# Homology other than singular?

Usually, one defines $n$-th homology functor on topological spaces as the composite functor $$\mathbf{Top} \to [\Delta^\mathrm{op},\mathbf{Set}] \to [\Delta^\mathrm{op},R\!-\!\mathbf{Mod}] \overset C \to \mathfrak{Ch}(R\!-\!\mathbf{Mod}) \to R\!-\!\mathbf{Mod}$$ where the first arrow is $X \mapsto \hom (\Delta^\mathrm{top}_\bullet, X)$ mapping a topological space to its simplicial set of singular simplexes, the second arrow is induced by the free $R$-module functor, and the last one is the homology functor of chain complexes. Finally, $C$ maps $M_\bullet$ to $$\dots \to C_n \overset {d_n} \to C_{n-1} \to \dots$$ with $d_n = \sum_{i=0}^n (-1)^i \partial^n_i$ where the $\partial^n_i$ are the face operators.

So, defining the functor $H_n \colon \mathbf{Top} \to R\!-\!\mathbf{Mod}$, every step is pretty natural and kind of canonical except for the choice of the $d_n$ : we have to choose the functor $C$. I understand the motivation behind the chosen derivation, but I wonder if there are other functors $C$ which give interesting homology spaces. I have been told more or less that there is not a lot of choice for such a $C$. Why is that ? Is there theoretically very few choices (and how can I see it) ? Or is it that for a general functor $C$, we can't tell anything (in fact, is it even an homology theory according to Eilenberg-Steenrod axiomatic) ?

(I am aware of the normalization functor (Dold-Kan correspondence) that is a good choice of $C$, but then the homology is the usual one, so...)

• I think your last comment about E-S axioms is right on the money. Since the homology theory is determined by coefficients there is really not much choice and the Dold-Kan correspondence is as a good a choice as any. Commented Sep 29, 2013 at 11:04

The book "Nonabelian algebraic topology:filtered spaces, crossed complexes, cubical homotopy groupoids" (published by the EMS, 2011), takes a nontraditional approach to the border between homology and homotopy.

The usual way, as you say, is to replace a space $X$ by its simplicial singular complex $SX$ and then make constructions on that. A key tool is simplicial approximation. After some work, one is able to define the cellular chains of a $CW$-complex, and by applying this to a universal cover of a reduced $CW$-complex $X$ to obtain a chain complex with the fundamental group of $X$ as a group of operators.

Now the geometric realisation $Y=|SX|$ is a CW-complex and so is filtered by its skeleta, i.e. is a filtered space, i.e. has an increasing sequence $Y_n$ of subspaces.

The approach in the book is to define a (actually somewhat classical) functor $\Pi: FTop \to Crs$ where $FTop$ is the category of filtered spaces, and $Crs$ is the category of what are called crossed complexes . This functor is defined homotopically using the fundamental groupoid $\pi_1(Y_1,Y_0)$ and the relative homotopy groups $\pi_n(Y_n,Y_{n-1},y), y \in Y_0$. This is a stronger structure than a chain complex with a group(oid) of operators. It has nonabelian features in dimensions $1,2$ and in higher dimensions gives $\pi_1$-modules. No use is made of simplicial approximation, but a key use is made of cubical methods: they are convenient because they allow "algebraic inverses to subdivision", leading to higher order Seifert-van Kampen Theorems, and because of the rule $I^m \times I^n \cong I^{m+n}$. This leads to a monoidalk closed structure for crossed complexes.

There is also a functor $\nabla$ from crossed complexes to chain complexes with a groupoid of operators; this functor has a right adjoint, and for a $CW$-complex $X$ with its skeletal filtration, $\nabla \Pi (X_*)$ is indeed the chain complex of universal covers of $X$ at various base points, with the fundamental groupoid $\pi_1(X,X_0)$ as groupoid of operators. However $\Pi X_*$ has better realisation properties that the corresponding chain complex.

A number of these ideas go back to J.H.C. Whitehead in his 1949 paper "Combinatorial homotopy II", but the use of cubical methods in this way, of colimit methods for calculation, and of monoidal closed categories, all enable a better setting for Whitehead's ideas.

There are many issues, for example the use of $R$-modules for a ring $R$, and the book concludes with a list of 32 problems and problem areas.

Also included is history, motivation, and many calculations, for example of homotopy $2$-types via crossed modules, not available to traditional methods.

The work for the book started in the 1970s, and involved many colleagues  and research students.  The aim of the publication was to put all this work in one place in a consistent style so that it would be easier to evaluate.