Homology theory constructed in a homotopy-invariant way

Question

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological spaces.

But the usual definition of singular homology is on the category of topological spaces, and you can show that it is homotopy-invariant only after having defined it on the category of topological spaces.
For example, the definition uses the free abelian group on the underlying set of the space of singular $n$-simplexes, and taking the underlying set of a space do not make sense in the homotopy category.

I would like to have a construction of singular homology that can be entirely carried out in the homotopy category.

I was thinking of categorifying the usual construction:
Take the free spectrum ($(\infty,1)$ equivalent of abelian group?) on the space of singular n-simplexes, show that this is an "$(\infty,1)$ chain complex" and compute its "$(\infty,1)$ homology".

But I have the impression it will not work as is, because simplexes are only interesting as topological spaces, not as homotopy types.

Is there a way to construct singular homology (or in fact any homology theory) directly in the homotopy category without using the category of topological spaces?

Answer 1

Throughout the following, I'll say "homotopy category" when I really mean the weak homotopy category.

For a space $X$, the homology of $X$ is canonically isomorphic to the reduced homology of $X_+$, which is $X$ with a disjoint basepoint. Therefore, it suffices to give a definition of the reduced homology of a based space.

The smash product $\wedge$ descends to a well-defined operation on the homotopy category of based topological spaces. For any $n \geq 0$, we have an object $K(\mathbb{Z},n)$ in the homotopy category, and weak equivalences $K(\mathbb{Z},n) \to \Omega K(\mathbb{Z},n+1)$ which are adjoint to maps $S^1 \wedge K(\mathbb{Z},n) \to K(\mathbb{Z},n+1)$. For any integer $m$, we can therefore define a directed system of sets $$ [S^{m+k}, K(\mathbb{Z},k) \wedge X ] \to [S^{m+k + 1}, K(\mathbb{Z},k+1) \wedge X ] \to \cdots $$ The colimit $colim_k \pi_{m+k}( K(\mathbb{Z},k) \wedge X)$ is isomorphic to the $m$'th reduced homology group of $X$ in a canonical way.

This kind of definition produces generalized homology theories, and this all falls into the subject of stable homotopy theory.

The method you suggest of taking the free spectrum in this $\infty$-categorical sense will give a homology theory, but instead of producing the homology groups of $X$ it will produce the stable homotopy groups. The "free abelian topological group" on the topological space $X$ can be used instead, and shown to give a good notion on the homotopy category; this will produce the homology of $X$, as a result of the work of Dold and Thom.

Answer 2

I don't know what you are looking for, but here is my idea:

You could use the universal property of the homotopy category among "higher categories" as follows: For any "higher category" $C$ and any object $c$ of $C$, there is up to isomorphism one functor from the homotopy category of spaces - let's call it $H$ - to $C$ sending the point to $c$. Here, "higher categories" could mean homotopy categories of model categories, where the functors we allow are derived left Quillen functors, or (I guess) $\infty$-categories.

The derived category of $\mathbb Z$ is such a category, so we find a unique functor $F: H \rightarrow D(\mathbb Z)$ sending the point to the chain complex $U$ which has only $\mathbb Z$ in degree $0$. Now $D(\mathbb Z)$ is triangulated and we can define $H_{\star}(X) = [U, F(X)]_*$ for $X \in H$. It is easy to check that this is a homology theory and has the correct behaviour on the point. (This approach also has the advantage that the Künneth formula follows from abstract nonsense.)

To me, something like this seems to be the only way to avoid the mentioning of topological spaces or simplicial set, since the category $H$ either has to be given explicitly or characterised by some property as above. And introducing "higher category" structures also seems natural since you cannot properly define a homology theory on some random category; you need some additional structure such as cofiber and fiber sequences and suspension etc.

There is also the notion of an algebraic model category (and probably there is the same notion for $\infty$-categories), and our functor $F$ is the initial functor from $H$ to the homotopy category of an algebraic model category. In this way, one could view singular homology as the universal algebraic invariant of a homotopy type, but this is rather vague.

Answer 3

I think your intuition of founding algebraic topology through homotopical methods is sound, and I am pleased to report that this has been done in the two papers:

R. Brown and P.J. Higgins, ``On the algebra of cubes'', J. Pure Appl. Algebra 21 (1981) 233-260.

R. Brown and P.J. Higgins, ``Colimit theorems for relative homotopy groups'', J. Pure Appl. Algebra 22 (1981) 11-41.

In the space of less than 60 pages, and without using simplicial approximation or singular homology, these papers prove

1. the Brouwer Degree Theorem (the $n$-sphere $S^n$ is $(n-1)$-connected and the homotopy classes of maps of $S^n$ to itself are classified by an integer called the degree of the map);

2. the Relative Hurewicz Theorem, which is seen here as describing the morphism $$\pi_n(X,A,x) \to \pi_n(X \cup CA,CA,x) \cong \pi_n(X \cup CA,x)$$ when $(X,A)$ is $(n-1)$-connected, and so does not require the usual involvement of homology groups;

3. Whitehead's theorem (1949) that $ \pi_2(X \cup \{e^2_{\lambda} \},X,x)$ is a free crossed $\pi_1(X,x)$-module.

The last theorem allows, for the usual identification of a square $\sigma$ to give the Klein bottle, to write the nonabelian formula:

$$ \delta \sigma = a+b -a +b.$$ Whitehead's theorem is sometimes mentioned in texts but rarely proved. For more on the background to the theorem, see also http://ncatlab.org/nlab/show/free+crossed+module.

Actually one has here a method for direct calculation of some homotopy $2$-types, as crossed modules.

Historical background and intuitions for this method are given in an expository paper showing the origins of the idea of cubical higher homotopy groupoid. Not surprisingly, the method needs a number of new ideas, and departures from tradition, particularly the generalisation to higher dimensions of the fundamental groupoid and the Seifert-van Kampen Theorem.

The method is not to start with just a topological space but with a filtered space: i.e. a space $X$ and an increasing sequence of subspaces $X_n, n \geqslant 0$. This enables one to define a crossed complex $\Pi X_*$ using the fundamental groupoid of $X_1$ on the set $X_0$ of base points and for $n \geqslant 2$ the relative homotopy groups $\pi_n(X_n,X_{n-1},x), x \in X_0$, with the operations of $\pi_1$ and the standard boundary operations. The functor $\Pi$ satisfies a form of the Seifert-van Kampen Theorem, i.e. it can be computed on certain pushouts (and more generally) of filtered spaces. This gives the above results, and more. So one get new nonabelian calculations of second relative homotopy groups; and of higher relative homotopy groups as modules over a fundamental group, without using covering spaces.

However this result is not proved directly, but via a related (and not so trivial to establish!) construction of a cubical higher homotopy groupoid $\rho X_*$ of a filtered space. The first paper above establishes many key algebraic properties of these gadgets. These higher homotopy groupoids are strict structures, like the relative homotopy groups, and so a colimit type theorem allows for precise calculations.

The use of filtered spaces may be a surprise, though they are very common. However Grothendieck in Section 5 of "Esquisse d'un programme" argues that more structured concepts than a topological space are needed for the purposes of geometry: he likes stratifications. We use filtered spaces because in that context we can make these strict higher homotopy groupoids work, and express intuitions such as "algebraic inverses to subdivisions", for which simplicial or globular theories are not so well suited.

This work has been developed since those papers were published and some details of exposition modified: a long account is published as

R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011). (referred to below as NAT).

(more details on NAT including a downloadable pdf) on NAT.

The above papers and most of the book have been written to be checkable by a graduate student with some knowledge of general topology and the fundamental groupoid, such as in my Topology and groupoids.

I have to give this plug for my work with Philip (and others) since the notion of a Higher Homotopy Seifert-van Kampen Theorem is to my knowledge not mentioned in any text in algebraic topology (except mine).

I have no idea how to do an analogue of this work for a filtered topos!

Edit: 23 April, 2014: I should mention that this work develops that of J.H.C. Whitehead in his highly original 1949 paper "Combinatorial Homotopy II", which introduced free crossed modules, and is also relevant to my answer to this mathoverflow question.

Edit 24 April: To answer some doubts of Johannes, the first step is to define homotopically the fundamental crossed complex functor $\Pi: ($filtered spaces$) \to ($crossed complexes$)$, where the latter objects have analogies to chain complexes but are nonabelian in dimensions 1 and 2 and have operations of dimension 1 on higher dimensions, and of course a number of axioms. Their "homology" is a group(oid) in dimension 1 and modules over this in higher dimensions. On a CW-complex skeletal filtration these modules are the homology of the universal covers at various base points. This is a fairly classical construction in the single base point case, Blakers (1948), Whitehead (1949).

To conjecture and prove a number of theorems on $\Pi$ without using singular chains, whose algebra is inadequate for the nonabelian aspects, we use a cubical higher homotopy groupoid, $\rho(X_*)$, which has an underlying Kan cubical set, and $n$ groupoid compositions in dimension $n$. Part of the work is to show $\rho(X_*)$ contains $\Pi(X_*)$ as a substructure, and that the former can be recovered from the latter; the algebraic structures are equivalent! But the cubical $\rho(X_*)$ is better for conjecturing and proving local-to-global theorems. From those one can prove that for a CW-filtration, $\Pi(X_*)$ is free in each dimension with basis the characteristic maps of the cells.

Whitehead remarks in his paper "Combinatorial Homotopy II" (but with a different notation!) that for a CW-filtration, $\Pi(X_*)$ has better realisation properties than the cellular chains of the universal cover, which are chain complexes with a group(oid) of operators.

Adding later a further point is that this approach avoids the use of "formal sums", i.e. free abelian groups, as introduced by Poincar'e, and replace them by actual compositions, based on homotopy theory.

Hope that helps.

October 27, 2020 I hope the following comments will help to explain my references.

My general attitude is that we are interested in particular applications of homology to spaces. That means we need to calculate something, and so we need information on the space. That information will have some structure, and so we need to be able to exploit that. A methodology for this is laid out in the paper 2018 Modelling and computing homotopy types: I.

An example is cellular homology, which is conveniently studied in terms of filtered spaces $X_*$; these give rise to a homotopically defined fundamental crossed complex $\Pi( X_*)$, which goes back (with different terminology) to A.L. Blakers (!948) and to JHC Whitehead in his paper "Combinatorial homotopy II" (1949), where the filtration is the skeletal filtration of a CW-complex.

There is a higher Van Kampen Theorem for $\Pi X_*$ which allows for some specific calculations and proofs of for example relative Hurewicz Theorems. See the book NAT.