Category of groups

In mathematics, the category Grp 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

U:Grp → Set

Where M has two adjoints:

One right; I:Mon→Grp

One left; K:Mon→Grp

Here 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.

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).

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