To quote [Kerodon](https://kerodon.net/tag/0004):



> 
> 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](https://kerodon.net/tag/00V2) 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).



Algebraic topology and homotopy theory with simplicial sets instead of topological spaces
It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them are just an application of $\operatorname{Sing}$ away from the world of simplicial sets).


But if one concentrates oneself to the "purely homotopical" statements (like, say, the Freudenthal suspension theorem, the Whitehead theorem, the Brown representability theorem and the Blakers-Massey theorem) they can all be stated in terms of simplicial sets (or, better, Kan complexes).


Indeed there is a [textbook](https://link.springer.com/book/10.1007/978-3-0346-0189-4) by Goerss and Jardine that does most elementary homotopy theory in terms of simplicial sets.


