Algebraic topology and homotopy theory with simplicial sets instead of topological spaces

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).

• If you only care about things to homotopy then bigger experts than I will say all you need is simplcial sets and say stuff about model categories and Quillen equialence. If you cannot replace maps by homotopy equivalent maps you lose information. A good reference to simplicial sets is Peter May's classic Simplicial Objects in Algebraic Topology Nov 23 at 19:13
• It's a fair question, but there's a circular or self-referential aspect to this kind of thing. Many people would more or less define mainstream 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. Nov 23 at 20:09
• @TomLeinster Actually, using modern shape theory, I would argue that simplicial sets are excellent for studying the homotopy theory of pathological spaces :). You just gotta use them in a creative way... Nov 23 at 20:38
• @Denis Nardin. That was true as well in now classical shape theory, an area that does not get as good mention as it deserves. e.g. see the classic lecture notes by Edwards and Hastings. Creative use of simplicial sets in Shape and Strong Shape was there from the start. Nov 24 at 10:46
• I would add the classic paper by Curtis to the list of places that the original contributor could look. Nov 24 at 10:48

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 by Goerss and Jardine that does most elementary homotopy theory in terms of simplicial sets.

• Thanks! Who first proved the analogues of Freudenthal suspension, Whitehead, Brown, Blakers-Massey for Kan complexes? Nov 23 at 21:08
• @user469290 I don't think there's "someone" who proved them first. Most of the point is that one can translate the original proofs seamlessly with some practice. Nov 23 at 21:11
• That is a very good answer, but I think we could add that all the basic knowledge we have of the homotopy groups of spheres (e.g. $\pi_n(S^n)=\mathbb Z$, $\pi_i(S^n)=0$ for $i<n$, Serre's finiteness theorem) can be formulated and proved with simplicial sets. This counts as Algebraic Topology made possibly without spaces, I think. Nov 24 at 9:51

It certainly used to be the case that if you went to an algebraic topology research talk, when the speaker said, "Let $$X$$ be a space," then there was about a 50% chance that they really meant "Let $$X$$ be a simplicial set": simplicial sets are that intertwined with homotopy theory.

More precisely, Quillen's model category framework allows you to start with a category $$C$$, say simplicial sets or topological spaces, add some extra structure, and then define what is called the "associated homotopy category," $$\text{Ho}\,C$$. Quillen showed further that the homotopy category for simplicial sets is equivalent to the homotopy category for topological spaces, and therefore if you want to study homotopy theory, you can use either topological spaces (with CW complexes as a distinguished subcategory) or simplicial sets (with Kan complexes as a distinguished subcategory) interchangeably. Even better, you can switch back and forth depending on which setting is more convenient for proving the result you happen to be interested in at the time.

• Thanks very much! Nov 24 at 15:08