Inmathematics, thecategoryGrp (orGp^[1]) has theclass of allgroups for objects andgroup homomorphisms formorphisms. As such, it is aconcrete category. The study of this category is known asgroup theory.

There are twoforgetful functors fromGrp, M:Grp →Mon from groups tomonoids and U:Grp →Set from groups tosets. M has twoadjoints: one right, I:Mon→Grp, and one left, K:Mon→Grp. I:Mon→Grp is thefunctor sending every monoid to the submonoid of invertible elements and K:Mon→Grp the functor sending every monoid to theGrothendieck 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 thefree functor; this functor assigns to every setS thefree group onS.

Themonomorphisms inGrp are precisely theinjective homomorphisms, theepimorphisms are precisely thesurjective homomorphisms, and theisomorphisms are precisely thebijective homomorphisms.

The categoryGrp is bothcomplete and co-complete. Thecategory-theoretical product inGrp is just thedirect product of groups while thecategory-theoretical coproduct inGrp is thefree product of groups. Thezero objects inGrp are thetrivial groups (consisting of just an identity element).

Every morphismf :G →H inGrp has acategory-theoretic kernel (given by the ordinarykernel of algebra ker f = {x inG |f(x) =e}), and also acategory-theoretic cokernel (given by thefactor group ofH by thenormal closure off(G) inH). Unlike in abelian categories, it is not true that every monomorphism inGrp is the kernel of its cokernel.

Thecategory of abelian groups,Ab, is afull subcategory ofGrp.Ab is anabelian category, butGrp is not. Indeed,Grp isn't even anadditive category, because there is no natural way to define the "sum" of two group homomorphisms. A proof of this is as follows: The set of morphisms from thesymmetric groupS₃ of order three to itself,${\displaystyle E=\mathrm{Hom}(S_3,S_3)}$ , has ten elements: an elementz whose product on either side with every element ofE isz (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms. IfGrp were an additive category, then this setE of ten elements would be aring. In any ring, the zero element is singled out by the property that 0x=x0=0 for allx in the ring, and soz would have to be the zero ofE. However, there are no two nonzero elements ofE whose product isz, so this finite ring would have nozero divisors. Afinite ring with no zero divisors is afield byWedderburn's little theorem, but there is no field with ten elements because everyfinite field has for its order, the power of a prime.

The notion ofexact sequence is meaningful inGrp, and some results from the theory of abelian categories, such as thenine lemma, thefive lemma, and their consequences hold true inGrp.Thesnake lemma however is not true inGrp.^{[dubious –discuss][citation needed]}

Grp is aregular category.

• Goldblatt, Robert (2006) [1984].Topoi, the Categorial Analysis of Logic (Revised ed.).Dover Publications.ISBN 978-0-486-45026-1. Retrieved2009-11-25.