Dávid Natingga

Is V4 the free product of the cyclic groups C2 and C2?

In algebraic topology: can a retraction map induce a homomorphism between fundamental groups of the original space and the retract if the retract is not a deformation retract?

Are a direct product and a cartesian product of vector spaces the same? Thanks.

@DavidWheeler I understand now, thank you very much for your answer on category theory question.

In algebraic topology, when attaching a cell to the skeleton, does an attachment map need be continuous?

@DavidWheeler But $x \to 2x$ can be an identity arrow if we have a category with objects being sets? Or every identity arrow has to be an identity map if I have chosen my arrows to be maps?

Does identity arrow in categories need to be an identity function or not? E.g. is 1R:R->R, 1R(r)=2*r considered an identity arrow although it is just bijection and not an identity on a set of the reals?

Let f:A->B, g:C->B be functions in topological spaces. For f and g to be equal is it necessary that dom(f)=A=dom(g)=C or it is sufficient that A would be homeomorphic to C?

Actually I have a better definition. Is it correct? A category is a set of objects with a reflexive and transitive relationship on them.

With a set of objects being the elements of the monoid.

I have a quick question: Is a category a monoid?