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. (more…)