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?
Does the homology, homotopy, and geometric realization functors of a simplicial group preserve colimits?
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algebraic-topology
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1 Answers
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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,
- As for the homology functor, it preserves filtered colimits.
- Homotopy groups preserve filtered colimits too. This you can find in J.P. May's "A Concise Course in Algebraic Topology".
- 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).
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0@a.r. Homology commutes with filtered colimits, but not with all of them. – 2016-02-01