To quote Kerodon:

In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces.

A similar quote can be found in the mathscinet review for Kan's *On c. s. s. complexes*:

In recent years it has become evident that for most purposes in homotopy theory it is more convenient to use semi-simplicial complexes instead of topological spaces.

For instance, I know that to special simplicial sets called Kan complexes one can assign higher homotopy groups and prove and analogue of Whitehead's theorem. This certainly demonstrates that one can do *some* homotopy theory with simplicial sets / Kan complexes.

*If I open an introductory book on algebraic topology or homotopy theory (such as Hatcher's), do all the main theorems admit analogues in the world of simplicial sets or Kan complexes (replacing topological spaces)?*

I'd be totally happy if you could give me, say, four theorems in algebraic topology / homotopy theory that can be phrased for simplicial sets, together with the original reference. I'd also be interested in whether these theorems are more algebraic topology or more homotopy theory (I don't really know the difference).

definemainstream algebraic topology to be that which can be captured by simplicial sets. If one were to argue that simplicial sets are an inadequate tool for the algebraic topology of certain non-standard classes of spaces (e.g. non-Hausdorff, or totally disconnected, or fractal) then the reply would probably be "that's not what I mean by algebraic topology". The subject is defined by its standard tools, of which simplicial sets are one. $\endgroup$4more comments