Deformation theory and DGLAs II

Let be a field of characteristic zero. In the previous post, we introduced the category (i.e., -category) of formal moduli problems over . A formal moduli problem over is a moduli problem, taking values in spaces, that can be evaluated on the class of “derived” artinian -algebras with residue field : this was the category introduced in the previous post.

In other words, a formal moduli problem was a functor

which was required to send itself to a point, and satisfy a certain cohesiveness condition: respects certain pullbacks in (which corresponded geometrically to pushouts of schemes).

The main goal of the series of posts was to sketch a proof of (and define everything in) the following result:

Theorem 7 (Lurie; Pridham) There is an equivalence of categories between and the -category of DGLAs over .

 

4. Overview

Here’s a rough sketch of the idea. Given a formal moduli problem , we should think of as something like a small space, concentrated at a point but with lots of “infinitesimal” thickening. (Something like a .) Moreover, has a canonical basepoint corresponding to the “trivial deformation.” That is, we can think of as taking values in pointed spaces rather than spaces.

It follows that we can form the loop space  of , which is a new formal moduli problem. However, has more structure: it’s a group object in the category of formal moduli problems — that is, it’s some sort of derived formal Lie group. Moreover, knowledge of the original is equivalent to knowledge of together with its group structure: we can recover as (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via ) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality.

 

The commutative cochain problem and the polynomial de Rham functor

Let be a field. The commutative cochain problem over is to assign (contravariantly) functorially, to every simplicial set , a commutative (in the graded sense) -algebra , which is naturally weakly equivalent to the algebra of singular cochains (with -coefficients). We also require that is a surjection whenever . Recall that is an associative algebra, but it is not commutative; the commutativity only appears after one takes cohomology. The commutative cochain problem attempts to find an improvement to .

If has finite characteristic, this problem cannot be solved, owing to the existence of nontrivial cohomology operations. (The answers at this MathOverflow question are relevant here.) However, there is a solution for , given by the polynomial de Rham theory. In this post, I will explain this.