Category of groups

In mathematics, the category Grp (or Gp[1]) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

There are two forgetful functors from Grp, M: GrpMon from groups to monoids and U: GrpSet from groups to sets. M has two adjoints: one right, I: MonGrp, and one left, K: MonGrp. I: MonGrp is the functor sending every monoid to the submonoid of invertible elements and K: MonGrp the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: GrpSet has a left adjoint given by the composite KF: SetMonGrp, where F is the free functor; this functor assigns to every set S the free group on S.

The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.

The category Grp is both complete and co-complete. The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element).

The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. The snake lemma however is not true in Grp.[dubious ][citation needed]