Category of groups

The category $\operatorname{Gr}$ whose objects are all groups and whose morphisms are all group homomorphisms. It is sometimes assumed that all groups studied belong to a given universal set. The category of groups is a locally small bicomplete category with null morphisms. It has a unique bicategory structure in which the admissible epimorphisms are normal (cf. Normal epimorphism), while all monomorphisms are admissible (cf. Monomorphism). Normal epimorphisms are in fact surjective homomorphisms, while monomorphisms are really injective homomorphisms. The projective objects of the category of groups are precisely the free groups (cf. Projective object of a category); the only injective objects are the unit groups, and these are also the null objects as well (cf. Injective object; Null object of a category). An axiomatic description of the category of groups was given by P. Leroux [3].

The category of groups is a special case of the general definition of a category of groups over an arbitrary category $K$. The category $\operatorname{Gr} K$ consists of all group objects (cf. Group object) in $K$ and the homomorphisms between them; this category has some of the properties of $K$; in particular, it is complete if $K$ is complete.

References

[1]	A.G. Kurosh, A.Kh. Livshits, E.G. Shul'geifer, "Foundations of the theory of categories" Russian Math. Surveys , 15 : 6 (1960) pp. 1–46 Uspekhi Mat. Nauk , 15 : 6 (1960) pp. 3–52
[2a]	B. Eckmann, P.J. Hilton, "Group-like structures in general categories I. Multiplications and comultiplications" Math. Ann. , 145 : 3 (1963) pp. 227–255
[2b]	B. Eckmann, P.J. Hilton, "Group-like structures in general categories II. Equalizers, limits, lengths" Math. Ann. , 151 : 2 (1963) pp. 150–186
[2c]	B. Eckmann, P.J. Hilton, "Group-like structures in general categories III. Primitive categories" Math. Ann. , 150 : 2 (1963) pp. 165–187
[3]	P. Leroux, "Une charactérisation de la catégorie des groupes" Canad. Math. Bull. , 15 : 3 (1972) pp. 375–380