Glossary of category theory

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

Contents 1 Categories 2 Morphisms 3 Functors 4 Objects 5 References

[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.
• 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 provided that there is an isomorphism between them.
• equivalent to a category B provided that there is an equivalence between them.
• concrete provided that there is a faithful functor from A to Set; e.g., Vec, Grp and Top.
• discrete provided that each morphism is an identity morphism (of some object).
• thin category provided that there is at most one morphism between any pair of objects.
• a subcategory of a category B provided that there is 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 object A there is only a set of pairwise non-isomorphic subobjects.
• complete provided that all small limits exist.
• cartesian closed provided that it has a terminal object and that any two objects have a product and exponential.
• abelian provided that it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
• normal provided that every monic is normal. ^[3]

[edit] Morphisms

A morphism f in a category is called:

• an epimorphism provided that g = h whenever . 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, and .
• an inverse to a morphism g if is defined and is equal to the identity morphism on the domain of f, and 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 (also called monic) provided that g = h whenever ; e.g., an injection in Set. 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 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.

[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, such as a trivial group in Grp.

[edit] References