The next step (in our discussion, started yesterday, of the cotangent complex) will be to define a model structure on the category of algebras over a fixed ring. Model structures allow one to define derived functors in a non-abelian setting. The key idea is that, when you want to derive an additive functor $F$ on an object $X$ in some abelian category, you replace $X$ by a projective resolution and evaluate the functor $F$ on this resolution. (And then, take its homology; in the setting of derived categories, though, one usually just takes $F$ of the projective resolution and leaves it at that.) Because $F$ on projective resolutions is much better behaved than $F$ simply on modules, the derived functor is a nice replacement.

The intuition is that a projective resolution is a cofibrant approximation to the initial object, in the language of model categories (which is often seen as a non-abelian version of classical homological algebra). This is actually precisely true if one imposes the usual model structure on bounded-below chain complexes for modules over a ring, for instance.

In constructing the cotangent complex, we are trying to derive the (highly non-abelian) functor of abelianization, which as we saw was closely related to the construction of differentials. This functor was defined on rings under a fixed ring $A$ and over a fixed ring $B$, which is not anywhere near an abelian category. So we will need the language of model categories, and today we shall construct a model structure on a certain class of categories.

In deriving an additive functor, one ultimately applies it not on the initial abelian category, but the larger category of chain complexes. Here the analogy extends again: by the Dold-Kan correspondence (which I recently talked about), this is equivalent to the category of simplicial objects in that category. The appropriate approach now seems to be to define a model structure not on $A$-algebras over $B$, but on the category of simplicial $A$-algebras over $B$. (more…)

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…)