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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]]

Revision as of 18:53, 12 September 2013

This is a glossary of properties and concepts in category theory in mathematics.

Categories

A category A is said to be:

  • small if the class of all morphisms is a 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.[1] (NB other authors use the term "quasicategory" with a different meaning.[2])
  • isomorphic to a category B if there is an isomorphism between them.
  • equivalent to a category B if there is an equivalence between them.
  • concrete if there is a faithful functor from A to Set; e.g., Vec, Grp and Top.
  • discrete if each morphism is an identity morphism (of some object).
  • 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 subobjects.
  • complete if all small limits exist.
  • cartesian closed if it has a terminal object and that any two objects have a product and exponential.
  • abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
  • normal if every monic is normal.[3]
  • balanced if every bimorphism is an isomorphism.
  • 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 if it is enriched over the monoidal category of abelian groups.

Morphisms

A morphism f in a category is called:

  • an epimorphism if g=h whenever g∘f=h∘f. In other words, f is the dual of a monomorphism.
  • an identity if f maps an object A to A and for any morphisms g with domain A and h with codomain A, g∘f=g and f∘h=h.
  • an inverse to a morphism g if g∘f is defined and is equal to the identity morphism on the codomain of g, and f∘g is defined and equal to the identity morphism on the domain of g. The inverse of g is unique and is denoted by g−1. f is a left inverse to g if f∘g 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 g=h whenever f∘g=f∘h; e.g., an injection in Set. In other words, f is the dual of an epimorphism.
  • a retraction if it has a right inverse.
  • a coretraction if it has a left inverse.

Functors

A functor F is said to be:

  • a 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 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:

An object A in an abelian category is:

  • simple if it is not isomorphic to the zero object and any subobject of A is isomorphic to zero or to A.
  • 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.[4]

Notes

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  3. ↑ http://planetmath.org/encyclopedia/NormalCategory.html
  4. ↑ Template:Harvnb
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