I'm given two simplicial sets $X,Y : \Delta^{\operatorname{op}} \to Set$. Of, course, if I study their geometric realisations $\lvert X \rvert$ and $\lvert Y \rvert$, I might find a homeomorphism, or maybe I can show via homology that they aren't homeomorphic. But how do I find that out just given the combinatorial data? Is there a combinatorial condition? Or is there an algorithm that can decide it for simplicial sets with finitely many nondegenerate simplices?
Edit: Is there a set of moves (similar to Pachner moves for triangulations) that can be applied to go from one simplicial set to another?