GLOSSARY OF CATEGORY THEORY

(Redirected from Locally small category)
This is a glossary of properties and concepts in category theory in mathematics.

Contents

Categories

Morphisms

Functors

Objects

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.

Morphisms

A morphism ''f'' in a category is said to be:

★ an 'epimorphism' provided that $g=h$ whenever $gcirc\; f=hcirc\; f$. 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'', $gcirc\; f=g$ and $fcirc\; h=h$.

★ an 'inverse' to a morphism ''g'' if $gcirc\; f$ is defined and is equal to the identity morphism on the domain of ''f'', and $fcirc\; g$ 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 $fcirc\; g=fcirc\; h$. In other words, ''f'' is the dual of an epimorphism.

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.

★

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'.