Category of groups
In mathematics, the category Grp (or Gp) 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: Grp → Mon from groups to monoids and U: Grp → Set from groups to sets. M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set→Mon→Grp, where F is the free functor; this functor assigns to every set S the free group on S.
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 ]