is an identity as well.
	* '''reflect isomorphisms''' provided that if ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.

	~~==Objects==~~
	~~An object ''A'' in a category is said to be:~~
	* '''isomorphic''' to an object B if there is an isomorphism between ''A'' and ''B''.
	* '''[[initial object\|initial]]''' if there is exactly one morphism from ''A'' to each object B; e.g., [[empty set]] in '''[[Category of sets\|Set]]'''.
	* '''[[terminal object\|terminal]]''' if there is exactly one morphism from each object B to ''A''; e.g., [[singleton (mathematics)\|singleton]]s in '''[[Category of sets\|Set]]'''.
	* a '''[[zero object]]''' if it is both initial and terminal, such as a [[trivial group]] in '''[[Category of groups\|Grp]]'''.

	~~An object ''A'' in an [[abelian category]] is~~:
	* <span id="simple object"></~~span>'''simple''' if it is not isomorphic to the zero object and any [[subobject]] of ''A'' is isomorphic to zero or to ''A''.~~
	* <span id="finite length"></~~span>'''finite length''' if it has a [[composition series]]~~. ~~The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.<ref>{{harvnb\|Kashiwara\|Schapira\|2006\|loc=exercise 8~~.~~20}}<~~/~~ref>~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{Cite document
	~~\| last=Kashiwara~~
	~~\| first=Masaki~~
	~~\| last2=Schapira~~
	~~\| first2=Pierre~~
	~~\| title=Categories and sheaves~~
	~~\| year=2006~~
	~~\| ref=harv~~
	~~\| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}~~
	}}
	* {{cite book \| last=Mac Lane \| first=Saunders \| authorlink=Saunders Mac Lane \| title=[[Categories for the Working Mathematician]] \| edition=2nd \| series=[[Graduate Texts in Mathematics]] \| volume=5 \| location=New York, NY \| publisher=[[Springer-Verlag]] \| year=1998 \| isbn=0-387-98403-8 \| zbl=0906.18001 }}
	* {{cite book \| editor1-last=Pedicchio \| editor1-first=Maria Cristina \| editor2-last=Tholen \| editor2-first=Walter \| title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory \| series=Encyclopedia of Mathematics and Its Applications \| volume=97 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2004 \| isbn=0-521-83414-7 \| zbl=1034.18001 }}

	~~{{DEFAULTSORT:Glossary Of Category Theory}}~~
	~~[[Category:Category theory\| ]]~~
	~~[[Category:Glossaries of mathematics\|Category theory]~~]

en>BabbaQ at 16:53, 12 September 2013

2013-09-12T16:53:52Z

← Older revision		Revision as of 18:53, 12 September 2013
Line 1:		Line 1:
	~~Hello~~. ~~Allow me introduce~~ the ~~writer~~. ~~Her name~~ is ~~Refugia Shryock~~. ~~Years~~ in the ~~past we moved~~ to ~~North Dakota~~ and ~~I love every working day residing right here~~. ~~I am~~ a ~~meter reader but I strategy~~ on ~~changing it~~. The ~~preferred hobby for my kids~~ and me is to ~~play baseball but I haven~~'~~t produced~~ a ~~dime with it~~.<br><br>~~my web site~~ :: [~~http~~://~~nemoonehfilm~~.ir/~~index~~.~~php?do~~=~~/profile~~-~~6876/info/ http~~:~~//nemoonehfilm.ir~~]		This is a glossary of properties and concepts in [[category theory]] in [[mathematics]].

			==Categories==
			A [[category (mathematics)\|category]] '''A''' is said to be:
			* '''small''' if the class of all morphisms is a [[Set (mathematics)\|set]] (i.e., not a [[proper class]]); otherwise '''large'''.
			* '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set.
			* Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a conglomerate.<ref>{{cite book \|last=Adámek \|first=Jiří \|coauthors=Herrlich, Horst, and Strecker, George E \|title=Abstract and Concrete Categories (The Joy of Cats) \|origyear=1990 \|url=http://katmat.math.uni-bremen.de/acc/ \|format=PDF \|year=2004 \|publisher= Wiley & Sons \|location=New York \|isbn=0-471-60922-6 \|page=40}}</ref> (NB other authors use the term "quasicategory" with a different meaning.<ref>{{cite journal\|doi=10.1016/S0022-4049(02)00135-4\|last=Joyal\|first=A.\|title=Quasi-categories and Kan complexes\|journal=Journal of Pure and Applied Algebra\|volume=175\|year=2002\|issue=1-3\|pages=207–222\|ref=harv}}</ref>)
			* '''isomorphic''' to a category '''B''' if there is an isomorphism between them.
			* '''equivalent''' to a category '''B''' if there is an [[equivalence of categories\|equivalence]] between them.
			* '''[[concrete category\|concrete]]''' if there is a faithful functor from '''A''' to '''[[Category of sets\|Set]]'''; e.g., '''[[category of vector spaces\|Vec]]''', '''[[category of groups\|Grp]]''' and '''[[category of topological spaces\|Top]]'''.
			* '''[[discrete category\|discrete]]''' if each morphism is an identity morphism (of some object).
			* '''[[thin category\|thin]]''' category if there is at most one morphism between any pair of objects.
			* a '''subcategory''' of a category '''B''' if there is an inclusion functor given from '''A''' to '''B'''.
			* a '''full subcategory''' of a category '''B''' if the inclusion functor is full.
			* '''wellpowered''' if for each object ''A'' there is only a set of pairwise non-isomorphic [[subobject]]s.
			* '''[[complete category\|complete]]''' if all small limits exist.
			* '''[[Cartesian closed category\|cartesian closed]]''' if it has a terminal object and that any two objects have a product and exponential.
			* '''[[Abelian category\|abelian]]''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
			* '''[[Normal category\|normal]]''' if every monic is normal.<ref>http://planetmath.org/encyclopedia/NormalCategory.html</ref>
			* '''balanced''' if every bimorphism is an isomorphism.
			* <span id="linear category"></span>'''''R''-linear''' (''R'' is a [[commutative ring]]) if '''A''' is locally small, each hom set is an ''R''-module, and composition of morphisms is ''R''-bilinear. The category '''A''' is also said to be '''over ''R'''''.
			* '''[[preadditive category\|preadditive]]''' if it is [[enriched category\|enriched]] over the [[monoidal category]] of [[abelian group]]s.

			==Morphisms==
			A [[morphism]] ''f'' in a category is called:
			* an '''[[epimorphism]]''' if <math>g=h</math> whenever <math>g\circ f=h\circ f</math>. In other words, ''f'' is the dual of a monomorphism.
			* an '''[[identity (mathematics)\|identity]]''' if ''f'' maps an object ''A'' to ''A'' and for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', <math>g\circ f=g</math> and <math>f\circ h=h</math>.
			* an '''[[Inverse (mathematics)\|inverse]]''' to a morphism ''g'' if <math>g\circ f</math> is defined and is equal to the identity morphism on the codomain of ''g'', and <math>f\circ g</math> is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a '''left inverse''' to ''g'' if <math>f\circ g</math> is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.
			* an '''[[isomorphism]]''' if there exists an ''inverse'' of ''f''.
			* a '''[[monomorphism]]''' (also called '''monic''') if <math>g=h</math> whenever <math>f\circ g=f\circ h</math>; e.g., an [[Injective function\|injection]] in '''[[Category of sets\|Set]]'''. In other words, ''f'' is the dual of an epimorphism.
			* a '''[[section (category theory)\|retraction]]''' if it has a right inverse.
			* a '''[[section (category theory)\|coretraction]]''' if it has a left inverse.

			==Functors==
			A [[functor]] ''F'' is said to be:
			* a '''[[constant functor\|constant]]''' if ''F'' maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''.
			* '''faithful''' if ''F'' is injective when restricted to each [[hom-set]].
			* '''full''' if ''F'' is surjective when restricted to each hom-set.
			* '''isomorphism-dense''' (sometimes called '''essentially surjective''') if for every ''B'' there exists ''A'' such that ''F''(''A'') is isomorphic to ''B''.
			* an '''[[equivalence of categories\|equivalence]]''' if ''F'' is faithful, full and isomorphism-dense.
			* '''amnestic''' provided that if ''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.
			* '''reflect identities''' provided that if ''F''(''k'') is an identity then ''k'' is an identity as well.
			* '''reflect isomorphisms''' provided that if ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.

			==Objects==
			An object ''A'' in a category is said to be:
			* '''isomorphic''' to an object B if there is an isomorphism between ''A'' and ''B''.
			* '''[[initial object\|initial]]''' if there is exactly one morphism from ''A'' to each object B; e.g., [[empty set]] in '''[[Category of sets\|Set]]'''.
			* '''[[terminal object\|terminal]]''' if there is exactly one morphism from each object B to ''A''; e.g., [[singleton (mathematics)\|singleton]]s in '''[[Category of sets\|Set]]'''.
			* a '''[[zero object]]''' if it is both initial and terminal, such as a [[trivial group]] in '''[[Category of groups\|Grp]]'''.

			An object ''A'' in an [[abelian category]] is:
			* <span id="simple object"></span>'''simple''' if it is not isomorphic to the zero object and any [[subobject]] of ''A'' is isomorphic to zero or to ''A''.
			* <span id="finite length"></span>'''finite length''' if it has a [[composition series]]. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.<ref>{{harvnb\|Kashiwara\|Schapira\|2006\|loc=exercise 8.20}}</ref>

			==Notes==
			{{reflist}}

			==References==
			*{{Cite document
			\| last=Kashiwara
			\| first=Masaki
			\| last2=Schapira
			\| first2=Pierre
			\| title=Categories and sheaves
			\| year=2006
			\| ref=harv
			\| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
			}}
			* {{cite book \| last=Mac Lane \| first=Saunders \| authorlink=Saunders Mac Lane \| title=[[Categories for the Working Mathematician]] \| edition=2nd \| series=[[Graduate Texts in Mathematics]] \| volume=5 \| location=New York, NY \| publisher=[[Springer-Verlag]] \| year=1998 \| isbn=0-387-98403-8 \| zbl=0906.18001 }}
			* {{cite book \| editor1-last=Pedicchio \| editor1-first=Maria Cristina \| editor2-last=Tholen \| editor2-first=Walter \| title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory \| series=Encyclopedia of Mathematics and Its Applications \| volume=97 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2004 \| isbn=0-521-83414-7 \| zbl=1034.18001 }}

			{{DEFAULTSORT:Glossary Of Category Theory}}
			[[Category:Category theory\| ]]
			[[Category:Glossaries of mathematics\|Category theory]]

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes, added orphan tag

2009-12-16T11:14:34Z

remove Erik9bot category,outdated, tag and general fixes, added orphan tag

New page

Hello. Allow me introduce the writer. Her name is Refugia Shryock. Years in the past we moved to North Dakota and I love every working day residing right here. I am a meter reader but I strategy on changing it. The preferred hobby for my kids and me is to play baseball but I haven't produced a dime with it.<br><br>my web site :: [http://nemoonehfilm.ir/index.php?do=/profile-6876/info/ http://nemoonehfilm.ir]