Homology and (co)Limits

Singular homology already fails to commute with pushouts. A pushout of spaces doesn't give a pushout of homology groups, but instead gives (maybe under niceness conditions) a long exact sequence. For an explicit example consider the pushout of

$$D^2 \leftarrow S^1 \rightarrow D^2$$

which is $S^2$. This doesn't induce a pushout on $H_2$.

Singular homology also fails to commute with products. (Note that the tensor product is not the product in the category of abelian groups, or of graded abelian groups, so even if we're working over a field the Kunneth formula is not a response to this claim.)

The first fact is in some sense a reflection of a failure to be suitably higher categorical. There is a very abstract description of what it means to compute the homology (not the homology groups, but "the homology") of a space, namely tensoring it with some spectrum, and this construction preserves all homotopy colimits (in fact it is a left adjoint in a higher categorical sense). It's very natural to think about homotopy colimits rather than colimits because taking singular homology is homotopy-invariant, but taking colimits is not, while taking homotopy colimits is.

Then you have to figure out how to compute homotopy colimits of spaces, and also figure out what a homotopy colimit of spectra buys you once you pass to homotopy groups (e.g. long exact sequences, or more generally spectral sequences).