I just started learning algebraic topology, with almost no background in homology.
My supervisor mentioned the following result to me:
If $f: X \to Y$ is a continuous map between simplicial spaces (manifolds equipped with triangulation),
then there exists a refined triangulation with respect to $f$ (i.e. vertices mapped to vertices, edges mapped to edges, faces mapped to faces).
However after a bit of googling, I am still not sure what this theorem is called and how to prove it.
I thought since triangulation approximates the manifold, perhaps I can use some standard analysis tools about the convergence of sequences in each space.
I am not sure how much homology I need in order to understand this theorem.
Any hints will be greatly appreciated and thank you in advance!