The notion of mapping cone and cylinder for chain complexes is typically motivated by the corresponding notions in topology. For example, take the mapping cone $C_f$ of some map $f\colon X \to Y$. Using Mayer-Vietoris, I think it's straightforward to prove that this gives rise to a long exact sequence in homology. But why does the chain complex of Cf have the form described, for example, here? https://en.wikipedia.org/wiki/Mapping_cone_(homological_algebra)
I imagine with singular homology it's impossible to see this, so here's my question.
If X and Y are simplicial complexes and f is a simplicial map, where can I find the triangulation of $C_f$ giving rise to that chain complex?
Same question for suspension and mapping cylinder.
Similarly, if X and Y are cell complexes, is there a cell structure on $C_f$ giving rise to the correct chain complex (via cellular homology)?
Again, same question for suspension and mapping cylinder.