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

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

2012-07-17T14:02:39Z

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

2012-07-17T15:23:01Z

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 His viewpoint has been superceded by the Cisisnski-Maltsiniotis publications and Dugger's but it has the benefit of being fairly simple.

Answer by David Roberts for Is there a sense in which the homotopy theory of simplicial sets is the "paradigmatic" one?

2012-07-18T01:01:46Z

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

2012-07-18T09:58:20Z

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:

Algebraic inverses to subdivision (multiple compositions);
Tensor products.

These properties are exploited in the book "Nonabelian algebraic topology", to prove a Higher Homotopy Seifert-van Kampen Theorem, and to prove 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

(see also http://mathoverflow.net/questions/3656/cubical-vs-simplicial-singular-homology/83666#83666) and saying to myself: surely there should be some mathematics which expresses that? Then Ehresmann's book "Categori\'es 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.

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

2012-07-18T10:55:52Z

a better title would have paradigmatic written correctly