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

In other words, a formal moduli problem was a functor

\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S} \ (= \text{spaces}),

which was required to send {k} itself to a point, and satisfy a certain cohesiveness condition: {F} respects certain pullbacks in {\mathrm{CAlg}_{sm}} (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 {\mathrm{Moduli}_k} and the {\infty}-category {\mathrm{dgLie}} of DGLAs over {k}.


4. Overview

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

It follows that we can form the loop space {\Omega F = \ast \times_F \ast} of {F}, which is a new formal moduli problem. However, {\Omega F} 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 {F} is equivalent to knowledge of {\Omega F} together with its group structure: we can recover {F} as {B \Omega F} (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via {B, \Omega}) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality. (more…)

Let {k} be a field. The commutative cochain problem over {k} is to assign (contravariantly) functorially, to every simplicial set {K_\bullet}, a commutative (in the graded sense) {k}-algebra {A(K_\bullet)}, which is naturally weakly equivalent to the algebra {C^*(K_\bullet, k)} of singular cochains (with {k}-coefficients). We also require that {A(K_\bullet) \rightarrow A(L_\bullet)} is a surjection whenever {L_\bullet \subset K_\bullet}. Recall that {C^*(K_\bullet, k)} 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 {C^*(K_\bullet, k)}.

If {k} 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 {k = \mathbb{Q}}, given by the polynomial de Rham theory. In this post, I will explain this. (more…)