After the effort invested in proving the general theorem on acyclic models, it is time to apply it to topology. First, let us prove:

Theorem 5 Suppose ${f, g: X \rightarrow Y}$ are homotopic. Then the maps ${f_*, g_*: H_*(X) \rightarrow H_*(Y)}$ are equal.

Proof: Suppose ${H: X \times [0,1]}$ is a homotopy with ${H(\cdot, 0) = f, H(\cdot, 1) = g}$. Then the maps ${f,g}$ factor as

$\displaystyle X \rightrightarrows^{i_0}_{i_1} X \times [0,1] \rightarrow^H Y$

so if we show that the inclusions ${i_0,i_1}$ sending ${x \in X}$ to ${(x, 0), (x,1)}$ induce equal maps on homology, we will be done.

Write ${I = [0,1]}$ for simplicity. For each space ${X}$, the maps ${i_0, i_1: X \rightarrow X \times I}$ are natural. More precisely, if ${X \rightarrow X, X \rightarrow X \times I}$ are the two functors ${\mathbf{Top} \rightarrow \mathbf{Top}}$, then ${i_0, i_1}$ are natural transformations between them.

I hope I’ll get a chance to continue with blogging about descent soon; for now, I’m swamped with other things and mildly distracted by algebraic topology.

There are various theorems in algebraic topology whose proofs can require significant computation. For instance, the homotopy invariance of singular homology, the Eilenberg-Zilber theorem (which relates the singular chain complex ${C_*(X \times Y)}$ of a product ${X \times Y}$ to the singular complexes ${C_*(X), C_*(Y)}$). On the other hand, there is also a strictly categorical framework in which these theorems may be proved. This is the method of acyclic models, to which the present post is dedicated. Let us start with the first example.

Theorem 1 Suppose ${f, g: X \rightarrow Y}$ are homotopic. Then the maps ${f_*, g_*: H_*(X) \rightarrow H_*(Y)}$ are equal.

One way to give an explicit proof is to argue geometrically, decomposing the space ${\Delta^n \times I}$ into a bunch of ${n}$-simplices. I always found this confusing. So I will explain how category theory does this magically and gives a natural chain homotopy. To start, note that it is enough to show that the two inclusions ${X \rightarrow X \times I}$ sending ${x \rightarrow (x,0), (x,1)}$ induce the same maps on homology. This is a standard argument. (more…)