Homomorphism of Group Object: Are all the axioms necessary?

Question

This was a question brought up in class(not by me), and I figured I would ask it here.

In a category, we may define a group object as the following:

With this definition, we may use also define homomorphism of group object.

The question that was brought up is that the in a regular group, we define homomorphism as functions that respect the group multiplication, and the fact that it preserves identity and inverses follows from the fact that it respects group multiplication.

But between group objects, we impose that the homomorphism respects group multiplication AND identity AND inverses.

So the question is, what are some examples of category in which homomorphism respects the group multiplication but does not respect inverses or identity?

(For example, if we take the category of Sets, it's obvious that a homomorphism respecting group multiplication implies it preserves identity and inverses)

Thanks!

Answer 1

in a regular group, we define homomorphism as functions that respect the group multiplication, and the fact that it preserves identity and inverses follows from the fact that it respects group multiplication.

This is, in my opinion, a bad definition. You should define homomorphisms to respect the multiplication and identity and inverses, and then present as a theorem that it suffices to check that it respects multiplication. The reason is that this statement is false if inverses are dropped (that is, for monoids), so it gives you the wrong idea about what a homomorphism is in general. For example, to check that a map is a ring homomorphism you really do have to check that it respects the multiplicative identity.

Nevertheless:

So the question is, what are some examples of category in which homomorphism respects the group multiplication but does not respect inverses or identity?

There aren't any. If a morphism $f : G \to H$ between group objects respects multiplication then it automatically respects identity and inverses, just like for ordinary groups. The point is that you can check these conditions on generalized points $\text{Hom}(-, f) : \text{Hom}(-, G) \to \text{Hom}(-, H)$, where you've reduced to reasoning about ordinary groups, then appeal to the Yoneda lemma.