Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a functor from local Artin CDGA's to homotopy types satisfying a certain sheaf condition. If the commutativity condition is weakened to an $E\_n$ condition, any formal moduli problem is (uniquely) representable by an $E\_n$ algebra; in the commutative case, a formal moduli problem is not necessarily representable by a CDGA, but rather by a (derived) Lie algebra. 


There's an intuitive picture I like for this in a special case, and I want to understand how it fits into the general picture. Namely, say that $G$ is an affine algebraic Lie group. Let $BG$ be its classifying stack. Then the deformation problem of maps to $BG$ (relative to a choice of point $\*\to BG$) is classified by the Lie group $\mathfrak{g}$.


Of course this picture involves a group rather than a "formal stack", which is my intuition for (the opposite category to) formal moduli problems, and I am curious how hard it is for a formal moduli problem to be representable by a group object. 


From the point of view of *spaces*, going from groups to spaces "is not very hard": the category of groups is equivalent (via the delooping functor) to the category of connected pointed spaces.


On the other hand, for $E\_n$ algebras, the category of cogroup objects is equivalent to the category of (suitably defined) $(n,1)$-commutative Hopf algebras, which is more closely related (via Koszul duality) to the category of $E\_{n+1}$ algebras than to $E\_n$ algebras; in particular, the delooping functor from group objects to $E\_n$ algebras is far from fully faithful. 


Now by formal nonsense, there is a loop-deloop pair of adjoint functors $$B:GFMP\leftrightarrows FMP:\Omega,$$ where $FMP$ is the category of ($E\_\infty$) formal moduli problems and $GFMP$ is the category of cogroup objects in $FMP$. It seems natural to ask the following questions, to which I don't know the answer: 


1. Is $B$ fully faithful, and if so what is its image?
- Is there an analogue on the level of (co)group objects in Lie algebras to the Koszul duality functor from $(n,1)$ Hopf algebras to $E\_{n+1}$ algebras?


A natural further question, which I suspect is harder, is to ask whether there is an "algebraic point of view" via Lie-like objects for $E\_\infty$ objects in $FMP$. 



Connectedness, loops and formal moduli problems
The presentation of the formal moduli problems story in Gaitsgory-Rosenblyum *A Study in Derived Algebraic Geometry, Vol 2* may be what you are looking for. We review it here (in the case over $\mathrm{Spec}\, k$ for a field $k$ of characteristic zero, that the question concerns):


**1. Looping/delooping equivalence in formal DAG**


Just like the familiar adjoint equivalence in the homotopy theory of spaces
$$\mathrm B: \mathrm{Grp}\_{\mathbb E\_1}(\mathcal S)\simeq \mathcal S\_\*^{\ge 1}:\Omega,$$
there is an analogous adjoint equivalence in formal DAG
$$\mathrm B:\mathrm{Grp}\_{\mathbb E\_1}(\mathrm{FMP}\_k) \simeq \mathrm{FMP}\_k:\Omega,$$
where the loop space functor is in both cases given by $\Omega X = \mathrm{pt}\times\_X \mathrm{pt}$, as per usual in homotopical/$\infty$-categorical things. Note that formal moduli problems are already inherently pointed, by the assumption that $X(k)$ is contractible.


**2. Formal groups and Lie algebras**


Now, $\mathrm{Grp}\_{\mathbb E\_1}(\mathrm{FMP}\_k) = \mathrm{FGrp}\_k$ is (an incarnation) of the $\infty$-category of (derived) formal groups over $k$. Thanks to the characteristic zero assumption, there is a further equivalence
$$
\mathrm{Lie}:\mathrm{FGrp}\_k \simeq \mathrm{LieAlg}\_k : \exp
$$
with derived Lie algebras (as modelled for instance by dg Lie algebras). Just like expected, the derived Lie algebra $\mathfrak g$ corresponding to the formal group $G$ is $\mathfrak g = T\_{G, e}$, the tangent fiber at the unit.


**3. Formal moduli problems and Lie algebras**


The celebrated Lurie-Pridham identification between formal moduli problems and derived Lie algebras is precisely the composite of these two equivalences of $\infty$-categories
$$
\mathrm{FMP}\_k\simeq \mathrm{Grp}\_{\mathbb E\_1}(\mathrm{FMP}\_k)=\mathrm{FGrp}\_k\simeq \mathrm{LieAlg}\_k.
$$
That is


* It sends a formal moduli problem $X$ to
$$
\mathrm{Lie}(\Omega X) = T\_{\Omega X, e} = T\_{X, x\_0}[-1],
$$
where $x\_0$ is the base-point of $X$, unique up to a contractible space of choices $X(\kappa)$. This shifted tangent fiber carries a canonical Lie algebra structure coming from (i.e. as the Lie algebra of) the $\mathbb E\_1$-group structure of $\Omega X$.
* The inverse equivalence $\Psi: \mathrm{LieAlg}\_k \simeq \mathrm{FMP}\_k$ is then given by $\Psi(\mathfrak g)= \mathrm B\exp(\mathfrak g)$.


**4. Lurie's formula for $\Psi$**


You may justifiably complain that this description of the inverse functor $\Psi$ does not look the same as the one in Lurie's writing. Let's see how to get it in that form.


Let's assume that the formal moduli problem $\Psi(\mathfrak g)$ is formally affine (true under some finiteness assumptions on $\mathfrak g$), in the sense that
$$
\mathrm B\exp(\mathfrak g)= (\mathrm{Spf}\,k)/\exp(\mathfrak g)\simeq \mathrm{Spf} \,k^{\mathfrak g}.
$$
Here of course the formal spectrum is the usual functor $\mathrm{Spf}:(\mathrm{CAlg}^\mathrm{aug}\_k)^\mathrm{op}\to \mathrm{FMP}\_k$. The derived Lie algebra invariants may be computed via the Chevalley-Eilenberg complex, thus $k^\mathfrak g\simeq {C}^\*(\mathfrak g)$.
Then for any Artinian $k$-algebra $A$ we have
$$ (\Psi(\mathfrak g))(A) \simeq \mathrm{Map}\_{\mathrm{CAlg}^\mathrm{aug}\_k}({C}^\*(\mathfrak g), A). \qquad \quad(1)
$$
This is why Lurie tells us to consider the "Koszul duality functor" $\mathfrak D: (\mathrm{CAlg}\_k^\mathrm{aug})^\mathrm{op}\to \mathrm{LieAlg}\_k,$ right-adjoint to the Chevalley-Eilenberg cochains functor. Indeed, (through a little use of the finiteness of $\mathfrak g$) we get
$$(\Psi(\mathfrak g))(A) \simeq \mathrm{Map}\_{\mathrm{LieAlg}\_k}( \mathfrak D(A), \mathfrak g),\qquad \qquad(2)
$$
which is how Lurie tells us to define $\Psi$.


Note that this is going the other way than the $C^\*\dashv\mathfrak D$ adjunction. Indeed, without finiteness assumptions on $\mathfrak g$, the formal moduli problem $\Psi(\mathfrak g)$ will not necessarily be formally affine, and the formula (1) will not necessarily work. On the other hand, as Lurie teaches us, formula (2) *will always work*.


**5. Loose ends**


If I understand the original question correctly, this is presenting the formal moduli problem story precisely like the intuitive picture mentioned. Indeed: any formal moduli problem may be written as $X\simeq \mathrm B G$ for a derived formal group $G$, and the formal moduli problem (of mapping into) $\mathrm BG$ is classified by the Lie algebra $\mathfrak g$.


That said, the question makes analogy with the $\mathbb E\_n$-algebra version of this story too. I am a little confused about the points raised - in particular, it seems like the following is asserted: a formal moduli problem on $\mathbb E\_n$-algebras is *represented* by an $\mathbb E\_n$-algebra. That is, so far as I understand, incorrect. Instead, any FMP on $\mathbb E\_n$-algebras in *classified* by an $\mathbb E\_n$-algebra, in a way somewhat analogous to the way that the functor $\mathfrak D$ presents formal moduli problems in the commutative case with Lie algebras.


In particular, there seems to be no need to think about cogroup objects, as the equivalence of $\infty$-categories $\mathrm{FMP}\_k^{\mathbb E\_n}\simeq \mathrm{Alg}^\mathrm{aug}\_{\mathbb E\_n}$ is covariant.


$\qquad$




---


**Edit:** A question was raised in the comments whether group objects in $\mathrm{FMP}\_k^{\mathbb E\_n}\simeq \mathrm{Alg}\_{\mathbb E\_n}^\mathrm{aug}$ correspond to $\mathbb E\_{n+1}$-algebras. Unless I am misunderstanding the question, the answer is negative.


A monoid structure (of which a group structure is a particular example of) on an $\mathbb E\_n$-algebra $A$ is given by a map $A\times A\to A$, plus coherence data. On the other hand, an additional $\mathbb E\_1$-algebra structure on $A$ (which is equivalent to making it into an $\mathbb E\_{n+1}$-algebra by Dunn Additivity) is given by a map $A\otimes\_k A\to A$, plus coherence data. So a group object in $\mathbb E\_n$-algebras, and an $\mathbb E\_{n+1}$-algebra are different structures.


This might feel a little weird because we're used to tensor products corresponding to products of schemes. Alas, unlike the contravariant $\mathrm{Spec}$, the equivalence $\Psi:\mathrm{Alg}^{\mathrm{aug}}\_{\mathbb E\_n}\simeq \mathrm{FMP}\_k^{\mathbb E\_n}$  is covariant. (PS: another difficulty: the tensor product only becomes the coproduct on the level of $\mathbb E\_\infty$-algebras, not $\mathbb E\_n$-algebras, so even a contravariant equivalence wouldn't necessarily do the trick).


Rather, the situation is the same as in the $\mathbb E\_\infty$-case above: there is an equivalence
$$\mathrm B:\mathrm{Grp}\_{\mathbb E\_1}(\mathrm{FMP}\_k^{\mathbb E\_n})\simeq \mathrm{FMP}\_k^{\mathbb E\_n}:\Omega.$$
This is a special case of the following general phenomenon: for any operad $\mathcal O$, the loops functor $\Omega: \mathrm{Alg}\_{\mathcal O}(\mathrm{Mod}\_k)\to \mathrm{Grp}\_{\mathbb E\_1}(\mathrm{Alg}\_{\mathcal O}(\mathrm{Mod}\_k))$ is an equivalence of $\infty$-categories (*A Sudy in Derived Algebraic Geometry, Vol 2*, Chapter 6, Proposition 1.6.4). The above claims are special cases for $\mathcal O$ the Lie operad and the $\mathbb E\_n$-operad respectively.


