# Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering...

It's not a secret that nowadays homotopy theory is not just a branch of algebraic topology, it's a actually an enourmous mathematical field of research subsuming category theory, homological algebra and having deep connections with algebraic geometry and commutative algebra.

I wonder if it's worth it to think about introducing homotopy theory as an independent subject, for example, build everything axiomatically relying on a category theory machinery rather than introduct it as a subject of algebraic topology.

I'm not sure such references do exist, maybe, Emily Riehl's book "Categorical homotopy theory" can be thought as a introduction to homotopy theory for category theorists(thought she advises to read on simplicial homotopy theory and homotopy limits and colimits before approaching the book, but says it's not a formal prerequisiste ).

P.G. Goerss, J.F. Jardine "Simplicial homotopy theory" seems to assume algebraic topology, though I haven't seriously researched whether it truly does so.

So, to sum it up, it is a question about both whether such approach can be viable and whether it's already used in some literature. Of course, anything else close to this topic will be appreciated as well.

• The use of simplicial sets simply takes away those annoying point-set complications (e.g. which topology do you put on mapping spaces?), nothing more. I dream of an introductory book which uses them because of this but do not expect the shape of the subject to change sensibly. Nov 21 '15 at 12:52
• You forgot to mention the connections of homotopy to foundations of mathematics. I have had students who learned homotopy theory through type theory. It does strange things to their minds – and they think classical texts on homotopy theory are "clumsy" and "annoying". Nov 22 '15 at 22:50
• It sounds like you'd like to learn about model categories, as a start... Nov 23 '15 at 0:06
• @DmitriPavlov Sure, I love simplicial sets. There's also the fact that defining the Kan complex parametrizing tubular neighbouroods or embeddings of manifolds is completely trivial, while defining the appropriate topologies is a bit of a pain (and I can never remember how it's done anyway). That's why I really think that they should be taught in an introductory course instead of simplicial complexes and similar stuff. Still, this is not going to change the fact that you have to prove, say, the Blackers-Massy theorem or Poincarè duality. Nov 23 '15 at 12:53
• For Blakers-Massey at least, there is a nice model independent proof described in a note of Rezk.
– AAK
Nov 23 '15 at 17:13

One of the starting points for homotopy was Poincaré's notion of the fundamental group, in which one took homotopy classes of loops at a base point. Later homology was proved homotopy invariant, and then Cech, and later more forcefully, Hurewicz, introduced the homotopy groups. One of the advantages of taking homotopy classes is that could reduce uncountable models to countable, or even finite ones. Another was modelling the intuitive idea of deformation.

These both reside on the idea of composition: you compose paths, you compose homotopies. And in category theory, you compose morphisms.

Thus it is reasonable to look for higher compositions. And this was the reason why Dan Kan's thesis and first paper were cubical. However cubical sets were found to have to have two disadvantages:

1. Cubical groups were not necessarily Kan, unlike simplicial groups.

2. The realisation of the cartesian product of cubical sets had the wrong homotopy type, though that of a tensor product was OK.

So cubical sets were kind of abandoned for the development of algebraic topology.

However cubical methods still have advantages some of which are discussed in this mathoverflow discussion on $n$-fold categories. One is the rule that $I^m \times I^n \cong I^{m+n}$; another is the easy formulation of the idea of multiple higher composition, and so of "algebraic inverses to subdivision", and hence applications to "local-to-global" questions. These advantages are exploited in the book I refer to; the main results on Higher Seifert-van Kampen Theorems would hardly have been conjectured, let alone proved, in traditional terms. The results do require the notion of cubical set with connections. This innovation has led to the above specific difficulties being resolved to some extent.

There is a large literature on simplicial sets and their applications, and it seems that all talk of $(\infty,n)$-categories is in simplicial terms. Also $n$-fold categories need a geometric aspect which is broader than the usual cubical theory, in which the faces in different directions are all of the same type.

It seems to me that young people should try to be eclectic, and assess and evaluate different approaches. It all may depend on the problem at hand, or the next problems to come. Or simply for expressing intuitions.

Nov 23: For a discussion of "Anomalies in algebraic topology" see this presentation in Galway, Dec 2014.