Glossary of category theory

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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.
• preadditive if it is enriched over the monoidal category of abelian groups. More generally, it is R-linear if it is enriched over the monoidal category of R-modules, for R a commutative ring.
• skeletal if isomorphic objects are necessarily identical.

Morphisms

A morphism f in a category is called:

• an epimorphism if $g=h$ whenever $g\circ f=h\circ f$. 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, $g\circ f=g$ and $f\circ h=h$.
• an inverse to a morphism g if $g\circ f$ is defined and is equal to the identity morphism on the codomain of g, and $f\circ g$ 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 $f\circ g$ 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 $f\circ g=f\circ h$; e.g., an injection in Set. In other words, f is the dual of an epimorphism.
• a bimorphism is a morphism that is both an epimorphism and a monomorphism.
• 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:

• isomorphic to an object B if there is an isomorphism between A and B.
• initial if there is exactly one morphism from A to each object B; e.g., empty set in Set.
• terminal if there is exactly one morphism from each object B to A; e.g., singletons in Set.
• a zero object if it is both initial and terminal, such as a trivial group in Grp.

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

1. ^ Adámek, Jiří; Herrlich, Horst; Strecker, George E (2004) [1990]. Abstract and Concrete Categories (The Joy of Cats) (PDF). New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6.
2. ^ Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra 175 (1-3): 207–222. doi:10.1016/S0022-4049(02)00135-4.
3. ^ http://planetmath.org/encyclopedia/NormalCategory.html
4. ^ Kashiwara & Schapira 2006, exercise 8.20