Does the homology, homotopy, and geometric realization functors of a simplicial group preserve colimits?

Question

Given a based simplicial group, you can find its reduced homology with coefficients in a field, homotopy, and geometric realization. These are functors. If I have a free product of based simplicial groups, does these associated functors split into coproducts? More generally, does these functors preserve colimits?

Answer 1

The answer to all of your questions is "yes", in general, if you're talking about filtered colimits -but the free product is NOT a colimit of that kind. More specifically,

1. As for the homology functor, it preserves filtered colimits.
2. Homotopy groups preserve filtered colimits too. This you can find in J.P. May's "A Concise Course in Algebraic Topology".
3. The realization functor preserves all kind of colimits. This follows from the fact that it is left adjoint to the total singular complex functor -see, for instance, Simplicial objects in Algebraic Topology, page 61, also by J.P. May-, and functors which are left adjoints preserve all colimits (S. Mac Lane, "Categories for the working mathematician", first edition, page 115).