Is there a sense in which the homotopy theory of simplicial sets is the "paradigmatic" one?

Question

I could not come up with a better title for my question.

What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):

the category of simplicial sets has always been to me something distinguished, on multiple counts:

to begin with, it is simple to describe, almost childish, yet it has a seemingly unfathomable richness.

Its homotopy theory, although usually presented via the geometric realization functor (which is historically correct), is in fact entirely self-contained, and purely combinatorial in character.

Also, simplicial sets plays a very special role in category theory (after all, categories are just some almost trivial example of simplicial sets) and, even more important, in higher dimensional cats.

Somehow, I have the lingering feeling that, in abstract homotopy theory, simplicial sets (or, more properly simplicial objects in some ambient category ) should be, at least for some suitable notion of "regularity" of homotopy theories, paradigmatic. I am thinking of something like:

If an homotopy theory is "regular...." (whatever that may mean, fill the dots, the "regular" would stand for combinatorial in essence), then it is representable in the homotopy theory of simplicial objects for some ambient category.

Am I dreaming or there is something along these lines?

Answer 1

This requires some topos theory I'm afraid, but gives, I think, a satisfactory explanation.

The category $\mathrm{Set}^{\Delta^{\mathrm{op}}}$ classifies intervals: the simplicial set $\Delta[1]$ is linearly ordered, with smallest and largest elements $0,1:\Delta[0]\to\Delta[1]$, and for any topos $\mathscr E$ and any linearly ordered object $I\in\mathscr E$ with distinguished smallest and largest elements $\bot,\top:1\to I$ there is a unique up to isomorphism $f_I:\mathscr E\to\mathrm{Set}^{\Delta^{\mathrm{op}}}$ with $f_I^*(\Delta[1])=I$. In particular, one can form various $\mathscr E$ containing one or other category of spaces, and for the corresponding interval $I$ there, $f_I^*(S)$ is the geometric realization of the simplicial set $S$, while for a space $X$, ${f_I}_*(X)$ is the singular simplicial set of $X$.

Thus simplicial sets are determined (up to equivalence of categories) by the fact that they classify all possible notions of a continuous path, hence of a homotopy.

Reference - e. g. in ncatlab; first occurrence of this I've seen is in Johnstone's first topos theory book, where it is attributed to Joyal.

Answer 2

I will as usual act as Devil's advocate (or cubical advocate) by saying that there are some things we can do with cubical sets (with connections) which we cannot (maybe others can!) do with simplicial sets, namely:

1. Algebraic inverses to subdivision (multiple compositions);

2. Tensor products.

These properties are exploited and used (paradigmatically!) in the book Nonabelian algebraic topology: there is a Higher Homotopy Seifert-van Kampen Theorem, and some theorems on homotopy classification of maps (in the non simply connected case), and so rewrite a portion of algebraic topology without using singular homology or simplicial approximation. Also these results would not, I believe, even have been conjectured simplicially.

This is not to deny the advantages of the simplicial approach, which include its large literature, or some disadvantages of the cubical approach.

I started in the 1960s with drawing many times the diagram of a subdivided square pictured as

(source)

(see also Cubical vs. simplicial singular homology ) and saying to myself: surely there should be some mathematics which expresses that? Then Ehresmann's book "Catégories et structure" came out and the answer was clear: double categories (or groupoids)! In these one can describe the above as an array $(a_{rs})$ or as a composition $[a_{rs}]$ and this makes sense for the singular cubical complex, and also for double categories or groupoids. So one expresses "algebraic inverses to subdivision", while the corresponding globular or simplicial expressions are difficult or not available.

This role of cubical sets for intuition and conjecture is also suggested by the fact that Dan Kan's first results were cubical. Then problems were found with realisations and so, instead of trying to fix the cubical, they simply gave it up.

We came across connections for cubical sets for other reasons, namely to describe commutative cubes in terms of double groupoids.

January 9, 2015: See also this presentation at Galway, Dec, 2014, on "A homotopical approach to algebraic topology via compositions of cubes". Note that a simplicial approach of a similar type seems a non starter. However many aspects of a cubical approach have not yet been developed, because of the apparent success of the simplicial methods.

May 24, 2020 I can now refer to a more developed presentation of the philosophy of, background to, and development of the cubical approach in a paper Modelling and Computing Homotopy Types: I preprint which appeared in Indagationes Math. in 2017 in a special issue to honor L.E.J. Brouwer. One main point is that the cubical theory has been enriched by the notion of connections which, it has been found, rescue it from some problems of the older theory, and which make the expanded theory applicable in places not available to the simplicial theory.

Answer 3

Cisinski has some very detailed thoughts here explaining how the homotopy theory of simplicial sets arises naturally using derivators as the free completion of the trivial category by homotopy colimits. That derivators arise he also puts in a very natural way (and of course Heller knew all about). But perhaps all this is contained in the monograph that Tim linked to.

Answer 4

There is an interesting perspective on the role of simplicial sets in the Mem. AMS of Alex Heller (Homotopy Theories, number 383 in Memoirs Amer. Math. Soc. 1988). I suggest you look there as well as some of the more recent sources suggested in the previous answer mentioned above. His viewpoint has been superseded by the Cisinski-Maltsiniotis publications and Dugger's but it has the benefit of being fairly simple.