Groups and Categories

Groups in a Category

Group Object

Let be a category with finite products. A group object in consists of objects and arrows as: satisfying the following conditions:

• is associative, that is the following commutes: where is the canonical associativity isomorphism for products.
• is a unit for , that is both triangles in the following commute:
• is an inverse with respect to , that is both sides of this commute:

Homomorphism

A morphism of groups in is called homomorphism if

• preserves
• preserves
• preserves

e.g. The idea of a group in a category captures the familiar notion of a group with additional structure:

• A group in the usual sense is a group in the category .
• A topological group is a group in , the category of topological spaces.

Category of Groups

Prop If is a normal subgroup of , then the following forms a coequalizer: Proof

Groups as Categories

Prop A group is a category with one object, in which every arrow is an isomorphism.

Def Congruence A congruence on a category is an equivalence relation on morphisms such that:

• implies and .
• implies for all morphisms and , where and .

Prop The intersection of a family of congruences is a congruence.

Def Congruence Category Let be a congruence on the category , and define the congruence category by

Def Quotient Category We define the quotient category as follows:

• .
• .
• The morphisms have the form where .
• , and .

Prop There is an evident quotient functor . It then makes the following a coequalizer of categories: Proof

Def Kernel of Functor Any functor determines a congruence on by settingAnd we define as the kernel of .

Thrm For all functor , given any congruence on , one has: if and only if there is a factorization such that the following commutes: Proof

Corollary Every functor factors as : where is bijective on objects and surjective on Hom-sets, and is injective on Hom-sets.

Finitely Presented Categories

e.g. The category with two uniquely isomorphic objects is not free on any graph, since it’s finite, but has loops. But it is finitely presented with graph and relations and .