Glossary of category theory

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

[edit] Categories

A category A is said to be:

small provided that the class of all morphisms is a set (i.e., not a proper class); otherwise large.
locally small provided that the morphisms between every pair of objects A and B form a set.
quasicategory provided that objects in A may not form a class and morphisms between objects A and B may not form a set.
isomorphic to a category B provided that there exists an isomorphism between them.
equivalent to a category B provided that there exists an equivalence between them.
concrete provided that there exists a faithful functor from A to Set; e.g., Vec, Grp and Top.
discrete provided that each morphism is the identity morphism.
thin category provided that there is at most one morphism between objects A and B.
a subcategory of a category B provided that there exists an inclusion functor from A to B.
a full subcategory of a category B provided that the inclusion functor is full.
wellpowered provided for each A-object A there is only a set of pairwise nonisomorphic subobjects.

A morphism f in a category is said to be:

an epimorphism provided that $g = h$ whenever $g\circ f=h\circ f$ . In other words, f is the dual of a monomorphism.
an identity provided that 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 domain of f, and $f\circ g$ is defined and equal to the identity morphism on the codomain of g. The inverse of g is unique and is denoted by g ^-1
an isomorphism provided that there exists an inverse of f.
a monomorphism provided that $g = h$ whenever $f\circ g=f\circ h$ . In other words, f is the dual of an epimorphism.

A functor F is said to be:

a constant provided that F maps every object in a category to the same object A and every morphism to the identity on A.
faithful provided that F is injective when restricted to each hom-set.
full provided that F is surjective when restricted to each hom-set.
isomorphism-dense (sometimes called essentially surjective) provided that for every B there exists A such that F(A) is isomorphic to B.
an equivalence provided that 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.

An object A in a category is said to be:

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