Talk:Category of groups

Other than having to be careful about what is meant by the cokernel in Gp is there actually anything that prevents the Snake Lemma from working?

Well, the standard proof certainly doesn't work in Grp since it uses the difference of morphisms. AxelBoldt (talk) 00:14, 26 October 2008 (UTC)[reply]

The complicated endomorphism example shows that no definition of addition makes the hom sets into a group suitable for an additive category. The 1+1=2 assumes a definition of addition (the one used for the near ring of endormorphisms, aka the natural definition). The whole point of the example is to show that even if the addition is not natural, it cannot exist. JackSchmidt 21:02, 3 August 2007 (UTC)[reply]

"The study of this category is known as group theory." I find this to be quite an imperialist statement; group theory is the study of groups, not of the category of groups. 220.235.100.46 (talk) 17:31, 21 July 2022 (UTC)[reply]

"K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid."

Under "Relation to other categories";

This seems wrong as stated, perhaps requiring a citation. AFAIK The Grothendieck Group is defined for commutative monoids only, and this is reflected in the linked article on that subject. A functor cannot act partially; a functor K:Mon -> Grp must assign to every monoid a group.

So, to what group do we assign a non-commutative monoid M? The trivial group? The Grothendieck group of the largest abelian submonoid of M? As a non-expert that seems a non-trivial choice to me, and at least requires mention or citation. 75.81.80.223 (talk) 17:51, 27 September 2023 (UTC)[reply]