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.

In characteristic zero, there is a classical equivalence between formal groups and Lie algebras, given by taking the tangent space at the identity. In the derived world, something like this still works: the (appropriately defined) tangent space to ${\Omega F}$ acquires the structure of a DGLA, and this is enough to determine ${\Omega F}$ with the group structure. In other words, we have a functor

$\displaystyle \Psi: \mathrm{Moduli}_k \rightarrow \mathrm{dgLie}, \quad \Psi(F) = T( \Omega F) = TF[-1],$

which implements the equivalence of categories.

Example 4 Given an ordinary ${k}$-scheme ${X}$ and a ${k}$-valued point ${x: \mathrm{Spec} k \rightarrow X}$, the formal completion ${\hat{X}}$ at ${x}$ is a formal moduli problem; the associated DGLA is the shift ${T_x X[-1]}$. Here ${T_x X}$ has to be interpreted in the derived sense. In other words, it’s ${\hom_k( L_{X/k}, k)}$ where ${L_{X/k}}$ is the cotangent complex. But when ${X}$ is smooth, ${T_x X}$ is concentrated in degree zero (it’s the ordinary tangent bundle) and the associated DGLA is concentrated in degree ${-1}$.

In general, the formal completions of ordinary schemes (or Deligne-Mumford stacks, or even derived schemes) always give DGLAs concentrated in strictly negative degrees. These have the property that the associated formal moduli problem, when evaluated on ordinary (discrete) artinian rings, gives discrete sets, as opposed to groupoids or fancier things—there are no infinitesimal automorphisms of the moduli problem when evaluated on ordinary rings. However, these moduli problems still give nontrivial spaces when evaluated on derived artinian rings such as ${k \oplus k[1]}$.

Example 5 Let ${G}$ be an algebraic group over the algebraically closed field ${k}$. In this case, the formal completion of ${BG}$ at the trivial torsor corresponds to the DGLA which is the ordinary Lie algebra of ${G}$, concentrated in degree zero. Here, there are infinitesimal automorphisms, even when evaluated on ordinary rings: the tangent complex (which is the desuspension of the DGLA) is concentrated in degrees ${(-\infty, 1]}$.

Example 6 The formal moduli problems associated to DGLAs concentrated in degrees ${[1, \infty)}$ are very different. By Quillen’s rational homotopy theory, such objects correspond precisely to simply connected rational spaces, and should correspond to moduli problems that are somehow “all stacky.” In particular, their tangent space is a coconnective spectrum.

As an example, let’s consider a commutative algebraic group, say ${\mathbb{G}_a}$. Then one can form ${B^2 \mathbb{G}_a}$ as some sort of higher stack. For an affine scheme ${X = \mathrm{Spec} R}$, the space of maps ${X \rightarrow B^2 \mathbb{G}_a}$ is exactly the space ${B^2 R}$. In other words, the moduli problem comes from the nonconnective ${E_\infty}$-ring ${\mathbb{Q}[x_{-2}]}$, where ${x_{-2}}$ is a free variable in degree ${-2}$. The associated space in rational homotopy theory is ${K(\mathbb{Q}, 2)}$.

The associated formal completion sends an ${A \in \mathrm{CAlg}_{sm}}$ to ${B^2 \mathfrak{m}_A}$, where ${\mathfrak{m}_A}$ is the “maximal ideal”—the fiber of the augmentation. The associated DGLA is the Lie algebra of homotopy groups of a ${K(\mathbb{Q}, 2)}$, shifted by one—so ${\mathbb{Q}}$ in degree ${1}$ and zero everywhere else.

5. Construction

In DAG X, Lurie uses various Koszul duality functors to write down the equivalence in question. It’s also possible (and more classical) to describe fairly explicitly the formal moduli problem associated to a DGLA in terms of solutions to the Maurer-Cartan equation.

Let ${\mathfrak{g}}$ be a DGLA over the field ${k}$ of characteristic zero. In the paper “DG coalgebras as formal stacks,” Hinich explicitly writes down a (formal moduli) functor

$\displaystyle \Sigma_{\mathfrak{g}}: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S}.$

In fact, Hinich does so at the level of ordinary categories itself.

Let ${\mathbf{dgart}}$ be the ordinary category of nonnegatively graded commutative dg-algebras ${A}$ (with homological grading conventions; the differential has degree ${-1}$), whose total dimension is finite and such that ${A_0}$ is local artinian with residue field ${k}$. Observe that these are not cofibrant in the usual model structure on cdgas. Let ${\mathbf{dglie}}$ be the ordinary category of dg-Lie algebras. Hinich writes down a functor

$\displaystyle \sigma: \mathbf{dgart} \times \mathbf{dglie} \rightarrow \mathcal{K}an,$

where ${\mathcal{K}an}$ is the (ordinary) category of Kan complexes. The functor has the property that it respects weak equivalences in each variable: that is, quasi-isomorphic (artinian) cdgas and quasi-isomorphic DGLAs map to homotopy equivalent Kan complexes. For each ${\mathfrak{g}}$, this defines a functor

$\displaystyle \Sigma_{\mathfrak{g}}: \mathbf{dgart} \rightarrow \mathcal{K}an ,$

which preserves weak equivalences, and, as we’ll see below, leads to a formal moduli problem in the sense previously described.

Definition 8 Given a DGLA ${\mathfrak{g}}$, a Maurer-Cartan element of ${\mathfrak{g}}$ is an element ${x \in \mathfrak{g}_{-1}}$ such that

$\displaystyle dx + \frac{1}{2}[x, x] = 0. \ \ \ \ \ (1)$

Definition 9 Given a DGLA ${\mathfrak{g}}$, the space ${MC(\mathfrak{g})}$ (the “space of solutions to the Maurer-Cartan equation”) is defined as the simplicial set ${MC( \mathfrak{g} \otimes \Omega_{\bullet})}$, where ${\Omega_{n}}$ is the algebra of polynomial differential forms on ${\Delta^n}$. It turns out to be a Kan complex.

The idea is that, given a DGLA ${\mathfrak{g}}$ and ${A \in \mathbf{dgart}}$, we take ${\mathfrak{m}_A}$ to be the maximal ideal of ${A}$. The DGLA ${\mathfrak{g} \otimes \mathfrak{m}_A}$ is a nilpotent DGLA (at least in some homotopical sense) and we take the space of solutions to the Maurer-Cartan equation, ${ \Sigma_{\mathfrak{g}}(A) \stackrel{\mathrm{def}}{=}MC( \mathfrak{g} \otimes \mathfrak{m}_A)}$. In other words, we use the pairing

$\displaystyle \sigma: \mathbf{dgart} \times \mathbf{dglie} \rightarrow \mathcal{K}an, \quad \sigma(A, \mathfrak{g}) = MC( \mathfrak{g} \otimes \mathfrak{m}_A).$

Note also that the functor ${\mathfrak{g} \mapsto MC( \mathfrak{g})}$ is what implements rational homotopy theory, for ${\mathfrak{g}}$ positively graded.

We need to use two non-obvious facts here:

• The formal localization ${\mathbf{dgart}[W^{-1}]}$ (where ${W}$ is the collection of quasi-isomorphisms) is equivalent to ${\mathrm{CAlg}_{sm}}$.
• The formal localization ${\mathcal{K}an[W^{-1}]}$ (where ${W}$ is the collection of homotopy equivalences) is the ${\infty}$-category ${\mathcal{S}}$ of spaces.

In particular, the functor ${\Sigma_{\mathfrak{g}}: \mathbf{dgart} \rightarrow \mathcal{K}an}$ descends (after applying localization) to a functor ${\mathrm{CAlg}_{sm} \rightarrow \mathcal{S}}$. It clearly sends ${k}$ itself to ${\ast}$. To see that we get a formal moduli problem, note that ${\Sigma_{\mathfrak{g}}}$ clearly preserves 1-categorical fiber products, and therefore (replacing a cartesian square of ${\mathbf{dgart}}$ by one where desired maps are surjective) preserves fiber products in a good derived sense.

To make this precise, we need to know a little about the properties of this functor ${MC}$, which is really only well-defined for nilpotent DGLAs:

• There is a good model structure on ${\mathbf{dglie}}$, where the weak equivalences are the quasi-isomorphisms and the fibrations are the surjections. (This is obtained by transfer from the model structure on chain complexes.)
• Given a fibration (resp. acyclic fibration) ${\mathfrak{g} \rightarrow \mathfrak{h}}$ of nilpotent DGLAs, the map ${MC(\mathfrak{g}) \rightarrow MC(\mathfrak{h})}$ is a fibration (resp. acyclic fibration). Consequently, the same is true for ${MC( \mathfrak{g} \otimes \mathfrak{m}_A) \rightarrow MC( \mathfrak{h} \otimes \mathfrak{m}_A)}$ (for ${A \in \mathbf{dgart}}$) without the nilpotence assumption.
• Even better, ${MC (\cdot \otimes \mathfrak{m}_A)}$ is homotopically well-defined — a quasi-isomorphism between DGLAs leads to a homotopy equivalence on ${MC( \cdot \otimes \mathfrak{m}_A)}$‘s. (Apply Ken Brown’s lemma.)
• Incidentally, given ${\mathfrak{g}}$, a Maurer-Cartan element of ${\mathfrak{g}}$ is exactly a primitive cycle of the cocommutative coalgebra ${C_*(\mathfrak{g})}$ of Chevalley-Eilenberg chains. This leads to the relation with Koszul duality — Lie algebra homology ${C_*}$ is what implements it.

Together, these assertions imply that ${\Sigma_{\mathfrak{g}}}$ is a well-defined formal moduli problem, and in fact gives a functor

$\displaystyle \mathrm{dgLie} \rightarrow \mathrm{Moduli}_k.$

Classically, it was observed that many formal moduli problems arose in this manner.