What is combinatorial homotopy theory?

Question

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below.

Many times I've heard people speaking about combinatorial homotopy theory, but every reference apparently related seems to deal with general concepts of algebraic topology like CW-complexes, homotopy groups, homology and so on.

So could anybody explain to me what exactly is combinatorial homotopy theory and in what is its relation with algebraic topology?

Answer 1

The term "Combinatorial homotopy" was coined by J.H.C. Whitehead in its use for the first two of his papers

1. "Combinatorial homotopy. I." Bull. Amer. Math. Soc. 55 (1949) 213--245.

2. "Combinatorial homotopy. II." Bull. Amer. Math. Soc. 55 (1949) 453--496.

3. "Simple homotopy types." Amer. J. Math. 72 (1950) 1--57.

These were rewrites and extensions of some of his very original papers which appeared in 1939-1941, in which he showed how some possible extensions of basic techniques in what became called "Combinatorial group theory", such as Tietze transformations, were in fact not generally possible, and that there were obstructions lying in what is now called the Whitehead group. So I'd be interested to know of a contrary view, but it seems to me that the impetus for the name came from combinatorial group theory.

The first paper listed covers what is now a standard basis for homotopy theory (CW-complexes, the Whitehead Theorem on homotopy equivalences, ...) and the 3rd has been very influential. The second is not so well reported on, but is one of the bases of the use of higher homotopy groupoids in homotopy theory, which I have reported in a web page under the term "Higher dimensional group theory", and which kind of extends aspects of Whitehead's approach. In particular, the free crossed modules which appeared in the second paper were a key to progress, and have a strong link with group theory and "Identities among relations".

One point which struck me recently was that you really need this concept of free crossed modules if you want to write for the boundary of the standard square diagram $\sigma$ giving the Klein bottle a non commutative formula:

$$\partial(\sigma) = a+b-a +b $$

rather than the usual $\partial (\sigma) = 2b$, which has lost a lot of information. So Whitehead argues in paper 2 that his "homotopy systems", now called "free crossed complexes", carry more information and have better realisation properties that then the still more widely used free chain complexes with a group of operators.

Of course language and the use of terms change with the decades, but it can be useful to trace some history to its roots, and try and see to what extent the original aims have been accomplished.

A relation of the word "combinatorial" to simplicial sets, and so another line on how the word might be interpreted, was given in the paper

Kan, Daniel M. A combinatorial definition of homotopy groups. Ann. of Math. (2) 67 1958 282–312.

Nov 12, 2014 I have just come across the preprint by Vezzani on motives which uses cubical rather than simplicial methods because of some technical advantages in dealing with derived functors in an algebraic geometry setting. See also this preprint by I.Patchkoria on cubical and simplicial derived functors (published in HHA).

Answer 2

The theory of simplicial sets is, for the purposes of homotopy theory, often an effective substitute for the category of topological spaces. At least, there is a well-defined "homotopy theory" of simplicial sets which is encoded in the standard (Quillen) model structure on simplicial sets, as your professor points out. There is a Quillen equivalence between spaces and simplicial sets, where the right adjoint sends a space to its singular complex and the left adjoint sends a simplicial set to its geometric realization (which is probably how you visualize a simplicial set in the first place).

The meaning of its being a Quillen equivalence is that one can compute homotopy classes of maps either simplicially or topologically. If $X, Y$ are CW complexes, then homotopy classes of maps $[X, Y]$ are the same as (simplicial) homotopy classes of maps $[\mathrm{Sing} X, \mathrm{Sing} Y]$. Similarly, if $A, B$ are Kan complexes (actually, you only need $B$ to be a Kan complex), then $[A, B] = [|A|, |B|]$ where $|\cdot | $ denotes geometric realization.

So, if you want to work with homotopy types (of CW complexes, at least) then you may as well work with simplicial sets as with topological spaces. But there are various reasons to prefer simplicial sets in certain cases:

• The category of simplicial sets is very nice: it is a presheaf category (and thus presentable. One can manipulate simplicial sets purely combinatorially; when dealing with topological spaces, one has often to worry about general-topological considerations which make no appearance simplicially. (For instance, in constructing the function space $\mathrm{Map}(X, Y)$ between two topological spaces $X, Y$, it will generally not be an exponential in the categorical sense; that is, $\mathrm{Top}$ is not cartesian closed. The usual remedy is to work in a convenient category.)
• According to the Dold-Kan correspondence, simplicial abelian groups are the same as nonnegatively graded chain complexes. There is no analog for topological spaces.
• In category theory, one frequently meets simplicial objects via the bar construction (in topology, a classical example of this is in delooping theory), and it's useful to be able to work with them homotopically.
• Categories can be efficiently encoded using the data of a simplicial set, via the nerve. One gets a fully faithful embedding of the category of (small) categories in the category of simplicial sets. Remarkably, by relaxing the characterization on simplicial sets that defines the nerve of a category, one gets a workable model for $(\infty, 1)$-categories.

I don't really know enough about algebraic topology to make any sweeping statements, but these are just a few examples that I'm aware of. You might enjoy the answers at this MO question.

Answer 3

Here I want to post the answer I've recived to this question by professor Dan Christensen, I hope it can be interesting for others

Here's a very brief answer: there are combinatorial objects called simplicial sets, and you can define all of the usual concepts of homotopy theory for simplicial sets in a way that makes the homotopy theory of simplicial sets equivalent to the homotopy theory of spaces. (One way to say that they are "equivalent" is to show that they are "Quillen equivalent", which is stronger than just saying that the homotopy categories are equivalent.) As a result, any homotopy-theoretic problem you would like to study for topological spaces can instead be studied in the combinatorial world of simplicial sets.

Btw since it midnight in here I want to wish everyone happy new year.