I've looked around on MSE and online only to find scattered results, which confuse me. I want to understand how homology behaves with (co)limits. I want to know in particular about singular homology, and in general about homology theories.
- Does singular homology commute with arbitrary colimits? Where can I find proof of this? Under what conditions does it behave well with limits?
- In A Concise Course in Algebraic Topology, the author proves using the additivity, weak equivalence, and MVS axioms that homology preserves direct limits. Unless I'm mistaken, preserving direct limits is equivalent to preserving filtered colimits. Is this correct? Is there a simpler proof for May's result?
Added: In particular, I am confused by the answer to this question according to which homology preserves all kinds of colimits.