Glossary of category theory

From Wikipedia, the free encyclopedia

You have new messages (last change).

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

1 Categories
2 Morphisms
3 Functors
4 Objects

[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.

[edit] Morphisms

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 f ^-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.

[edit] Functors

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 an A such that F(A) is isomorphic to B.
an equivalence provided that F is faithful, full and isomorphism-dense.
reflect identities provided that if F(k) is an identity then k is an identity as well.

[edit] Objects

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.

This page has been transwikied to Wiktionary.

Because this article has content useful to Wikipedia's sister project Wiktionary, it has been copied to there, and its dictionary counterpart can be found at either Wiktionary:Transwiki:Glossary of category theory or Wiktionary:Glossary of category theory. It should no longer appear in Category:Copy to Wiktionary and should not be re-added there.
Wikipedia is not a dictionary, and if this article cannot be expanded beyond a dictionary definition, it should be tagged for deletion. If it can be expanded into an article, please do so and remove this template.
Note that {{vocab-stub}} is deprecated. If {{vocab-stub}} was removed when this article was transwikied, and the article is deemed encyclopedic, there should be a more suitable category for it.