Formal moduli problems and formal derived stacks
Damien Calaque Julien Grivaux


Constructing exact representations of quantum

 many-body systems with deep neural networks
    Masatoshi Imada
    December 30, 2023
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To the memory of Jean-Louis Koszul
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.G. H. Hardy – A Mathematician's Apology


This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy, which gives a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasizing the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of Gaitsgory and Rozenblyum. 







§ INTRODUCTION


The aim of these lecture notes is to present a recent work, due independently to Lurie and Pridham, concerning an equivalence of infinity-categories between formal moduli problems and differential graded Lie algebras. The link between deformation theory and dg-Lie algebras has a long history, which is as old as deformation theory itself. One of the first occurrence of a relation between these appears in Kodaira–Spencer's theory of deformations of complex compact manifolds (see <cit.>): if X is such a manifold, then infinitesimal deformations of X over a local artinian ℂ-algebra A with maximal ideal m_A correspond to Maurer–Cartan elements of the dg-Lie algebra 𝔤_X ⊗_ℂ m_A modulo gauge equivalence, where 𝔤_X is the dg-Lie algebra Γ(X, 𝒜^0, ∙_X ⊗T_X), the differential being the ∂ operator. This very concrete example illustrates the following general principle: to any sufficiently nice dg-Lie algebra it is possible to attach a deformation functor (see e.g. <cit.>), which is defined on local artinian algebras as follows: 

    Def_𝔤 (A) = MC(𝔤⊗ m_A) /  gauge equivalence

This correspondence was carried out by many people, including Quillen, Deligne and Drinfeld. In a letter to Schechtman <cit.>, Drinfeld introduced in 1988 the Derived Deformation Theory (DDT) philosophy: 


Every (dg/derived) deformation problem is controlled by a dg-Lie (or L_∞-) algebra.


Since then, there has been a lot of work confirming this philosophy. We can refer the interested reader to the expository paper <cit.> for more details. Let us give now a few examples, which are related to derived algebraic geometry.



In 1997 Kapranov <cit.> studied deformations of local systems: given a affine algebraic group G and a G-local system E on a manifold S, deformations of E are encoded by a formal dg-scheme RDef(E) whose tangent complex 𝕋_[E]RDef(E) at the closed point [E] is RΓ(S, ad(E)) [1]. The next result is that 𝕋_[E]RDef(E) [-1] is naturally an L_∞-algebra, the corresponding Lie algebra structure on the cohomology groups H^i(S, ad(E)) being induced by the natural Lie structure on ad(E). 
This last observation is not at all a hazard, and reflects a far more general phenomenon: in loc. cit., Kapranov proved[This idea is also present in Hinich's <cit.>.] that for any smooth dg-scheme X, the shifted tangent complex 𝕋_xX[-1] at a point x :* → X carries an L_∞-algebra structure that determines the formal geometry of X around x. A modern rephrasing of Kapranov's result can be presented as follows: for any derived affine scheme X=Spec(A) with a -point x=Spec(A → k), we have that 


    𝕋_xX[-1]  ≃𝕋_A⊗_A [-1] ≃𝕋_ / A≃RDer_A (,)


and thus 𝕋_xX[-1] is a derived Lie algebra. 
Let us explain another folklore calculation[We don't know a precise reference for that, but it seems to be known among experts. ] confirming this result in another concrete example. Let X=[G_0/G_1] be a 1-stack presented by a smooth groupoid G_1 s⟶
t⟶ G_0, where  G_0 and G_1 are smooth affine algebraic varieties over a field . If x is a -point of G_0, then the tangent complex 𝕋_[x]X of X at the -point [x] is a 2-step complex V → W where
V=T^s_id_x G_1 sits in degree -1, W=T_xG_0 sits in degree 0, and the arrow is given by the differential of the target map t at id_x. 
According to Kapranov's result, there should exist a natural L_∞-structure on 𝕋_[x]X[-1]. 
Let us sketch the construction of such a structure. 
Observe that L:=(T^s G_1)_| G_0 has the structure of a (, A) Lie algebroid, where G_0=Spec(A). In particular we have a -linear Lie bracket Λ^2_ L → L and an anchor map L→ TG_0=Der_(A). 
Very roughly, Taylor components at x of the anchor give us maps S^n(W)⊗ V → W, and Taylor components of the bracket at x give us maps S^n(W)⊗Λ^2 (V) → V. 
These are the only possible structure maps for an L_∞-algebra concentrated in degrees 0 and 1. 
Equations for the L_∞-structure are guaranteed from the axioms of a Lie algebroid. 
As a particular  but illuminating case, we can consider the stack BG for some affine algebraic group G. Here G_0={*} and G_1=G. Then 𝕋_* BG[-1] ≃𝔤 where 𝔤 is the Lie algebra of G, and the L_∞-structure is simply the Lie structure of 𝔤.
Let us explain how all this fits in the DDT philosophy. Any derived affine stack Spec(A) endowed with a -point x defines a representable deformation functor B →Hom(B, A), the Hom functor being taken in the category of -augmented commutative differential graded algebras (cdgas). Hence it should be associated to a dg-Lie (or indifferently a L_∞-) algebra. This dg-Lie algebra turns out to be  exactly 𝕋_x   Spec(A)[-1], endowed with Kapranov derived Lie structure. In this way, we get a clear picture of the DDT philosophy for representable deformation functors. However, the work of Lurie and Pridham goes way beyond that; what they prove is the following statement: 


[<cit.>]
Over a base field of characteristic zero, there is an equivalence of ∞-categories between formal moduli problems and dg-Lie algebras. 

A very nice exposition of the above Theorem, with several examples and perspectives, is in Toën's <cit.>. 

Lurie's work <cit.> extends to E_n-deformation problems.
Pridham's work <cit.> has some extension to the positive characteristic setting. 
In a forthcoming paper, Brantner and Mathew <cit.> actually generalize the above result over any field, proving that there is an equivalence of ∞-categories between formal moduli problems and so-called partition Lie algebras. 

In these lectures, we will work in characteristic 0, and our goal is to provide a version of Lurie–Pridham Theorem in families. In other words, we are aiming at first at a statement “over a cdga A”, and build an extension from the affine case X=Spec(A) to an arbitrary derived Artin stack X. One shall be very careful as there are different meanings to “over a base”. We will provide two variants. The second one, which is the one we are interested in, will actually rather be named “under a base”. 
Split families of formal moduli problems
In the same year 1997, Kapranov <cit.> considered the family of all formal neighborhood of points in a smooth algebraic variety X, which is nothing but the formal neighborhood X̂×̂ ̂X̂ of the diagonal in X× X. He showed that the sheaf T_X[-1]≃𝕋_X/X× X is a Lie algebra object in D^b(X), whose Chevalley–Eilenberg cdga gives back the structure sheaf of X̂×̂ ̂X̂. It is important to observe here that we have a kind of formal stack (X̂×̂ ̂X̂) that lives both over X and under X. This example is a prototype for split families of formal moduli problems: a generalization of Lurie–Pridham Theorem has recently been proven by Benjamin Hennion <cit.> in 2013, in the following form: 
[<cit.>]
Let A be a noetherian cdga concentrated in nonpositive degrees, and let X be a derived Artin stack of finite presentation.


  – A-pointed A-linear formal moduli problems are equivalent (as an ∞-category) to A-dg-Lie algebras. 

  – X-pointed X-families of formal moduli problems are equivalent to Lie algebra objects in QCoh(X).
 


Formal moduli problems under a base
In 2013, Caldararu, Tu and the first author <cit.> looked at the formal neighborhood Ŷ of a smooth closed subvariety X into a smooth algebraic variety Y. They prove that the relative tangent complex 𝕋_X/Y is a dg-Lie algebroid whose Chevalley–Eilenberg cdga gives back the structure sheaf of Ŷ. Similar results had been previously proven by Bhargav Bhatt in a slightly different formulation (see <cit.>). We also refer to the work of Shilin Yu <cit.>, who obtained parallel results in the complex analytic context. Note that this time, the derived scheme Ŷ doesn't live anymore over X (although it still lives under X). Keeping this example in mind, we expect the following generalization of Hennion's result: 


  – A-pointed -linear formal moduli problems are equivalent to dg-Lie algebroids over A, which has recently been proven by Joost Nuiten in <cit.>. This appears as Theorem <ref> of the present survey. 

  – X-pointed formal moduli problems are equivalent to dg-Lie algebroids over X. A weaker version of this appears as Theorem <ref> in the present paper. 


These results are somehow contained in the recent book <cit.> of Gaitsgory–Rozenblyum, though in a slightly different formulation[Actually, Gaitsgory and Rozenblyum define Lie algebroids as formal groupoids. Our results, as well as Nuiten's one, thus somehow indirectly give a formal exponentiation result for derived Lie algebroids. ]. 

We finally observe that it would be very interesting to understand the results from <cit.>, describing the derived geometry of locally split first order thickenings X↪ S of a smooth scheme X, in terms of Lie algebroids on X. 



 §.§ Description of the paper




  1 We present the basic notions necessary to understand Lurie–Pridham result, relate them to more classical constructions in deformation theory and provide some examples. We finally give a glimpse of Lurie's approach for the proof.

  2 A general framework for abstract formal moduli problems is given in details. In 2.1, we recall Lurie's deformation contexts. In 2.2, we deal with dual deformation contexts. In 2.3, we introduce the useful notion of Koszul duality context, which is a nice interplay between a deformation context and a dual deformation context. In 2.4 we talk about morphisms between these, which is a rather delicate notion. In 2.5 we restate some results of Lurie using the notion of Koszul duality context, making his approach a bit more systematic. In 2.6 we discuss tangent complexes.  

  3 We extend former results from dg-Lie algebras to dg-Lie algebroids. We prove Hennion's result <cit.> in 3.1. Most of the material in 3.2, 3.3 and 3.4 happens to be already contained in the recent preprints <cit.> of Joost Nuiten. 3.5 presents some kind of base change functor that will be useful for functoriality in the next Section. 

  4 We explain how to globalize the results of 3. In 4.1 we introduce formal (pre)stacks and formal thickenings, mainly following <cit.> and <cit.>, and compare these with formal moduli problems. In 4.2 we state a consequence, for Lie algebroids, of the previous , and we sketch an alternative proof that is based on Koszul duality contexts and morphisms thereof. In 4.3 we show that formal thickenings of X fully faithfully embed in Lie algebroids on X, and we conjecture that this actually is an equivalence. 




 §.§ Acknowledgements
 

The first author thanks Mathieu Anel, Benjamin Hennion, Pavel Safronov, and Bertrand Toën for numerous discussions on this topic. 
This survey paper grew out of lecture notes, taken by the second author, of a 3 hours mini-course given by the first author at the session DAGIT of the États de la Recherche, that took place in Toulouse in June 2017. We both would like to thank the organizers for a wonderful meeting, as well as the participants to the lectures for their enthusiasm. 
Damien Calaque acknowledges the financial support of the Institut Universitaire de France, and of the ANR grant “SAT” ANR-14-CE25-0008. Julien Grivaux acknowledges the financial support of the ANR grant “MicroLocal” ANR-15-CE40-0007, and ANR Grant “HodgeFun” ANR-16-CE40-0011. 



 §.§ Credits




  – 1 actually doesn't contain more than the beginning of <cit.>, and 2 is somehow a nice re-packaging of the general construction presented in loc.cit. However, we believe the formalism of 2 is way more user-friendly for the reader wanting to apply <cit.> in concrete situations.

  – Even though we thought they were new at the time we gave these lectures, Joost Nuiten independently obtained results that contain and encompass the material that is covered in 3 (see the two very nice papers <cit.>, that appeared while we were writing this survey). These results must be attributed to him. 

  – Global results presented in 4 rely on the general theory of formal derived (pre)stacks from <cit.> (see also <cit.>). Some of these results seem to be new. 




 §.§ Notation

Below are the notation and conventions we use in this paper. 



  
Categories, model categories and ∞-categories



  – We make use all along of the language of ∞-categories. We will only use (∞,1)-categories, that is categories where all k-morphisms for k ≥ 2 are invertible. By ∞-categories, we will always mean (∞, 1)-categories.

  – Abusing notation, we will denote by the same letter an ordinary (i.e. discrete) category and its 
associated ∞-category.

  – If M is a given model category, we write 𝒲_M for its subcategory of weak equivalences, 
and M:=M[𝒲_M^-1] for the associated ∞-category (unless otherwise specified, 
localization is always understood as the ∞-categorical localization, that is Dwyer-Kan simplicial localization). 

  – Going from M to M is harmless regarding (co)limits: homotopy (co)limits in M correspond to ∞-categorical (co)limits in M.

  – Conversely, any presentable ∞-category can be strictified to a model category (i.e. is of the form M for some model category M). In the whole paper, we will only deal with presentable ∞-categories.

  – Let  be the ∞-category of (small) ∞-categories. It can be obtained as the 
∞-category associated with the model category  of (small) quasi-categories. 

  – We denote by  the model category of simplicial sets (endowed with Quillen's model structure). The associated ∞-category is denoted by , it is equivalent to ∞ Grpd. We denote by  its stabilization (see <ref>), it is the category of spectra.

  – When writing an adjunction horizontally (resp. vertically), we always write the left adjoint above (resp. on the left) and the right adjoint below (resp. on the right). This means that when we write an adjunction as F: C ⟶
⟵  D:G, F is the left adjoint.





  
Complexes



  – The letter  will refer to a fixed field of characteristic zero. 


  – The category _ is the model category of unbounded (cochain) complexes of -modules from <cit.> (it is known as the projective model structure), and _ is its associated ∞-category. For this model structure, weak equivalences are quasi-isomorphisms, and fibrations are componentwise surjective morphisms.

  – The category _^≤ 0 is the model category of complexes of -modules sitting in nonpositive degrees, and _^≤ 0 is its associated ∞-category. For this model structure, fibrations are componentwise surjective morphisms in degree ≤-1.

  – In the sequel, we will consider categories of complexes with an additional algebraic structure (like commutative differential graded algebras, or differential graded Lie algebras). They carry model structures for which fibrations and weak equivalences are exactly the same as the ones for complexes. It is such that the “free-forget” adjunction with _ (or, _^≤0) is a Quillen adjunction. We refer the reader to the paper <cit.> for more details. 




  
Differential graded algebras



  – The category _ denotes the model category of (unbounded) unital commutative differential graded -algebras (that is commutative monoids in _), and _ denotes its associated ∞-category.

  – A (unbounded) unital commutative differential graded -algebra will be called a “cdga”. 

  – The category _ is the slice category _/, i.e. the category of augmented -algebras. It is equivalent to the category _ of non-unital commutative differential graded algebras. This equivalence is actually a Quillen equivalence. 

  – The category ^≤ 0_ denotes the model category of cdgas over  sitting in nonpositive degree.
The associated ∞-category is denoted by  ^≤ 0_.

  – For any A∈_ we write _A for the model category of left A-modules, and _A 
for its associated ∞-category. 

  – Note that _A is a stable ∞-category. 
This can be deduced from the fact that _A is a triangulated dg category. The looping and delooping functors 
Ω_* and Σ_* are then simply given by degree shifting (*)[-1] and (*)[1], respectively. 

  – We define _A is then the model category of A-algebras (commutative monoids in _A), and _A is the 
corresponding ∞-category. 

  – All these definitions have relative counterparts: if A is a cdga and if B is an object of _A, then _A/B is the slice category (_A)_/B of relative A-algebras over B.




  
Differential graded Lie algebras and L_∞-algebras



  – We denote by _ the model category of (unbounded) diffential graded Lie algebras over  (i.e. Lie algebra objects in _). The associated ∞-category is denoted by _.

  – A differential graded Lie algebra over  will be called a “dgla”. 

  – Remark that _ is equivalent to the localization of the category of L_∞-algebras, with morphisms being ∞-morphisms, with respect to ∞-quasi-isomorphisms (see e.g. <cit.>). 






§ INTRODUCTION TO POINTED FORMAL MODULI PROBLEMS FOR COMMUTATIVE ALGEBRAS




 §.§ Small augmented algebras
 
For any dg-algebra A and any A-module M, we can form the square zero extension of A by M; we denote it by A ⊕ M where there is no possible ambiguity. We set first some crucial definitions for the rest of the paper: 
 Recall that _ is the ∞-category of augmented cdgas.


  – A morphism in _ is called elementary if it is a pull-back[Since we are working in the ∞-categorical framework, pullback means homotopy pullback.] of →⊕[n] for some n≥1, where →⊕[n] is the square zero extension of  by [n].

  – A morphism in _ is called small if it is a finite composition of elementary morphisms. 

  – An object in _ is called small if the augmentation morphism ϵ A → is small. 


We denote by _ the full[Hence we allow all morphisms in the category _, not only small ones.] sub-∞-category of small objects of _. Small objects admits various concrete equivalent algebraic characterizations: 

[<cit.>]
An object A of _ is small if and only if the three following conditions hold: 


  – H^n(A)={0} for n positive and for n sufficiently negative. 

  – All cohomology groups H^n(A) are finite dimensional over .

  – H^0(A) is a local ring with maximal ideal 𝔪, and the morphism H^0(A)/𝔪→ is an isomorphism.

Moreover, a morphism A → B between small objects is small if and only if H^0(A) → H^0(B) is surjective. 

 
Let us make a few comments on this statement, in order to explain its meaning and link it to classical results in commutative algebra and deformation theory.


  – Observe that small algebras are nothing but dg-artinian algebras concentrated (cohomologically) in nonpositive degree.

  – To get a practical grasp to the definitions of elementary and small morphisms, it is necessary to be able to compute homotopy pullbacks in the model category _. This is a tractable problem since the model structure on _ is fairly explicit, and _ is a slice category of _.

  – If a small object A is concentrated in degree zero, the theorem says that A is small if and only if A is a local artinian algebra with residue field . Let us explain concretely why this holds (the argument is the same as in the general case). If A is a local artinian algebra, then A can be obtained from the residue field as a finite sequence of (classical) small extensions, that is extensions of the form

    0 → (t) → R_2 → R_1 → 0

where R_1, R_2 are local artinian with residue field , and (t) is the ideal generated by a single element t annihilated by the maximal ideal of R_2 (hence it is isomorphic to the residue field ). In this way we get a cartesian diagram[The cdga structure we put on the cone is the natural one: we require that (t) has square zero.]

    R_2 [r]  [d]    [d] 
    cone( (t) → R_2 ) [r]    ⊕[1]

in _. Since the bottom horizontal map is surjective in each degree, it is in particular a fibration. Therefore this diagram is also cartesian in _, and is isomorphic to a cartesian diagram of the form

    R_2 [r]  [d]    [d] 
    
    R_1  [r]    ⊕[1]

in _. Hence the morphism R_2 → R_1 is elementary.

  – An important part of classical deformation theory in algebraic geometry is devoted to formal deformations of algebraic schemes. For a complete account, we refer the reader to the book <cit.>. Following the beginning of <cit.>, we will explain quickly how small morphisms fit in this framework. Given a algebraic scheme Z over  (that will be assumed to be smooth for simplicity), the formal deformation theory of Z deals with equivalence classes of cartesian diagrams

    Z [r] [d]    ℨ[d] 
    
    Spec() [r]     Spec(A)

where A is a local artinian algebra with residue field k. This construction defines a deformation functor Def_Z from the category of local artinian algebras to sets.
It is possible to refine the functor Def_Z to a groupoid-valued functor that associates to any A the groupoid of such diagrams. Since any groupoid defines a homotopy type (by taking the nerve), we get an enriched functor 𝒟ef_Z from local artinian algebras to the homotopy category of topological spaces, such that π_0 (𝒟ef_Z)=Def_Z.
The first important case happens when A=[t]/t^2. In this case, Kodaira-Spencer theory gives a bijection between isomorphism classes of deformations of X over Spec ([t]/t^2) and the cohomology group H^1(Z, T_Z). In other words, H^1(Z, T_Z) is the tangent space to the deformation functor Def_Z. The next problem of the theory is the following: when can an infinitesimal deformation θ in H^1(Z, T_Z) be lifted to Spec(k[t]/t^3)? The answer is: exactly when [θ, θ] vanishes in H^2(Z, T_Z). This can be interpreted in the framework of derived algebraic geometry as follows: the extension

    0 → (t^2) →[t]/t^3 →[t]/t^2 → 0

yields a cartesian diagram

    [t]/t^3 [r]  [d]    [d] 
    [t]/t^2 [r]    ⊕[1]

in _. This gives a fiber sequence of homotopy types

    𝒟ef_Z ([t]/t^3) →𝒟ef_Z ([t]/t^2)  →𝒟ef_Z (⊕[1])

hence a long exact sequence

    ⋯→π_0(𝒟ef_Z ([t]/t^3)) →π_0 (𝒟ef_Z ([t]/t^2)) →π_0 (𝒟ef_Z (⊕[1]).

It turns out that the set π_0 (𝒟ef_Z (⊕[1]) of equivalence classes of deformations of Z over the derived scheme Spec(⊕[1]) is isomorphic to H^2(Z, T_Z). Hence the obstruction class morphism

    Def_Z([t]/t^2)≅H^1(Z, T_Z)    ⟶   H^2(Z, T_Z) 
    θ   ⟼    [θ, θ]

can be entirely understood by writing [t]/t^3 →[t]/t^2 as an elementary morphism.





 §.§ The ∞-category of formal moduli problems

We start by introducing formal moduli problems in the case of cdgas:

A formal moduli problem (we write fmp) is an ∞-functor X: _→ satisfying the following two properties: 


  –X() is contractible.

  – X preserves pull-backs along small morphisms.


The second condition means that given a cartesian diagram

    N [r] [d]     A [d] 
    
    M [r]     B

in _ where A → B is small, then 

    X(N) [r] [d]     X(A) [d] 
    
    X(M) [r]     X(B)

is cartesian. 

Observe that the second condition is stable under composition and pullback. Hence it is equivalent to replace in this condition small morphisms with elementary morphisms. We claim that we can even replace elementary morphisms with the particular morphisms →⊕[n] for every n≥ 1. Indeed, consider a cartesian diagram

    N [d] [r]     A [d]^-f
    
    M [r]     B

where f is elementary, that is given by a cartesian diagram

    A [d]_-f[r]    [d] 
    
    B [r]    ⊕[n]

If we look at the diagram 

    X(N) [r] [d]     X(A) [r] [d]     X(*) [d] 
    
    X(M) [r]     X(B) [r]     X(⊕[n])

and assume that X preserves pullbacks along the morphisms →⊕[n], then the right square is cartesian, so the left square is cartesian if and only if the big square is cartesian (which is the case).


A functor X: _→ is a fmp if and only if X(k) is contractible and preserves pull-backs whenever morphisms in the diagram are surjective on H^0. 


Assume that X preserves pull-backs whenever morphisms in the diagram are surjective on H^0. 
Consider a cartesian diagram of the following type, with n≥1: 

    N [r] [d]    [d] 
    
    M [r]    ⊕[n]

Since M is an augmented -algebra, the map M →⊕[n] is surjective on H^0. Hence 

    X(N) [r] [d]     X() [d] 
    
    X(M) [r]     X(⊕[n])

is cartesian. According to the preceding remark, this implies that X is a fmp.

The reverse implication is obvious thanks to the last sentence of Proposition <ref>. 


We write FMP_ for the full sub ∞-category of Fun(_, ) consisting of formal moduli problems. 



 §.§ A glimpse at the description of FMP_


In this section, we explain some heuristical aspects of the proof of the following theorem:
[<cit.>, <cit.>] 
There is an equivalence of ∞-categories _→FMP_. 

 Again, we discuss various points in this theorem related to more classical material.


  – Let us first give a naive idea about how the ∞-functor can be defined on a “sufficiently nice” dgla. The procedure is rather classical; we refer the reader to <cit.> and <cit.> for further details. If 𝔤 is a dgla of over , we can consider its (discrete) Maurer–Cartan set

    MC(𝔤)={ x ∈𝔤^1   such that   dx+[x, x]_𝔤=0 }.

The “good” object attached to 𝔤 is the quotient of MC(𝔤) under gauge equivalence. This can be better formulated using a simplicial enrichment as follows: for any n ≥ 0, let Ω^∙(Δ_n) be the cdga of polynomial differential forms on the n-simplex Δ_n. The collection of the Ω^∙(Δ_n) defines a simplicial cdga. Then we define the simplicial set ℳ𝒞(𝔤) as follows[The π_0 of ℳ𝒞(𝔤) is exactly the quotient of MC(𝔤) under gauge equivalence.]: 

    ℳ𝒞(𝔤)_n=MC(𝔤⊗Ω^∙(Δ_n)).

This being done, we can attach to 𝔤 a deformation functor Def_𝔤→ defined by

    Def_𝔤(A)=ℳ𝒞(𝔤⊗_I_A)

where I_A is the augmentation ideal of A. This defines (again in good cases) a formal moduli problem.

  – 0.07cm The main problem of the Maurer–Cartan construction is that the functor 𝔤→Def_𝔤 does not always preserve weak equivalences. For dglas that satisfy some extra conditions (like for instance nipoltence conditions), Def_𝔤 will be exactly the fmp we are seeking for. We will see very soon how Lurie circumvents this problem using the Chevalley–Eilenberg complex.

  – To illustrate an example where the Maurer–Cartan construction appears, let us come back to deformation theory of algebraic schemes in a slightly more differential-geometric context: instead of algebraic schemes we deform compact complex manifolds. We can attach to a complex compact manifold Z the Dolbeault complex of the holomorphic tangent bundle T_Z, which is the complex

    0 →𝒞^∞(T_Z) 𝒜^0, 1(T_Z) ⋯

We see this complex as a dgla over ℂ, the Lie structure being given by the classical Lie bracket of vector fields and the wedge product on forms. Then it is well known (see e.g. <cit.>) that deformations of Z over an artinian algebra A yield I_A-points in the set-valued deformation functor associated with the dgla (𝒜^0, ∙(T_Z), ∂). 



Let us now explain the good construction of the equivalence from  _ to FMP_. First we start by recalling the following standard definitions:
 For any dgla 𝔤, we define the homological and cohomological Chevalley–Eilenberg complexes CE_∙(𝔤) and CE^∙(𝔤) as follows: 


  – As a graded vector space, CE_∙(𝔤)=S (𝔤[1]) is the (graded) symmetric algebra of 𝔤[1]. The differential is obtained by extending, as a degree 1 graded coderivation, the sum ot the differential 𝔤[1]→𝔤[2] with the Lie bracket S^2 (𝔤[1])→𝔤[2]. Jacobi identity and Leibniz rule ensure that this coderivation squares to zero. 
The complex CE_∙(𝔤) is actually an (coaugmented, counital, and conilpotent) cocommutative coalgebra object in the category of complexes.

  – CE^∙(𝔤) is the linear dual of CE_∙(𝔤), it is an augmented cdga.


 
 


  – Observe that the above definition still makes sense for an L_∞-algebra 𝔤. Indeed, an L_∞-algebra structure on 𝔤 is defined as a degree 1 graded codifferential that makes S (𝔤[1]) a coaugmented counital cocommutative differential graded coalgebra. 

  – It is possible to prove that CE_∙ (𝔤)≃𝕃⊗_U(𝔤) and CE^∙(𝔤) ≃RHom_U(𝔤)(, ). 

  – Let V be an object of _, and let free  (V) be the free dgla generated by V. Then CE_∙(free  (V)) and CE^∙(free  (V)) are quasi-isomorphic to the square zero extensions ⊕ V[1] and ⊕ V^*[-1] respectively.

  – 
For any cdga A and any dgla 𝔤 of finite dimension over , there is a map

    Hom__(CE^∙(𝔤), A) →MC(𝔤⊗ I_A)

which is in good cases an isomorphism (where I_A is the augmentation ideal of A). To see how the map is constructed, let φ be an element in Hom__(CE^∙(𝔤), A). Forgetting the differential, it defines an algebra morphism from the completed algebra S (𝔤^*[-1]) to A, and in fact to I_A (since the morphism is pointed).
In particular, we have a map ϕ𝔤^*[-1] → I_A, and since 𝔤 is finite-dimensional, we can see the morphism ϕ as a map [-1] →𝔤⊗ I_A, hence an element x of (𝔤⊗ I_A)^1. Now we have a commutative diagram

    𝔤^*[-1] [r]^-ϕ[d]^-[ .  ,   .  ]^*    A [r]^-d_A   A 
    𝔤^*[-1] ⊗_𝔤^*[-1] [rr]^-ϕ⊗ϕ       A ⊗_ A [u]^-⋆_A

Unwrapping what it means expressing ϕ with x, we end up exactly with the Maurer–Cartan equation dx+[x, x]=0. The reason why the map (<ref>) is not always bijective is that CE^∙(𝔤) is not the symmetric algebra of 𝔤^*[-1], but it is the completed symmetric algebra. 

  – There is a simplicially enrichement of (<ref>), given by 

    Hom__(CE^∙(𝔤), A) →ℳ𝒞(𝔤⊗ I_A).

Apart from the completion issue that we have already discussed, this morphism may not be an equivalence as CE^∙(𝔤) may not be a cofibrant object in _.



 


  – The Chevalley–Eilenberg construction preserves weak equivalences, hence defining an ∞-functor

    CE^∙ _^op→_.

The functor (CE^∙)^op commutes with small colimits (see <cit.>), so since _ is presentable, CE^∙ admits a left adjoint. We call this adjoint 𝔇. Hence we have an adjunction

    𝔇__^opCE^∙

that can be seen as some version of Koszul duality. The main point in this step is that the Chevalley–Eilenberg functor does only commute with small homotopy limits, not usual small limits. Hence the adjoint functor is only defined in the ∞-categorical setting (i.e. it does not come from a Quillen adjunction).

  – 
We define an ∞-functor from _ to 𝐅𝐮𝐧(_, ) as follows:

    Δ (𝔤)= Hom__^op(𝔤, 𝔇(-))=Hom__(𝔇(-), 𝔤).

The functor Δ will define the equivalence we are seeking for.

  – Let us explain why Δ factors through FMP_. We introduce the notion of good object: a dg-Lie algebra ℒ is good if there exists a finite chain
0=ℒ_0 →ℒ_1 →…→ℒ_n=ℒ such that each of these morphisms appears in a pushout diagram

    free [-n_i-1] [r][d]    ℒ_i [d] 
    {0}[r]    ℒ_i+1

in _, or equivalently a pullback diagram

    ℒ_i+1[r][d]    {0}[d] 
    ℒ_i [r]    free  [-n_i-1]

in _^op. We denote by _^gd the full subcategory of _^op consisting of good objects. We see that good objects are formally the same as small ones in _^op, using the sequence of objects free [-n-1] instead of ⊕[n]. This will be formalized using the various notions of deformation contexts developed in the next section.

  – The next step consists of proving that if 𝔤 is good, the counit morphism 𝔇CE^∙(𝔤) →𝔤 in _^op is an equivalence. This is the crucial technical input and will be proved in Proposition <ref>. By purely formal arguments (see Proposition <ref>), this implies that the adjunction 𝔇CE^∙ defines an equivalence of categories between _ and _^gd.

  – Using this, if we have a cartesian diagram

    N [r] [d]    [d] 
    
    M [r]    ⊕[n]

where N and M are small, then 

    𝔇(N) [r] [d]    {0}[d] 
    𝔇(M) [r]    𝔇(⊕[n])

is cartesian in _^op, and therefore

    Δ(𝔤)(N) [r] [d]    *[d] 
    Δ(𝔤)(M) [r]    Δ(𝔤)(⊕[n])

is also cartesian in . This implies that Δ is an object of FMP_. Hence Δ factors through the category FMP_.





§ GENERAL FORMAL MODULI PROBLEMS AND KOSZUL DUALITY




 §.§ Deformation contexts and small objects
 
In this section, we will explain how the notions of small and elementary morphism make sense in a broader categorical setting. We start by some general facts on ∞-categories.


  – If C is an ∞-category with finite limits[This hypothesis will be always implicit in the sequel. It implies among other things that C has a terminal object *.], its stabilization Stab( C) can be described as the ∞-category of spectrum objects (also called infinite loop objects) in C. An object of Stab( C) is a sequence E=(E_n)_n ∈ℤ of pointed objects[The ∞-category of pointed objects is the coslice ∞-category ^*/ C under the terminal object.] together with weak equivalences E_n →Ω E_n+1, where Ω:=Ω_* denotes the based loop functor: Ω(c)=*c×*. We often write E_n=Ω^∞-nE.

  – If C is stable, then Stab( C) is naturally equivalent to C via the map sending (E_n)_n to E_0.

  – If C is an ∞-category with finite limits and c is an object of C, then the stabilization Stab(^c/ C) of its coslice ∞-category ^c/ C is equivalent to Stab( C). 
Indeed, the ∞-categories of pointed objects in C and ^c/ C are themselves equivalent. 

  – If C is an ∞-category with finite limits and c is an object of C, then the stabilization Stab( C_/c) of its slice ∞-category C_/c is the category of spectrum objects in the ∞-category ^id_c/( C_/c) of sequences c→ d→ c such that the composition is the identity. 






  – If 𝐂=, then Stab(𝐂) is the ∞-category  of spectra (that is spectrum objects in spaces).



A pair ( C,E), where C is a presentable ∞-category with finite limits and E is an object of Stab( C), is called a deformation context. 
Given a deformation context ( C,E): 


  – A morphism in C is elementary if it is a pull-back of * →Ω^∞-nE for n≥ 1 (where * is a terminal object in the category C). 

  – A morphism in C is small if it can be written as a finite sequence of elementary morphisms. 

  – An object c is small if the morphism c → * is small. 

We let ( C,E)^sm be the full subcategory of C spanned by the small objects. When it is clear from the context, we may abuse notation and write C^sm:=( C,E)^sm. 

Let us give two examples of deformation contexts:

If C=_, which is already stable, then we have an equivalence 

    _   ⟶̃   Stab( C) 
    
    M    ⟼    (M[n])_n .

In this context we will mainly consider the spectrum object E=([n+1])_n∈ℤ.


Instead of working over the ground field , we can work over an arbitrary cdga A. 
Then we can take C=_A and E=(A[n+1])_n∈ℤ in Stab( C).

 
The category Stab(_) is equivalent to _. 
Indeed, we have an equivalence _≃_, and the based loop functor Ω_0 in _ (0 is the terminal nonunital cdga) sends a non-unital algebra R to R[-1] equipped with the trivial product; hence the equivalence 

    _   ⟶̃   Stab(_) 
    
    M    ⟼    (⊕ M[n])_n∈ℤ .

In this context we will mainly consider the spectrum object E=(⊕[n])_n∈ℤ. 


One can prove in a similar way that the stabilization Stab(_A/A) of the ∞-category _A/A of A-augmented A-algebras, where A is in _, is equivalent to _A. 
Moreover, if _/A is the ∞-category of A-augmented -algebras, then 

    Stab (_/A) ≃Stab (_A/A)≃_A.

In this case a natural spectrum object to consider is (A⊕ A[n])_n∈ℤ. 


We have the following obvious, though very useful, lemma, which allows to transfer deformation contexts along adjunctions: 

Suppose ( C, E) is a deformation context. If we are given an adjunction T'  C'  C  T,, then T preserves small limits, so that ( C', T(E)) is a deformation context whenever C' is presentable. 
Besides, in this case T induces a functor from C^sm to ( C')^sm. 


Let us give four examples of this transfer principle[In all these examples, though, the use of transfer is not strictly necessary. Indeed, all the categories involved have a stabilization that is equivalent to _A.]

Given an A-module L, the push-out functor -∐_L0: ^L/_A→_A along the zero map L→ 0 admits a right adjoint, being the functor sending an A-module M to the zero map L0→M. 
Hence the deformation context from Example <ref> and Remark <ref> can be transferred to ^L/_A. 



The relative cotangent complex functor 𝕃_A/-:_A→_A admits a right adjoint, being the split square zero extension functor M↦ A⊕ M. 
Hence the deformation context from Example <ref> and Remark <ref> can be obtained by transfer from the one given in Example <ref> and Remark <ref>. 



The relative cotangent complex functor 𝕃_A/-:_/A→ ^𝕃_A/_/_A admits a right adjoint, being the functor sending a morphism 𝕃_A/d→M in _A to the non-necessarily split square zero extension Ad⊕M[-1] that it classifies. 
Hence the deformation context from Example <ref> (with L=𝕃_A/) can be transferred to _/A. 



The forgetful functor _A→_/A is a right adjoint: its left adjoint is A⊗-. 
Hence the deformation context from Example <ref> can also be obtained by transfer from the one given in Example <ref>. 


The main observation is that formal moduli problems make sense with _ being replaced by any deformation context ( C,E). We write FMP( C,E) for the ∞-category of formal moduli problems associated with it. Then FMP(_, (⊕[n])_n)=FMP_. 





 §.§ Dual deformation contexts and good objects

We introduce the dual concept of a deformation context:

A pair ( D,F), where D is a presentable ∞-category and F is an object of Stab( D^op), is called a dual deformation context. 

0.07cm If ( D,F) is a dual deformation context, ( D^op,F) is not in general a deformation context because D^op is almost never presentable.

The first easy example is for D=_A. Nevertheless, this example is of high interest (see Lemma <ref> below). Since _A is stable, so is _A^op, and the looping and delooping functors are swapped when passing to the opposite category. Hence spectrum objects in _A^op are of the form (M[-n])_n for some object M. In particular, (_A, (A[-n-1])_n) is a dual deformation context. 

Just like deformation contexts, dual deformation contexts can be transported using adjunctions: assume to be given a dual deformation context ( D, F) as well as an adjunction 
T  D ⟶
⟵  D' ι.
Then it is easy to see that ( D', T(F)) is a dual deformation context on D' whenever D' is presentable, since T preserves colimits (hence limits in the opposite category). 
Using this, one can build a lot of dual deformation contexts starting from _A. 

For instance, the adjunction 

    free_A _A forget

yields the dual deformation context (_A, free(A[-n-1])_n). 


Let T be an A-module. Then the pull-back functor -×_T0:_A/T→_A along the zero morphism 0→ T admits a left adjoint: it is the functor sending an A-module S to the zero morphism S0→ T. 
This yields the dual deformation context (_A/T,(A[-n-1]0→T)_n). 



Given a dual deformation context ( D, F), an object (resp.  morphism) of D is good if it is small when considered as an object (resp. morphism) of D^op. More explicitely, if ∅ denotes the initial element of D, an object b of D is good if there is a finite sequence of morphisms 

    ∅=b_m ⋯ b_1  b_0≅ b

that are pushouts along 
F_n→∅ in D: i.e. each f_i fits into a pushout square 

    F_n [d][r]     [d]^f_i b_i  
    ∅[r]     b_i-1

in D for some n≥1. 
We let ( D,F)^gd be the full subcategory of D spanned by the good objects, 
which we might simply denote D^gd if there is no ambiguity. 

Let us give a somehow nontrivial example. Let A be a -algebra[A is concentrated in degree 0. ], and consider the dual deformation context (_A, (A[-n-1])_n).
 
An element of _A is good for the dual deformation context (_A, (A[-n-1])_n) if and only if it is perfect and cohomologically concentrated in positive degrees.


Recall that a perfect complex is the same as a dualizable object, which means a complex quasi-isomorphic to a finite complex consisting of projective A-modules of finite type. 
Let K be a perfect complex concentrated in positive degree and prove, by induction on the amplitude, that K is good. 
If the amplitude is 0 then K is quasi-isomorphic to P[-n], for n ≥ 1, with A^r=P⊕ A^q . We have the following push-out square in _A:

    A^q[-n-2] [d] [r]     0 [d] 
    
    0  [r]     A^q[-n-1]

Hence A^q[-n-1] is good. We then also have another push-out square: 

    A^r[-n-1] [d] [r]     A^q[-n-1] [d] 
    
    0  [r]     P[-n]

Hence the morphism A^q[-n-1]→ P[-n] is good, and since A^q[-n-1] is good, then P[-n] is good as well. 
Performing the induction step is now easy: let K be a positively graded complex of finitely generated projective A-modules of some finite amplitude d>0, let n>0 be the index where K starts and let P=K_n. 
We have a push-out square:

    P[-n-1] [d] [r]    τ^>n K [d] 
    
    0 [r]     K
 
where τ^>n is the stupid truncation functor, and using again that A^r=P⊕ A^q we get another push-out square: 

    A^r[-n-1] [d] [r]     A^q[-n-1]⊕τ^>n K [d] 
    
    0 [r]     K

Hence the morphism A^q[-n-1]⊕τ^>n K→ K is good. But τ^>n K is good (by induction on the amplitude) and A^q[-n-1] is good as well, thus so is K. This finishes the induction step.
For the converse statement, it suffices to observe that, given a push-out square

    A[-n-1] [r] [d]     K [d] 
    
    0 [r]     L

in _A, where n≥1, and K is perfect and concentrated in positive degrees, then so is L. .

Similarly, one can prove that small objects for the deformation context (_A, (A[n+1])_n) are perfect complexes of A-modules cohomologicaly concentrated in negative degrees. 
We leave it as an exercise to the reader. 
 
If we replace the -algebra A by a bounded cdga concentrated in non-positive degrees, then one can still prove that good objects are perfect A-modules that are cohomologically generated in positive degree. 
In other words, a good object is quasi-isomorphic, as an A-module, to an A-module P having the following property: as a graded A-module, P is a direct summand of A⊗ V, where V is a finite dimensional positively graded -module. 




 §.§ Koszul duality contexts

We now introduce the main notion that is needed to state Lurie's theorem on formal moduli problems in full generality: 


(1) A weak Koszul duality context is the data of :


  – a deformation context ( C,E), 

  – a dual deformation context ( D,F), 

  – an adjoint pair 𝔇: C ⟶
⟵  D^op:𝔇', 

such that for every n≥0 there is an equivalence E_n ≃𝔇' F_n. 

(2) A Koszul duality context is a weak Kozsul duality context satisfying the two additional properties: 


  – For every good object d of D, the counit morphism 𝔇𝔇' d → d is an equivalence.

  – The functor

    ΘHom_𝐃(F_n,-) 𝐃→Stab()=

is conservative and preserves small sifted colimits.

With a slight abuse of notation, we will denote the whole package of a (weak) Koszul duality context by 

    𝔇 ( C, E) ⟶
    ⟵ ( D^op, F) 𝔇'



The name Kosul duality context has been chosen in agreement with the first non-trivial example we will deal with (see Proposition refopinel), as it reflects the well-known Koszul duality between commutative and Lie algebras. Similarly, the characterization of associative formal moduli problems <cit.>, resp. 𝔼_n formal moduli problems <cit.>, can be proven with the help of a Koszul duality context that reflects the Koszul duality for associative algebras, resp. 𝔼_n-algebras. We actually expect that a pair (𝒪,𝒫) of Koszul dual operads always lead to a Koszul duality context between 𝒪-algebras and 𝒫-algebras, allowing to show that formal moduli problems for 𝒪-algebras are 𝒫-algebras.  

Observe that there may be more objects of D for which the counit morphism  is an equivalence, than just good objects. We call them reflexive objects. 

The only elementary Koszul duality context we can give at this stage is the following: if A is a bounded cdga concentrated in nonpositive degrees, 

    (-)^∨=Hom_A(-, A) (_A, (A[n+1])_n)  ⟶
    ⟵ (_A^op, (A[-n-1])_n) Hom_A(-, A)=(-)^∨.

To prove that it is indeed a Koszul duality context, we use Lemma <ref> and Remark <ref>: a good object in _A^op is a perfect A-module (generated in non-negative degrees), so it is isomorphic to its bidual. Lastly, the functor Θ is simply the forgetful functor

    _A →_≃Stab(_) →Stab() ≃

which is conservative. 
Note that there are strictly more reflexive objects than good ones: for instance, ⊕_n≥0A[-n] is reflexive, but not good (it is an example of an almost finite cellular object in the terminology of <cit.>). 

There is a peculiar refinement of the above example, that will be useful for later purposes. 

Let A be a bounded cdga concentrated in nonpositive degrees, and let L be an A-module. 
We have a weak Koszul duality context[Observe that, contrary to what one could think, it is not required that L is perfect, or even just reflexive. 
Indeed, for Lℓ→ M and Kk→ L^∨, the following commuting diagrams completely determine each other: 

    K [r]^-f^∨[d]^-k    M^∨[dl]^-ℓ^∨
    
    L^∨       and    
    L [r][d]^-ℓ    (L^∨)^∨[d]^-k^∨
    
    M [r]^-f     K^∨

] 

    (-)^∨(^L/_A, (L0→A[n+1])_n)  ⟶
    ⟵ ((_A/L^∨)^op, (A[-n-1]0→L^∨)_n) (-)^∨ ,

where the right adjoint sends Kk⟶ L^∨ to the composed morphism L→ (L^∨)^∨k^∨⟶ K^∨. 
We claim that this is actually a Koszul duality context. 
Indeed, one first observes that good objects are given by morphisms K→ L^∨ where K is a good object in _A. Hence they are again isomorphic to their biduals. 
Lastly, the functor Θ is the composition of the pull-back functor -L^∨×0:(_A/L^∨)^op→_A along the zero morphism 0→ L^∨, with the forgetful functor _A → from the previous example. They are both conservative, hence Θ is.  






There are a few properties that can be deduced from the definition, which are absolutely crucial.
[<cit.>] 
Assume that we are given a Koszul duality context

    𝔇 ( C, E)  ( D^op, F) 𝔇'



  (A) For every n≥0, 𝔇 E_n ≃ F_n.

  (B) For every small object ℳ in C, the unit map ℳ→𝔇' 𝔇(ℳ) is an equivalence.

  (C) There is an equivalence of categories
( C , E)^sm@<2pt>[r]^𝔇   @<2pt>[l]^𝔇' ( D^op , F)^gd.

  (D) Consider a pullback diagram

    N [r] [d]     A [d]^-f
    
    M [r]     B

where f is small and M is small. Then the image of this diagram by 𝔇 is still a pullback diagram.






 The proof is clever but completely formal, and doesn't require any extra input. 
(A)  This is straightforward: 𝔇 E_n ≃𝔇𝔇' F_n ≃ F_n since F_n is good. 
(B)  We proceed in two steps.
First we prove it if ℳ is for the form 𝔇'(𝒩) for some good object 𝒩. This is easy:
the composition

    𝔇' ≅id∘𝔇'  ⇒  (𝔇' ∘𝔇) ∘𝔇' ≅𝔇' ∘ (𝔇∘𝔇') ⇒𝔇' ∘id≅𝔇'

is an equivalence (it is homotopic to the identity). This means that the composition
𝔇' 𝔇 (𝔇' 𝒩) →𝔇' 𝒩 is obtained up to equivalence by applying 𝔇' to the equivalence 𝔇𝔇'(𝒩) →𝒩, hence it is also an equivalence.
Now we prove that every small object can be written as 𝔇'(X) where X is good. This will in particular imply that 𝔇 maps small objects to good objects[This is nontrivial: 𝔇 preserves colimits, and small objects are defined using limits.]. Every small
morphism can be written as a composition of a finite number i of elementary morphisms, we will argue by induction on the number i. If i=0, then since 𝒟' is a right adjoint it preserves limits. Hence 𝔇'(∅)=*, where ∅ is the initial element of D. Since ∅ is good, we are done. We now deal with the induction step. Let us consider a cartesian diagram

    ℳ_i+1[d] [r]    *[d] 
    ℳ_i [r]     E_n

in C such that ℳ_i → * is obtained by composition of at most i elementary morphisms. By induction, ℳ_i+1=𝔇'(𝒩) for some good object 𝒩. This implies that 𝔇(ℳ_i) is good. Let X be the pullback the object of D making the diagram

    X [r] [d]    *[d] 
    𝔇(ℳ_i) [r]    𝔇(E_n)

cartesian in D^op. Since 𝔇(ℳ_i) is good and 𝔇(E_n) ≃ F_n, X is good. Now we apply 𝔇', which preserves limits. We get a cartesian diagram isomorphic to 

    𝔇'(X) [r] [d]    *[d] 
    ℳ_i[r]     E_n

This proves that ℳ_i+1 is isomorphic to 𝔇'(X).
(C) This is a direct consequence of (B).
(D) We can reduce to diagrams of the form

    N [r] [d]    *[d] 
    
    M [r]     E_n

Then the property follows from (C).




 §.§ Morphisms of (weak) Koszul duality contexts


We will now introduce the notion of morphisms between weak Koszul duality contexts. It will be extremely useful in the sequel.
 
Let 
    𝔇_1  ( C_1, E_1) ⟶
    ⟵ ( D_1^op, F_1) 𝔇'_1

and

    𝔇_2  ( C_2, E_2) ⟶
    ⟵ ( D^op_2, F_2) 𝔇'_2

be two weak Koszul duality contexts. 
A weak morphism between these two duality contexts consists of two additional pairs of adjoint functors appearing (vertically) in the diagram

    @C=60pt@R=30pt C_1 @<-2pt>[d]_-S@<2pt>[r]^-𝔇_1   @<2pt>[l]^-𝔇'_1D_1^op@<-2pt>[d]_-Y
     C_2@<-2pt>[u]_-T@<2pt>[r]^-𝔇_2   D_2^op@<2pt>[l]^-𝔇'_2@<-2pt>[u]_-Z

such that:


  (A) The diagram consisting of right adjoints C_1    [l]_-𝔇'_1 D_1^op
 C_2 [u]^-T   [u]_-Z[l]^-𝔇'_2 D_2^op commutes.

  (B) We have equivalences Z(F_2, n) ≃ F_1, n.



Let L be an A-module. We then have the following weak morphism from the Koszul duality context of Example <ref> to the one of Example <ref>: 

    @C=60pt@R=30pt
    ^L/_A @<-2pt>[d]_-cofib@<2pt>[r]^-(-)^∨   @<2pt>[l]^-(-)^∨ (_A/L^∨)^op@<-2pt>[d]_-fib
    _A@<-2pt>[u]_-M↦ (L0→M)@<2pt>[r]^-(-)^∨   _A^op@<2pt>[l]^-(-)^∨@<-2pt>[u]_-K↦ (K0→ L^∨)
 
where cofib=0L∐- and fib=0L^∨×-. 

Notice that the natural equivalence -1ex
[>=stealth,->] (0.4,0.4) to (0,0.4) ;
[>=stealth,->] (0.4,0) to (0.4,0.4) ;
[>=stealth,thin,double,<->] (0.5,0.2) to (0.9,0.2);
[>=stealth,->] (1,0) to (1,0.4) ;
[>=stealth,->] (1.4,0) to (1,0) ;
 from condition (A) above has a mate -1ex
[>=stealth,->] (0.4,0.4) to (0,0.4) ;
[>=stealth,->] (0,0.4) to (0,0) ;
[>=stealth,thin,double,->] (0.5,0.2) to (0.9,0.2) node[above left]θ;
[>=stealth,->] (1.4,0.4) to (1.4,0) ;
[>=stealth,->] (1.4,0) to (1,0) ;
 which we define pictorially as the composition 

    [>=stealth,->] (0.4,0.4) to (0,0.4) ;
     [>=stealth,->] (0,0.4) to (0,0) ;
     [>=stealth,thin,double,->] (0.5,0.2) to (0.9,0.2) ;
     [>=stealth,->] (1.3,0.4) to (1,0.4) ;
     [>=stealth,->] (1,0.4) to (1,0) ;
     [>=stealth,->] (1.4,0.4) to (1.4,0) to (1.3,0) to (1.3,0.4) ;
     [>=stealth,thin,double,<->] (1.5,0.2) to (1.9,0.2) ;
     [>=stealth,->] (2.4,0.4) to (2.4,0) ;
     [>=stealth,->] (2.4,0) to (2.1,0) ;
     [>=stealth,->] (2.1,0) to (2.1,0.4) to (2,0.4) to (2,0) ;
     [>=stealth,thin,double,->] (2.5,0.2) to (2.9,0.2) ;
     [>=stealth,->] (3.4,0.4) to (3.4,0) ;
     [>=stealth,->] (3.4,0) to (3,0) ;

where 
-1ex
(A) at (-0.2,0.2) id;
[>=stealth,thin,double,->] (0.1,0.2) to (0.5,0.2);
[>=stealth,->] (0.7,0.4) to (0.7,0) to (0.6,0) to (0.6,0.4) ;
 and -1ex
(A) at (0.7,0.2) id;
[>=stealth,thin,double,->] (0.1,0.2) to (0.5,0.2);
[>=stealth,->] (0,0) to (0,0.4) to (-0.1,0.4) to (-0.1,0) ;
 are used to depict units and counits of vertical adjunctions. In other words, the mate θ is the composition 
S𝔇_1'⇒ S𝔇_1'ZY≅ ST𝔇_2'Y⇒𝔇_2'Y. 

The commutativity of the square of right adjoints implies the commutativity of the square of left adjoints. Hence there is a natural equivalence 𝔇_2S≅ Y𝔇_1 that pictorially reads -1ex
[>=stealth,->] (0,0.4) to (0.4,0.4) ;
[>=stealth,->] (0.4,0.4) to (0.4,0) ;
[>=stealth,thin,double,<->] (0.5,0.2) to (0.9,0.2);
[>=stealth,->] (1,0.4) to (1,0) ;
[>=stealth,->] (1,0) to (1.4,0) ;
, and the mate θ can also be identified with the following composition:

    S𝔇_1'⇒𝔇_2'𝔇_2S𝔇_1≅𝔇_2'Y𝔇_1𝔇_1'⇒𝔇_2'Y
        or, pictorially     [>=stealth,->] (0.4,0.4) to (0,0.4) ;
     [>=stealth,->] (0,0.4) to (0,0) ;
     [>=stealth,thin,double,->] (0.5,0.2) to (0.9,0.2) ;
     [>=stealth,->] (1.4,0.4) to (1,0.4) ;
     [>=stealth,->] (1,0.4) to (1,0.1) ;
     [>=stealth,->] (1,0.1) to (1.4,0.1) to (1.4,0) to (1,0) ;
     [>=stealth,thin,double,<->] (1.5,0.2) to (1.9,0.2) ;
     [>=stealth,->] (2.4,0.4) to (2,0.4) to (2,0.3) to (2.4,0.3) ;
     [>=stealth,->] (2.4,0.3) to (2.4,0);
     [>=stealth,->] (2.4,0) to (2,0) ;
     [>=stealth,thin,double,->] (2.5,0.2) to (2.9,0.2) ;
     [>=stealth,->] (3.4,0.4) to (3.4,0) ;
     [>=stealth,->] (3.4,0) to (3,0) ;
     .














One says that the above commuting square of right ajoints satisfies the Beck–Chevalley condition locally at d, where d is an object of 𝐃_1, 
if the mate θ_d:S𝔇_1'(d)→𝔇_2'Y(d) is an equivalence. 



A morphism of weak Koszul duality contexts is a weak morphism such that, borrowing the above notation: 


  (C) The functor Y is conservative, preserves small sifted limits, and sends good objects to reflexive objects.

  (D) The commuting square of right adjoints satisfies the Beck–Chevalley condition locally at good objects. 






 
The main feature of this definition is a result allowing to transfer Koszul duality contexts along morphisms:

[Transfer theorem] 
Assume to be given a morphism between two weak Koszul duality contexts. If the target deformation context is a Koszul duality context, then the source is also a Koszul duality context.



We take the notation of Definition <ref>. We have two functors Θ_1 and Θ_2, defined on D_1 and D_2 respectively and with values in , defined by

    Θ_1(d)_n=Hom_D_1(F_1, n, d) 
    Θ_2(d)_n=Hom_D_2(F_2, n, d)


For any object d in D_1, we have 

    Θ_1(d)_n   =Hom_D_1(F_1, n, d) ≃Hom_D_1^op(d, Z(F_2, n))  
       ≃Hom_D_2^op(Y(d), F_2, n) ≃Hom_D_2(F_2, n, Y^op(d)) 
       ≃Θ_2(Y^op(d))_n

So Θ_1=Θ_2 ∘ Y^op, and thus Θ_1 is conservative (because Y and Θ_2 are). 
Moreover, Y preserves small sifted limits, so Y^op preserves small sifted colimits, and hence so does Θ_1 (because Θ_2 does too). Lastly, we consider the following diagram in D_2^op, where d is still an object in D_1: 

    @C=60pt
    Y 𝔇_1 𝔇'_1 @<=>[d]_-∼@=>[r]^-Y ∘ co-unit     Y 
    𝔇_2 S  𝔇'_1 @=>[r]_-𝔇_2∘θ   𝔇_2 𝔇'_2 Y @=>[u]_-co-unit ∘ Y.

We claim that it commutes. Indeed, writing the mate explicitely gives a diagram

    @C=60pt
    Y 𝔇_1 𝔇'_1 @<=>[d]_-∼@=>[r]^-Y ∘ co-unit     Y 
    𝔇_2 S  𝔇'_1 @=>[d]_-𝔇_2∘ unit ∘ S  𝔇'_1   𝔇_2 𝔇'_2 Y @=>[u]_-co-unit ∘ Y
    𝔇_2𝔇_2'𝔇_2 S  𝔇'_1 @<=>[r]^-∼   𝔇_2𝔇_2'Y𝔇_1𝔇_1' @=>[u]_-𝔇_2𝔇_2'Y∘ co-unit@=>[uul]|-co-unit ∘ Y𝔇_1𝔇_1'

where the two triangles commute. 

Given now a good object d in D_1, we have that Y(d) is reflexive (after (C) in Definition <ref>): thus the co-unit morphism of the adjunction (𝔇_2, 𝔇'_2) is an equivalence on Y(d). Besides, θ_d is also an equivalence (thanks to (D) in Definition <ref>). Therefore Y∘co-unit(d) is an equivalence and thus, by conservativity of Y, the co-unit morphism 𝔇_1𝔇_1'(d)→ d is an equivalence.


Going back to Example <ref>, one sees that the weak morphism is a morphism if and only if L is reflexive (in which case, L^∨ is so as well). Indeed: 


  – The functor fib is conservative. 

  – If K→ L^∨ is good then K itself is good, thus perfect, in _A, and thus fib(K→ L^∨) is reflexive as soon as L^∨ is so.  

  – the mate θ_K→ L^∨ is the natural morphism cofib(L→ K^∨)→fib(K→ L^∨)^∨, which is an equivalence if and only if the unit morphism L→ (L^∨)^∨ is an equivalence. 

Hence the Koszul duality context from Example <ref> is in general not obtained by transfer from the one of <ref>, but it is in the case when L is reflexive.

 
The adjunction

    𝔇 (_, ⊕[n])  (_^op, free [-n-1]) CE^∙

defines a Koszul duality context.


We consider the diagram (see Remark <ref>):

    @C=80pt@R=40pt_   [l]_-CE^∙_^op
    _[u]^-V ↦⊕ V[-1]   _^op[l]_-Hom(-, )[u]_-free

We can fill it with left adjoints everywhere. This gives the following (nice!) diagram:

    @C=100pt@R=50pt_@<-2pt>[d]_-𝔏@<2pt>[r]^-𝔇   @<2pt>[l]^-CE^∙_^op@<-2pt>[d]_-forget
    _@<-2pt>[u]_-V ↦⊕ V[-1]@<2pt>[r]^-Hom(-, )   _^op@<2pt>[l]^-Hom(-, )@<-2pt>[u]_-free

where 𝔏(R) =𝕃_/R≃𝕃_R/⊗_R[1]. 

We claim that these four adjunctions define a morphism of weak Koszul duality contexts. Properties (A) and (B) are true, so that we have a weak morphism. 
We will now prove that it is actually a morphism, so that we get the result, using Proposition <ref> and the fact that the bottom adjunction is a Koszul duality context (Example <ref>).  
The functor forget is conservative. Let us now prove that, if 𝔤 is good, then it is equivalent to a very good dgla: a good dgla with underlying graded Lie algebra being generated by a finite dimensional graded vector space sitting in positive degrees. 

Any good dgla 𝔤 is quasi-isomorphic to a very good one. 


We first observe that 0 is very good. We then proceed by induction: assume that 𝔤 is very good, and consider a dgla 𝔤' obtained by a pushout 

    free  [-n-1]   [r] [d]    𝔤[d] 
    
    0 [r]   𝔤'

in _. The first trick is that free  [-n-1] is cofibrant and 𝔤 is fibrant (every object in _ is). Hence the morphism free  [-n-1] in the ∞-category _ can be represented by a honest morphism in _. The next step consists in picking a cofibrant replacement of the left vertical arrow. A cofibrant replacement is given by the morphism

    free  [-n-1] →free  {cone ()[-n-1] }.

Hence our pushout in _ is represented by the following honest non-derived pushout in _: 

    free  [-n-1]   [r] [d]    𝔤[d] 
    free  {cone ()[-n-1] }[r]   𝔤'

This pushout is obtained by making the free product of 𝔤 and free  {cone ()[-n-1] }, and then by taking the quotient by the image of the ideal generated by free  [-n-1]. This completes the induction step: 𝔤' is still very good. 

This in particular shows that forget sends good objects to reflexive objects for the dual deformation context (_, [-n-1]). Lastly, thanks to <cit.>, the forgetful functor from _^op to _^op preserves small sifted limits. Thus (C) holds. It remains to prove (D), which is the main delicate point of the proof. Recall for that purpose that, for a dgla 𝔤, θ_𝔤 is defined as the following composition, where we omit the forget functor (its appearance being obvious): 

    𝔏(CE^∙(𝔤))→𝔏(CE^∙(free 𝔤))→𝔏(⊕𝔤^*[-1])→𝔤^* .

At this point it is important to make a rather elementary observation: the composed cdga morphism CE^∙(𝔤)→CE^∙(free 𝔤)→⊕𝔤^*[-1] is nothing but the projection onto the quotient by the square ℐ^2 of the augmentation ideal ℐ=(CE^∙(𝔤)→) whenever 𝔤 is very good[Indeed, in this case the natural map S^k(𝔤^*[-1])→ S^k(𝔤[1])^* is an isomorphism. Thus, forgetting the differential, CE^∙(𝔤)=Ŝ(𝔤^*[-1]). ]. 

We now introduce the uncompleted Chevalley–Eilenberg cdga CE^∙(𝔤) as a sub-cdga S(𝔤^*[-1])⊂CE^∙(𝔤) (it is obviously a graded subalgebra, and it can be easily checked as an exercise that it is stable under the differential). The quotient by the square of its augmentation ideal is still ⊕𝔤^*[-1], and we have the following commuting diagram and its image through the left-most vertical adjunction of our square: 

    CE^∙(𝔤) [d]_ι_𝔤[rd]    
    CE^∙(𝔤) [r]    ⊕𝔤^*[-1]
        𝔏(CE^∙(𝔤)) [d]_𝔏(ι_𝔤)[rd]^θ_𝔤   
    𝔏(CE^∙(𝔤)) [r]_-θ_𝔤   𝔤^*

In order to prove that θ_𝔤 is an equivalence when 𝔤 is good, we will prove that both θ_𝔤 and 𝔏(ι_𝔤) are. 
Let us start with the following lemma: 

Let A be an augmented cdga, with augmentation ideal 𝒥, that is cofibrant as a cdga. 
Then the morphism 𝔏(A)→𝒥/𝒥^2[1] associated with the projection A→⊕𝒥/𝒥^2 is an equivalence. 


First of all, the projection A→⊕𝒥/𝒥^2 is an actual morphism in the category _, so that we have a factorization 

    𝔏(A)≃𝕃_A/⊗_A[1]→Ω^1_A/⊗_A[1]→𝒥/𝒥^2[1] ,

where: 


  – The morphism 𝕃_A/⊗_A→Ω^1_A/⊗_A is an equivalence because A is cofibrant. 

  – The second morphism is an isomorphism in _. 

The lemma is proved. 

We then observe that when 𝔤 is very good (which we can always assume without loss of generality when dealing with good dglas), then CE^∙(𝔤) is cofibrant and thus θ_𝔤 is an equivalence[Here is another approach, avoiding the use of very good models. If 𝔤 is a dgla then there is an L_∞-structure on H^*(𝔤) that makes it equivalent to 𝔤 in _. If moreover 𝔤 is good then H^*(𝔤) is finite dimensional and concentrated in positive degree, so that the L_∞-structure has only finitely many non-trivial structure maps. Thus the uncompleted Chevalley-Eilenberg cdga CE^∙(H^*(𝔤)) can be defined, is cofibrant, and is equivalent to CE^∙(𝔤). ]. 

Lastly, it can be shown that CE^∙(𝔤) is flat over CE^∙(𝔤), which implies that the natural map 

    𝕃_CE^∙(𝔤)/⊗_CE^∙(𝔤)CE^∙(𝔤)⟶𝕃_CE^∙(𝔤)/

is also an equivalence, which shows (after applying -⊗_CE^∙(𝔤)) that 𝔏(ι_𝔤) is an equivalence. 




 §.§ Description of general formal moduli problems

Assume that we are given a Koszul duality context

    𝔇 ( C, E)  ( D^op, F) 𝔇'

We can define a functor ΨD→Fun(C, ) by the composition

    ΨD𝐅𝐮𝐧(D^op, ) 𝐅𝐮𝐧(C, ).

where 𝒴 is the Yoneda functor d →Hom_D^op(d,-).
[Lurie <cit.>] 
Given a Koszul duality context

    𝔇 ( C, E)  ( D^op, F) 𝔇',

The functor Ψ factors through 𝐅𝐌𝐏(C, E) and the induced functor

    ΨD→𝐅𝐌𝐏(C, E)
 
is an equivalence.



The first point is a direct consequence of Proposition <ref> (D). The proof that Ψ is an equivalence proceeds on several steps.  



  – The first step consists of proving that Ψ is conservative. This follows almost immediately from the hypotheses. Indeed, let f  X → Y an arrow in D inducing isomorphic formal moduli problems. We have
Ψ(X)=Hom_D(𝔇(*), X) and similarly for Y. Since all E_n are small and 𝔇(E_n) ≃ F_n, f induces an equivalence of spectra

    Hom_D(F_n, X) ≃Hom_D(F_n, Y) .

Since Hom_𝐃(F_n,-) 𝐃→ is conservative, f is an equivalence.



  – The next step consists in proving that Ψ commutes with limits and accessible colimits, so it has a left adjoint. This part is elementary, and uses the same techniques as in Proposition <ref>. We denote this left adjoint by Φ.



  – The functor Ψ being conservative, it suffices to prove that the unit id_𝐅𝐌𝐏(C, E)⇒Ψ∘Φ is an equivalence. This is the most technical part in the proof: it involves hypercoverings to reduce to pro-representable moduli problems; this is where the condition on sifted colimits plays a role. Here pro-representable means small limit of fmp representables by small objects.



  – In the representable case, we can explain what happens: we have an adjunction diagram

    D@<-2pt>[rr]_-Ψ      @<-2pt>[ll]_-ΦFMP(C, E) 
       C^op[ur]_-𝒴@<-2pt>[lu]^-𝔇^op

where 𝒴 is the Yoneda functor c →Hom_C(c,-). We claim that Φ∘𝒴≃𝔇^op on small objects, that is on (C^sm)^op. Indeed, for any objects c and d of C^sm and D respectively,

    Hom_D(Φ(𝒴_c), d)    ≃Hom_FMP(C, E)(𝒴_c, Ψ(d)) 
       = Hom_Fun(C^sm, E)(𝒴_c, Ψ(d)) 
       = ψ(d)(c) 
       = Hom_D^op(d, 𝔇(c)) 
       =  Hom_D(𝔇^op(c), d)

Using this, the unit map of 𝒴_c is given by the unit map of the adjunction between 𝔇 and 𝔇' via the natural equivalence

    𝒴_c →Ψ∘Φ (𝒴_c) ≃Ψ (𝔇^op(c))=Hom_ D^op(𝔇(c), 𝔇(-)) ≃Hom_C(c, 𝔇' 𝔇(-)).

Using Proposition <ref> (B), we see that this map is an equivalence of formal moduli problems.





 §.§ The tangent complex

In this section, we introduce the tangent complex associated to a formal moduli problem. We start with a very general definition.


Let ( C, E) be a deformation context. For any fmp X in FMP(C, E), its tangent complex 𝕋_X is the spectrum X(E)=X(E_n)_n.



There is a slight subtelty in the definition of 𝕋_X, as X(E_n) is only defined for n ≥ 0. 
However, this suffices to define it uniquely as a spectra, by putting (𝕋_X)_-m:=Ω^m_*((𝕋_X)_0).

Assume now to be given a Koszul duality context

    𝔇 ( C, E)  ( D^op, F) 𝔇'.

Then we have the following result: 

 
The following diagram commutes: 

    D [rr]^-Ψ[rd]_-Θ      𝐅𝐌𝐏( C, E) [ld]^-𝕋
       ^




This is straightforward: given an object d in D, Ψ(d)=Hom_D^op(d, 𝔇(*)), so

    𝕋_Ψ(d)=Hom_D^op(d,𝔇(E_n))=Hom_D(F_n, d)=Θ(d).
 



If D is -linear (resp. A-linear), then Θ actually lifts to _ (resp. _A): indeed, replacing Hom_ D(F_n,-) by its enriched version HOM_ D(F_n,-) gives us the lift of Θ. 



Going back to Example <ref> we get that, for an A-module M, 

    Θ(M)=Hom__A(A[-n-1],M)_n .

In the _A-enriched version, we have 
Θ(M)=(M[n+1])_n which gives Θ(M)≃ M[1] via the canonical identification Stab(_A)≃_A. 

We now deal with functoriality: 

Assume to be given a weak morphism 

    @C=60pt@R=30pt C_1 @<-2pt>[d]_-S@<2pt>[r]^-𝔇_1   @<2pt>[l]^-𝔇_1'D_1^op@<-2pt>[d]_-Y
     C_2@<-2pt>[u]_-T@<2pt>[r]^-𝔇_2   D_2^op@<2pt>[l]^-𝔇_2'@<-2pt>[u]_-Z

between two Koszul duality contexts. Then there is an induced commuting diagram

    @C=40pt@R=20pt D_1 [r]^-∼[dd]^Y^op   FMP (C_1, E_1) [dd]^-∘ T[rd]^-𝕋   
          
     D_2 [r]^-∼   FMP (C_2, E_2) [ru]_-𝕋

In particular, the functor FMP (C_1, E_1) →FMP (C_2, E_2) is conservative.



We begin by proving that T^*:=-∘ T:𝐅𝐮𝐧( C_1,𝐬𝐒𝐞𝐭)→𝐅𝐮𝐧( C_2,𝐬𝐒𝐞𝐭) indeed defines a functor FMP (C_1, E_1)→FMP (C_2, E_2): 


  (1) First, for every n≥0: T(E_n,2)≃ T𝔇'_2(F_n,2)≃𝔇_1'Z(F_n,2)≃𝔇_1'(F_n,1)≃ E_n,1. 

  (2) Then, T being a right adjoint it preserves in particular pull-backs along * → E_n,2, and thus send them to pull-backs along *→ E_n,1 whenever n≥0. Hence it sends small objects to small objects. 

  (3) Finally, let F be an fmp for (C_1,E_1). Then F∘ T(*)≃ F(*)≃ * (because T is a right adjoint and F is an fmp), and F∘ T preserves pull-back along * → E_n,2 (thanks to the second point and that F is an fmp).  

Note that Z also sends good objects to good objects (the proof is the same as for the second point above). 

We now come to the proof of the commutativity of the square, which is essentially based on the following Lemma: 

There is a natural transformation 𝔇_1T⇒ Z𝔇_2 that is an equivalence on small objects. 


The commutativity of the square of right adjoints tells us there is a natural equivalence T𝔇_2'≅𝔇_1'Z. Hence we have a mate 𝔇_1T⇒ Z𝔇_2 defined as the composition 

    𝔇_1T⇒𝔇_1T𝔇_2'𝔇_2≅𝔇_1𝔇_1'Z𝔇_2⇒ Z𝔇_2 ,

that can also be depicted as -1ex 
[>=stealth,<-] (0.4,0.4) to (0,0.4) ;
[>=stealth,<-] (0,0.4) to (0,0) ;
[>=stealth,thin,double,->] (0.5,0.2) to (0.9,0.2) ;
[>=stealth,<-] (1.4,0.4) to (1,0.4) ;
[>=stealth,<-] (1,0.4) to (1,0.1) ;
[>=stealth,<-] (1,0.1) to (1.4,0.1) to (1.4,0) to (1,0) ;
[>=stealth,thin,double,<->] (1.5,0.2) to (1.9,0.2) ;
[>=stealth,<-] (2.4,0.4) to (2,0.4) to (2,0.3) to (2.4,0.3) ;
[>=stealth,<-] (2.4,0.3) to (2.4,0);
[>=stealth,<-] (2.4,0) to (2,0) ;
[>=stealth,thin,double,->] (2.5,0.2) to (2.9,0.2) ;
[>=stealth,<-] (3.4,0.4) to (3.4,0) ;
[>=stealth,<-] (3.4,0) to (3,0) ;
 . 
We then observe that 


  – on small objects, the unit id⇒𝔇_2'𝔇_2 is an equivalence. 

  – Z𝔇_2 sends small objects to good objects (𝔇_2 realizes an equivalence between smalls and goods, and Z preserves the goods). 

  – on good objects, the co-unit 𝔇_1𝔇_1'⇒ id is an equivalence. 

Hence the mate 𝔇_1T⇒ Z𝔇_2 is an equivalence on small objects. 

The commutativity of the square then reads as follows (recall that we are reasoning on the category of small objects): 

    T^*Ψ_1     =    Hom_ D_1^op(-,𝔇_1T(-))  (by definition)
       ≃   Hom_ D_1^op(-,Z𝔇_2(-))  (using the mate)
       ≃   Hom_ D_2^op(Y(-),𝔇_2(-))  (by adjunction)
        =    Ψ_2Y^op  (by definition).

Finally, we have to prove that the triangle commutes. This is obvious: X∘ T(E_n,2)≃ X(E_n,1). 


Let us see what it implies for our preferred morphism of Koszul duality context:

    @C=100pt@R=50pt_@<-2pt>[d]_-𝔏@<2pt>[r]^-𝔇   @<2pt>[l]^-CE^∙_^op@<-2pt>[d]_-forget
    _@<-2pt>[u]_-V ↦⊕ V[-1]@<2pt>[r]^-Hom(-, )   _^op@<2pt>[l]^-Hom(-, )@<-2pt>[u]_-free

Note that we are in the -linear situation, hence viewing the tangent complex as an object in _. Let ϑ denote the functor V ↦⊕ V[-1]. 
If X is in FMP_, then X∘ϑ belongs to FMP(_,([n+1])_n), and thus 𝕋_X=𝕋_X∘ϑ. Now: 

    𝕋_X∘ϑ=Θ_2Φ_2(X∘ϑ)=Φ_2(X∘ϑ)[1]=forget Φ_1(X)[1] .
 
Hence we obtain the following beautiful result, explaining one of the phenomenon mentionned in the introduction: 


The underlying complex of the dgla 𝔤_X:=Φ_1(X) attached to X is 𝕋_X[-1].



The following proposition describes the tangent complex of a representable moduli problem: 

Let A be a representable element in FMP_. Then its tangent complex 𝕋_A, considered in _, is equivalent to 𝕋_A⊗_A, with 𝕋_A:=(𝕃_A/k)^∨. 


It is a simple calculation: 

    (𝕋_A)_n:=Hom__(A, ⊕[n])    ≃Hom__(𝕃_/A, [n+1]) 
       ≃Hom__(𝕃_A/⊗_A[1], [n+1]) 
       ≃𝕋_A⊗_A[n]

Hence, using the canonical identification Stab(_)≃_ we get that 𝕋_A≃𝕋_A⊗_A. 

As a consequence, we get that 𝕋_/A≃𝕋_A⊗_A[-1] carries a dgla structure. 



§ DG-LIE ALGEBROIDS AND FORMAL MODULI PROBLEMS UNDER SPEC(A)


In this part, A will denote a fixed cdga over  that is concentrated in non-positive degree and cohomologically bounded. In what follows the boundedness hypothesis is important; we will explain precisely where it has to be used.



 §.§ Split formal moduli problems under Spec(A)
 

One of the main purpose of the work <cit.> (in the local case) is to prove that the equivalence provided by Theorem <ref> can be extended when replacing the ground field  by A. We have (see Lemma <ref>)

    Stab (_A/A) ≃_A.

We consider the deformation context (_A/A, (A ⊕ A[n])_n). Then Hennion's result runs as follows:
[Hennion <cit.>]
If A is a cohomologically bounded cdga concentrated in non-positive degree, there is an isomorphism

    FMP_A/A≃_A ,

where FMP_A/A:=FMP(_A/A, (A ⊕ A[n])_n).


In <cit.>, there is the additional assumption that H^0(A) is noetherian, but it does not appear to be necessary. 


The strategy is to produce a Koszul duality context

    (_A/A, A ⊕ A[n])  (_A^op, free   A[-n-1])

and then to apply Theorem <ref>. The Chevalley–Eilenberg construction can be performed over A, but we must take its derived version (because the functor M →S_A(M) does not respect weak equivalences). Concretely, assuming that A is cofibrant, the Chevalley Eilenberg is defined in the usual way on A-dg-Lie algebras that are projective as A-modules. Since every element in _A is isomorphic to a dg-Lie algebra whose underlying A-module is projective, we get a
functor CE^∙_A _A^op→_A/A, and it admits a left adjoint 𝔇_A. We consider again a diagram of the form

    @C=100pt@R=50pt_A/A@<-2pt>[d]_-ℒ_A@<2pt>[r]^-𝔇_A   @<2pt>[l]^-CE^∙_A_A^op@<-2pt>[d]_-forget
    _A@<-2pt>[u]_-V → A ⊕ V[-1]@<2pt>[r]^-Hom_A(-, A)   _A^op@<2pt>[l]^-Hom_A(-, A)@<-2pt>[u]_-free

where 𝔏_A(R)=𝕃_A/R and try to construct a morphism between weak Koszul duality contexts. This works exactly as in the proof of Proposition <ref>, but some aspects have to be taken care of in condition (C) of morphisms between weak Koszul duality contexts: the fact that theforgetful functor commutes with with sifted limits is explained in the proof of <cit.>. On the other hand, the cohomological boundedness of A is crucial, otherwise the forgetful functor (although conservative) wouldn't send good objects to reflexive objects. 
More precisely, even if Lemma <ref> still holds in this context[A very good dgla over A is a good dgla over A such that 


  – the underlying A-module is projective. 

  – the underlying graded Lie algebra is freely generated over A by finitely many generators in positive degree. 
], very good dglas over A are not necessarily reflexive as A-modules (for instance if A is not a bounded -algebra, then free   A[-2] is not quasi-isomorphic to its double dual as a complex of A-modules). But they are whenever A is bounded. 


All results of the previous Section remain true if one replaces  with a bounded A. For instance, we have the following analog of Proposition <ref>: 

Let B be a representable element in FMP_A/A. Then its tangent complex 𝕋_B, considered in _A, is equivalent to 𝕋_B/A⊗_BA, with 𝕋_B/A:=(𝕃_B/A)^∨. 

In particular if B=A ⊗_ A, with A-augmentation being given by the product, then 

    𝕃_A ⊗_ A/A⊗_A ⊗_ AA≃𝕃_A

and thus 𝕋_A ⊗_ A≃𝕋_A. 
 
The A-module 𝕋_A[-1] is a Lie algebra object in _A. 

In <cit.>, Hennion proves global versions of the above results, and shows in particular that if X is an algebraic derived stack locally of finite presentation, then 𝕋_X[-1] is a Lie algebra object in 𝐐𝐂𝐨𝐡(X). This again explains (and generalizes) a phenomenon that we mentioned in the introduction. 



 §.§ DG-Lie algebroids


In this section, we introduce the notion of dg-Lie algebroids, which is the dg-enriched version of Lie algebroids. This will be used to construct a Koszul duality context for A-augmented k-algebras in the next section. Informally, a (dg)-Lie algebroid is a Lie algebra L over  such that L is an A-module, A is a L-module, both structures being compatible. More precisely: 

A (, A)-dg-Lie algebroid is the data of a dgla L over  endowed with an A-module structure, as well as an action of L on A, satisfying the following conditions: 


  – L acts on A by derivations, meaning that the action is given by a A-linear morphism of -dglas ρ L →Der_(A), called the anchor map.

  – The following Leibniz type rule holds for any a∈ A and any ℓ_1,ℓ_2∈ L: 

    [ℓ_1 , aℓ_2]=(-1)^|a| × |ℓ_1|  a[ℓ_1 , ℓ_2] + ρ(ℓ_1)(a)ℓ_2


Morphisms of (, A)-dg-Lie algebroids are A-linear morphisms of dglas over  commuting with the anchor map.

  

  – Every A-linear dgla defines a dg-Lie algebroid: it suffices to keep the same underlying object and to set the anchor map to zero. On the other hand, if L is a (, A)-dg-Lie algebroid, the kernel of the anchor map is a true A-linear dgla. These two constructions are adjoint. 

  – Given a pair (, A), it is possible (see <cit.>) to attach to any object V of _A / Der_(A) a free (, A)-dg-Lie algebroid, denoted by free  (V). The functor V →free  (V) is the left adjoint to the forgetful functor from (, A)-dg-Lie algebroids to A-modules lying over Der_(A). 

  – The Chevalley–Eilenberg construction can be performed for (, A)-dg-Lie algebroids. If L is a (, A)-dg-Lie algebroid, then 
the A-module underlying CE_/A^∙(L) is the A-dual of S_A(L[1]) and the differential reads as follows (omitting signs):  

    d_CEω(ℓ_0,…,ℓ_n)=    ±∑_i=0^nω(ℓ_0,…,d_L(ℓ_i),…,ℓ_n)
       ±∑_i=0^nρ(ℓ_i)(ω(ℓ_0,…ℓ̂_̂î,…,ℓ_n))±∑_0≤ i<j≤ nω([ℓ_i,ℓ_j],ℓ_0,…ℓ̂_̂î,…,ℓ̂_̂ĵ,…,ℓ_n) .

However, the same problem appearing for the Chevalley-Eilenberg complex of A-dg-Lie algebras happens: this functor must be derived. Hence, the above definition is set only on (, A)-dg-Lie algebroids that are projective as A-modules (again A is assumed to be cofibrant), and since every (, A)-dg-Lie algebroid is quasi-isomorphic to one whose underlying A-module is projective, this defines a functor CE_/A^∙ from (, A) dg-Lie algebroids to A-augmented -cdgas.

  – If two cdgas A and A' are equivalent, then the ∞-categories _A and _A' are equivalent. It is impossible to expect this kind of result for (, A)-dg-Lie algebroids, due to the presence of the derivations of A (which is not an ∞-functor). This motivates the next forthcoming definition. 



A derived (, A)-dg-Lie algebroid is a (, QA)-dg-Lie algebroid, where QA is a cofibrant replacement of A. We denote by _/A the category of (, QA)-dg-Lie algebroids.

It can be proved that the category _/A has a naturel model structure obtained by transferring the model structure of _QA/Der_(QA) 
via the adjunction 

    free_QA/Der_(QA)_/Aforget

(see <cit.>). 
Hence we get an ∞-category _/A together with an adjunction 

    _A/𝕋_A ⟶
    ⟵ _/A .

where 𝕋_A≃Der_(QA) is the tangent complex of A. Fibrant (, A)-dg-Lie algebroids are those with surjective anchor map.
[Corrigendum] 
Actually, the transferred structure is only a semi-model structure[The main problem is that the initial object 0 is not fibrant.], as shown in <cit.>. One can consider the under-category ^𝔥/_/A, where 𝔥 is a fibrant-cofibrant replacement of the initial Lie algebroid 0. It happens to be a genuine combinatorial model category that is obviously Quillen equivalent to _/A (see <cit.>). Hence, from our point of view everything is still fine, as the ∞-category _/A is indeed presentable. 

In the sequel, we will always assume that A is cofibrant (which is possible after taking a cofibrant replacement). In this way, derived (,A)-dg-Lie algebroids will be usual (,A)-dg-Lie algebroids. 

Observe that _/A can be made into a dual deformation context: namely, we transfer the dual deformation context (_A/𝕋_A,(A[-n-1]0→𝕋_A)_n) from Example <ref> along the adjunction

    forget  _/A^op_A/𝕋_A^opfree .




 §.§ Formal moduli problems under Spec(A)


In this section, we are looking at a more general situation as in <ref>: we look at A-augmented -algebras, where A is a bounded cdga in non-positive degrees, which we assume to be cofibrant. 
Recall from Remark <ref> that we have a deformation context (_/A,(A ⊕ A[n])_n). The aim of this section is to give a description of formal moduli problems for A-augmented cdgas over . 
The motto behind what follows will be the following one:


Going from A-augmented A-algebras to A-augmented -algebras corresponds via Koszul duality to go from Lie algebras over A to (, A) Lie algebroids. 


The precise result we want to prove is: 
[Nuiten <cit.>]
Given a cdga A as above, there is a Koszul duality context 

    (_ /A, (A ⊕ A[n])_n) (_/A^op, (free (A[-n-1]0→𝕋_A)_n)

This in particular implies that there is an equivalence 

    FMP(_/A, A ⊕ A[n]) ≃_/A .




The first step consists in building the weak Koszul duality context. A reasonable candidate for the right adjoint functor is the Chevalley–Eimenberg functor 

    CE^∙_/A_/A^op⟶_/A .

Here are a few properties of the Chevalley–Eilenberg functor that carry on to the Lie algebroid setting: 


  – CE^∙_/A(L)≃RHom_U(A,L)(A,A), where U(A,L) is the universal envelopping algebra of L (see e.g. <cit.> for the definition of U(A,L))[In <cit.>, (,A)-Lie algebroids are named (,A)-Lie algebras, and everything that is done makes sense in the differential graded setting as well.]. 

  – To every Mf→𝕋_A=Der(A) one can associate a derivation d_f:A→ M^∨ and thus a non-split square zero extension Ad_f⊕M^∨[-1]. It is a fact that CE^∙_/A(free (f))≃ Ad_f⊕M^∨[-1]. 

  – CE^∙_/A commutes with small limits (viewed as a functor on _/A^op) and thus, since _/A is presentable, admits a left adjoint, that we denote by 𝔇_/A. 

Thus we have a weak Koszul duality context 

    𝔇_/A(_ /A, (A ⊕ A[n])_n) (_/A^op, (free (A[-n-1]0→𝕋_A)_n)CE^∙_/A

The next step is to prove that it is a Koszul duality context, by following the strategy of Proposition <ref>. We first consider the diagram 

    @C=80pt@R=40pt_ /A   [l]_-CE^∙_/A_/A^op
    
    ^𝕃_A/_A [u]_-(𝕃_Ad→M)↦ Ad⊕M[-1]   [l]_-(-)^∨  (_A/𝕋_A)^op[u]_-free

that one can fill everywhere with left adjoints, giving rise to the following weak morphism of weak Koszul duality contexts:

    @C=80pt@R=40pt_ /A@<-2pt>[d]_-𝔏: (B→ A) ↦ (𝕃_A→𝕃_A/B)@<2pt>[r]^-𝔇_/A   @<2pt>[l]^-CE^∙_/A_/A^op@<-2pt>[d]_-forget
    
    ^𝕃_A/_A   @<2pt>[r]^-(-)^∨@<-2pt>[u]_-d↦ Ad⊕M[-1]   @<2pt>[l]^-(-)^∨  (_A/𝕋_A)^op@<-2pt>[u]_-free

Knowing, from Example <ref>, that the bottom weak Kozul duality context is actually a Koszul duality context, according to Proposition <ref> it remains to prove that this weak morphism is a morphism. The functor forget is conservative, and it is proved in <cit.> that it preserves sifted limits. Let us now show that, if L is good, then it is equivalent to a very good dg-Lie algebroid: i.e. a good dg-Lie algebroid that is of the form 𝔤0→𝕋_A for a very good dgla 𝔤 over A. 

Any good dg-Lie algebroid L is quasi-isomorphic to a very good one. 


We first observe that 0 is very good. We then proceed by induction: assume that L is very good, and consider a dg-Lie algebroid L' obtained by a pushout 

    free (A[-n-1]0→𝕋_A)   [r] [d]     L [d] 
    
    0 [r]     L'

in _/A. Since free (A[-n-1]0→𝕋_A) is cofibrant, then a morphism free (A[-n-1]0→𝕋_A)→ L in _/A is represented by a morphism 
free (A[-n-1]0→𝕋_A)→L in _/A, where L is a chosen fibrant replacement of L. Now, we observe that one can choose L=L⊕𝔥, where 𝔥 is a fibrant replacement of 0, so that the replacement morphism L→L splits. All in all, any morphism free (A[-n-1]0→𝕋_A)→ L in _/A can be represented by an actual morphism free (A[-n-1]0→𝕋_A)→ L in _/A. Next, a cofibrant replacement of the left vertical arrow is given by the morphism

    free (A[-n-1]0→𝕋_A)→free {cone (A  A)[-n-1]0→𝕋_A }.

Hence our pushout in _/A is represented by the following honest non-derived pushout in _/A: 

    free (A[-n-1]0→𝕋_A)   [r] [d]     L [d] 
    free {cone (A  A)[-n-1]0→𝕋_A)  }[r]     L'

Now recall that, by induction, L=(𝔤0→𝕋_A) for some very good dgla 𝔤 over A. 
We finally observe that the functor 

    (-)0→𝕋_A_A⟶_/A

commutes with colimits, so that L'=(𝔤'0→𝕋_A) for a very good dgla over A. 

This in particular shows that, again under the boundedness assumption on A, forget sends good objects to reflexive objects; hence (C) holds. It remains to prove (D). 

We consider the natural morphism θ_L defined as the following composition: 

    𝔏(CE^∙_/A(L))→𝔏(CE^∙_/A(free forget L))→𝔏(Ad_ρ⊕(forget L)^∨[-1])→ (forget L)^∨ ,

where ρ is the anchor map[In the above, (forget L)^∨ shall be understood as 𝕃_Aρ^∨→L^∨. ] and 𝔏 is the left adjoint to the twisted square zero extension functor given by the formula 𝔏(B→ A) = (𝕃_A→𝕃_A/B). Again, the cdga morphism 

    CE^∙_/A(L)→CE^∙_/A(free forget L)→ Ad_ρ⊕L^∨[-1]

is nothing but the obvious projection (onto the quotient by the square of the augmentation ideal whenever L is very good). 

As in the proof of Proposition <ref> we consider the uncompleted Chevalley–Eilenberg sub-cdga CE^∙_/A(L)↪CE^∙_/A(L). 
The quotient by the square of its augmentation ideal is again Ad_ρ⊕L^∨[-1], and we have the following commuting diagram and its image through the left-most vertical adjunction of our square: 

    CE_/A^∙(L) [d]_ι_L[rd]    
    CE^∙_/A(L) [r]     Ad_ρ⊕(forget L)^∨[-1]
        𝔏(CE^∙_/A(L)) [d]_𝔏(ι_L)[rd]^θ_L   
    𝔏(CE^∙_/A(L)) [r]_-θ_L    (forget L)^∨

In order to prove that θ_L is an equivalence when L is good, it suffices to prove that both θ_L and 𝔏(ι_L) are. 
Let us start with the following variation on Lemma <ref>: 

Let B be an A-augmented -cdga, with augmentation ideal 𝒥, that is cofibrant as a -cdga, and such that the augmentation morphism is a fibration. 
Then 𝒥/𝒥^2[1] is naturally an A-module under 𝕃_A, and there is an equivalence 
𝔏(B)≃(𝕃_A→𝒥/𝒥^2[1]) in ^𝕃_A/_A. 


First of all, observe that the exact sequence

    0 →𝒥/𝒥^2 → B/𝒥^2 → A → 0

splits in _. 
Hence there is a -derivation δ :A→𝒥/𝒥^2[1] such that B/𝒥^2 is isomorphic to Aδ⊕𝒥/𝒥^2 in _/A. 
Hence, 

    Hom__/A(B, B/𝒥^2) 
    = Hom__/A(B,  A δ⊕𝒥/𝒥^2)
    ≃Hom_  ^𝕃_A/_A(𝔏(B), 𝒥/𝒥^2[1]) .

Therefore, the quotient morphism B→ B/𝒥^2 gives us a morphism 𝔏(B)→ (𝕃_A→𝒥/𝒥^2[1]). 
To show that this is an equivalence it is sufficient to prove that the morphism 𝕃_A/B→𝒥/𝒥^2[1] is an equivalence. Now, since A and B are cofibrant, and B→ A is fibrant, 

    𝕃_A/B≃ cone(Ω^1_B⊗_BA→Ω^1_A)≃(Ω^1_B⊗_BA→Ω^1_A)[1] .

The morphism 𝕃_A/B→𝒥/𝒥^2[1] is thus modeled by the shift of the natural map 

    (Ω^1_B⊗_BA→Ω^1_A)→𝒥/𝒥^2 ,

which is an isomorphism. 













We then observe that 


  – the augmentation map CE^∙_/A(L)→ A is obviously a fibration. 

  – when L is very good (which we can always assume without loss of generality when dealing with good dg-Lie algebroids), then CE^∙_/A(L) is cofibrant. 

Thus θ_L is an equivalence if L is good. 
We again conclude with the very same flatness argument as in the proof of Proposition <ref> in order to get that 𝔏(ι_L) is an equivalence. 




 §.§ The relative tangent complex


We want to compute the underlying anchored module of the dg-Lie algebroid associated with a representable fmp under Spec(A). 
We have an equivalence

    Φ_/A𝐅𝐌𝐏(_/A)⟶̃_/A


Let B be the representable fmp associated with an A-augmented -algebra B. Then the image of Φ_/A(B) along the forgetful functor _/A→_A/𝕋_A is 𝕋_A/B→𝕋_A. 


Using Proposition <ref> together with Theorem <ref>, one sees that the image of Φ_/A(B) along the forgetful functor is equivalent to the image of the representable fmp 𝕃_A→𝕃_A/B on ^𝕃_A/_A along the equivalence 𝐅𝐌𝐏(^𝕃_A/_A)→̃_A/𝕋_A. One easily sees that the image of this representable fmp is indeed 
(𝕃_A→𝕃_A/B)^∨=𝕋_A/B→𝕋_𝔸. 

According to the above proposition, the image of Φ_/A(X) of an fmp under Spec(A) along the forgetful functor _/A→_A/𝕋_A deserves to be denoted 𝕋_Spec(A)/X and called the “relative tangent complex of Spec(A) over X”. 

Unsurprisingly, an easy calculation allows to prove that the tangent complex to an fmp under Spec(A) is 

    𝕋_X≃cofib(𝕋_Spec(A)/X→𝕋_A) ,

where 𝕋_X can be viewed in _A because _/A is A-linear. 

In the case A=k, we get back the Lie structure[A dg-Lie (,)-algebroid is nothing but a dgla. ] on 

    𝕋_X[-1]≃fib(0→𝕋_X)=fib(𝕋_→𝕋_X)≃𝕋_Spec()/X .



Even if A is not  itself, we have a fiber sequence of Lie algebroids

    𝕋_X[-1]→𝕋_Spec(A)/X→𝕋_A

in _A, where 𝕋_Spec(A)/X and 𝕋_A are (,A)-dg-Lie algebroids, and 𝕋_Spec(A)/X→𝕋_A is a morphism in _/A. 
The forgetful functor _/A→_A/𝕋_A being a right adjoint, there is a (,A)-dg-Lie algebroid structure on 𝕋_X[-1] that allows to upgrade the above sequence into a fiber sequence of (,A)-dg-Lie algebroids. One can then show that the functor “taking the fiber of the anchor” actually factors through dglas over A. Here is a question we would like to answer : 


What is the split formal moduli probem under Spec(A) having 𝕋_X[-1] as associated dgla?



We consider the commuting diagram: 

    @C=80pt@R=40pt_ /A   [l]_-CE^∙_/A_/A^op
    _A/A[u]_-forget   [l]_-CE^∙_A_A^op[u]_-𝔤↦(𝔤0→𝕋_A)

It can be filled with left adjoints in the following manner: 

    @C=80pt@R=40pt_ /A@<-2pt>[d]_-V → V ⊗_ A@<2pt>[r]^-𝔇_/A   @<2pt>[l]^-CE^∙_/A_/A^op@<-2pt>[d]_-Ξ
    _A/A@<2pt>[r]^-𝔇_A@<-2pt>[u]_-forget   @<2pt>[l]^-CE^∙_A_A^op@<-2pt>[u]_-𝔤↦(𝔤0→𝕋_A)

Here the right-most vertical adjunction is obtained from a Quillen adjunction: in particular Ξ is the left[Recall that we are working with opposite categories: Ξ:_/A^op→_A^op. ] derived functor of the functor given on models by the kernel of the anchor map. 

Note that the forgetful functor 

    _A/A→_k/A
 
maps _A/A^sm to _k/A^sm; however it is not an equivalence: _k/A^sm is way bigger than _A/A^sm. Typically, non-split square-zero extensions do not belong to _A/A^sm. 


The above square of adjunctions is a weak morphism of Koszul duality contexts. 


The only thing left to prove is condition (B), which is obvious: indeed, forget (A⊕ A[n]) is precisely A⊕ A[n], viewed as a -algebra. 

Consequently, Proposition <ref> tells us that we have the following commuting diagram 

    @C=40pt@R=20pt_/A[r]^-Ψ_/A[dd]^Ξ   FMP_/A[dd]^-∘forget[rd]^-𝕋   
          _A 
    _A [r]^-Ψ_A   FMP_A/A[ru]_-𝕋

Let X be a formal moduli problem under Spec(A), and let L_X:=Φ_/A(X) be its relative tangent Lie algebroid. We then have: 

    Ξ(L_X)≃𝔤_X∘forget .

Hence the Lie dgla structure on 𝕋_X[-1] obtained by taking the fiber of the anchor map of the relative tangent Lie algebroid L_X coincide with the one given on 𝕋_X∘forget. 
This answers our question: the split formal moduli problem under Spec(A) having Ξ(L_X) as tangent Lie algebra is X∘forget. 

Geometrically, and following the intuitive ideas presented in the introduction, one shall understand X as a formal thickening Spec(A)→ X of Spec(A). 
The geometric interpretation of X∘forget is then as the split formal thickening Spec(A)→ Spec(A)× X→ Spec(A) given by the graph[Actually, the formal neighborhood of the graph. ] of the previous one. 
Making this idea more precise is a bit complicated as Spec(A) is initial for the two ∞-categories of fmp's that we are considering. 
Hence Spec(A)×- does nothing, and shall be understood as composing with the forgetful functor. 




 §.§ Generalizations


Before going to the last, more geometric, Section of this survey, let us mention a few unnecessary assumptions that we have made for the sake of exposition: 


  – one can replace  with any cdga sitting (cohomologically) in non-positive degree and containing ℚ (as a sub-cdga). 

  – one can replace A with any -cdga with a reflexive cotangent complex 𝕃_A/ in _A. 

  – one can replace  and A with presheaves of such cdgas on some given ∞-category. 



One could ask if all this still works for commutative algebra objects in a gentle enough stable model category M, e.g. satisfying the standing assumptions of <cit.>. 
Ideally, we would like to be able to allow M to be the category of graded mixed modules (see <ref>), and cover the following situation: =𝔻_X_DR is the mixed crystalline structure sheaf of a derived Artin stack X and A=ℬ_X is the mixed principal parts of X (after <cit.>), which are graded mixed cdgas (see <ref>, again). We will come back to the comparison with the <cit.> in the next Section. 

In particular, let → B→ A be a sequence of cdgas, with  and B sitting cohomologically in non-positive degree, and both 𝕃_A/B and 𝕃_A/ reflexive in _A. One can show in a very similar fashion that the weak morphism of Koszul duality contexts from the previous subsection generalizes as follows: 

    @C=80pt@R=40pt_ /A@<-2pt>[d]_-V → V ⊗_B A@<2pt>[r]^-𝔇_/A   @<2pt>[l]^-CE^∙_/A_/A^op@<-2pt>[d]_--𝕋_A/×𝕋_A/B
    _B/A@<2pt>[r]^-𝔇_B/A@<-2pt>[u]_-forget   @<2pt>[l]^-CE^∙_B/A_B/A^op@<-2pt>[u]_-(L→𝕋_A/B)↦(L→𝕋_A/)

This leads (again after Proposition <ref>) to a commuting square 

    @C=40pt@R=20pt_/A[r]^-Ψ_/A[dd]^-𝕋_A/×𝕋_A/B   FMP_/A[dd]^-∘forget[rd]^-𝕋   
          _A 
    _B/A[r]^-Ψ_B/A   FMP_B/A[ru]_-𝕋

If X lies in FMP_/A, then it has a relative tangent Lie algebroid 𝕋_Spec(A)/X→𝕋_A/. 
It follows from the above that the relative tangent Lie algebroid[Note that we slightly abuse the notation here. As we have seen in Proposition <ref>, 𝕋_A/B is the underlying A-module of the (,A)-dg-Lie algebroid Ψ_/A(B). ] 𝕋_Spec(B)/X∘forget of X∘forget, viewed as a (,A)-dg-Lie algebroid using the Lie algebroid morphism 𝕋_A/B→𝕋_A/, is equivalent to the pull-back 

    𝕋_Spec(A)/X𝕋_A/×𝕋_A/B .

In the spirit of Remark <ref>, one geometrically interprets X∘forget as the following “Spec(B)-split formal thickening of Spec(A)”: 

    Spec(A)→ Spec(B)× X .


One can actually show that _B/A≃ (_/A)_/𝕋_A/B. 
In particular, this shows that FMP_B/A≃ (FMP_/A)_/B. 

This shall be interpreted geometrically as follows. If we have a Spec(B)-split formal thickening of Spec(A), then the map X→ Spec(B) uniquely factors through X→B, which is terminal among such formal thickenings (being the formal completion of the map Spec(A)→ Spec(B)). 


There is yet another geometric situation one may want to understand in terms of Lie algebroids: given a fmp Spec(B)→ X under Spec(B), one can look at it under Spec(A) by just composing[And, maybe, completing afterwards. ] with Spec(A)→ Spec(B). On the level of anchored modules, we have a functor 

    _B/𝕋_B/   ⟶   _A/𝕋_A/
    
    L    ⟼    L⊗_BA𝕋_B/⊗_BA⊕𝕋_A/ .

Assume that B→ A is a cofibration betwen cofibrant cdgas. Hence the morphism 𝕋_B/⊗_BA→𝕋_A/ in _A is represented by dϕ:Der_(B)⊗ A→Der_(A) in _A, which is moreover a fibration. Hence the above fiber product in _A can be computed as an ordinary fiber product in _A: more precisely, it is (ρ⊗_BA-dϕ), where ρ:L→Der_(B) is the anchor map. In this situation, it has been shown by Higgins and Mackenzie <cit.> that one can put a Lie bracket on L that turns it into a Lie algebroid[In <cit.> the authors work in the differential setting and nothing is dg. Nevertheless, their construction carry on without any problem. Their only technical assumption is that the map along which they pull-back is submersive, which is fine in our setting thanks to the cofibrancy of B→ A. ]. We therefore get a functor _/B→_/A that actually factors as 

    _/B⟶  ^𝕋_A/B/_/A⟶_/A


The main upshot of this sketchy discussion is that if we are given a commuting square of cdgas

    B_1 [r] [d]     B_2 [d] 
    
    A_1 [r]     A_2
 
where A_1 and B_1 cohomologically sit in non-positive degree and contain ℚ, and 𝕃_A_2/B_1 and 𝕃_A_2/A_1 are reflexive in _A_2, then we have a pull-back functor 

    _B_1/B_2→_B_1/A_2→_A_1/A_2 .




§ GLOBAL ASPECTS




 §.§ Formal derived prestacks and formal thickenings


We start with several definitions and statements, mainly extracted from <cit.>. 

Let us denote by 𝐝𝐀𝐟𝐟_^f.p. the opposite ∞-category of non-positively graded -cdgas A that are almost finitely presented: H^0(A) is finitely generated as a -algebra and H^i(A) is a finitely presented H^0(A)-module. We then define the ∞-category 𝐝𝐏𝐫𝐒𝐭_^f.p. of locally almost finitely presented derived prestacks over  to be the ∞-category of presheaves on 𝐝𝐀𝐟𝐟_^f.p.. 

We say that a locally almost finitely presented derived prestack F is nilcomplete (convergent in the terminology of <cit.>) if the canonical map 

    F(A)→nlimF(A_≤ n)

is an equivalence, where A_≤ n denotes the n-th Postnikov truncation of A. 

The full sub-∞-category 𝐝𝐏𝐫𝐒𝐭_^f.t. of 𝐝𝐏𝐫𝐒𝐭_^f.p. spanned by nilcomplete prestacks[The superscript “f.t.” stands for (locally almost of) finite type. ] is equivalent to the essential image of the right Kan extension from the full sub-∞-category 𝐝𝐀𝐟𝐟_^f.t.⊂𝐝𝐀𝐟𝐟_^f.p. spanned by eventually coconnective cdgas (see <cit.>), where a cdga A is called eventually coconnective if H^n(A)=0 for n<<0. 

A formal derived prestack is a nilcomplete locally almost finitely presented derived prestack[Nilcomplete locally almost finitely presented prestacks are called said to be locally almost of finite type (laft) in <cit.>. ] F with the following to properties: 


  – F is infinitesimally cohesive: for any cartesian square of almost finitely presented non-positively graded -cdgas 

    A [r][d]     A_1 [d] 
    
    A_2 [r]     A_0

such that each H^0(A_i)→ H^0(A_0) is surjective with nilpotent kernel, then the induced square 

    F(A) [r][d]     F(A_1) [d] 
    
    F(A_2) [r]     F(A_0)

is cartesian as well. 

  – F admits a procotangent complex. Here we say that F admits a procotangent complex if it admits a procotangent complex 𝕃_F,x at every point x:Spec(A)→ F, and for every morphism of points 

    Spec(A) [rr]^u [rd]^x        Spec(B)[ld]^y 
        F

the induced morphism u^*:𝕃_F,y⊗_BA→𝕃_F,x is an equivalence. We say that F admits a procotangent complex at a given point x:Spec(A)→ F if the functor 𝔻er_F(x,-) of derivations is co-prorepresentable. 


We denote by FdPrSt_ the ∞-category of formal derived prestacks over . 

For a derived prestack F, we denote by F_red its associated reduced prestack F_red:=Spec(A)→ FcolimSpec(A_red), where A_red denotes the quotient of H^0(A) by its maximal nilpotent ideal.  We also consider the de Rham prestack F_DR of F, that is defined as follows: F_DR(A):=F(A_red). We now summarize the properties of reduced and de Rham prestacks: 


  – there are canonical maps F_red→ F→ F_DR, inducing equivalences (F_red)_DR→̃F_DR and F_red→̃(F_DR)_red. This implies in particular that a morphism F→ G induces an equivalence F_red→̃G_red between their  reduced prestacks if and only if it induces an equivalence  F_DR→̃G_DR between their de Rham prestacks. 

  – (-)_red is left adjoint to (-)_DR. 



For any formal derived prestack X, one can consider the sub-∞-category 𝐓𝐡𝐢𝐜𝐤(X) of ^X/FdPrSt spanned by those morphisms X→ F that induce an equivalence X_red→ F_red between the associated reduced prestacks. We call it the ∞-category of formal thickenings of X. 

[<cit.>, Chapter 5, Proposition 1.4.2]
Let A be a non-positively graded cdga that is almost finitely presented and eventually coconnective (i.e. A is almost of finite type). Then there is an equivalence of ∞-categories 

    FMP_/A≃𝐓𝐡𝐢𝐜𝐤(Spec(A)) .
 


We have a pull-back functor 𝐓𝐡𝐢𝐜𝐤(Spec(A))→FMP_/A along 

    _/A^sm→_^f.t.:=(𝐝𝐀𝐟𝐟_^f.t.)^op ,

that consists in evaluating our prestack on Spec(A)→ Spec(B) with B a small A-augmented -cdga. This pull-back functor has a left adjoint, given by a left Kan extension[Observe that the left Kan extension of a functor on small A-augmented cdgas is a priori just a derived prestack locally almost of finite type. It is an exercise to check that left Kan extensions of fmps are sent to formal thickenings. The only non-obvious property to check that it admits a procotangent complex, which follows from the prorepresentability of fmps. ]. 
One can prove (using infinitesimal cohesiveness) that a formal thickening is completely determined by its restriction on Spec(B)'s for which B_red≃ A_red. We conclude by observing that the functor _/A^sm→_^f.t. obviously factors through the category of those cdgas of finite type with reduced algebra equivalent to A_red, and that this has a left adjoint: sending a B such that B_red≃ A_red to A×_A_redB. 

As a consequence, we have the following useful Corollary: 

For any non-positively graded cdga A almost of finite type, there is an equivalence 

    FMP_/A≃ ^A/FMP_/A_red .

Moreover, for any formal derived prestack X→ Spec(A) over Spec(A) that exhibits Spec(A) as a formal thickening of X, we have an equivalence 

    𝐓𝐡𝐢𝐜𝐤(X)_/Spec(A)≃ ^X/FMP_A/A_red .


In the second part of the Corollary, we abusively denote by the same symbol the formal derived prestack X and its associated formal moduli problem under Spec(A_red). 

Let X→ Y be a morphism of formal derived prestacks that induces an equivalence X_red→̃Y_red between their associated reduced prestacks. Then we obviously have an equivalence 𝐓𝐡𝐢𝐜𝐤(Y)≃ ^Y/𝐓𝐡𝐢𝐜𝐤(X). 

To prove the first part of the Corollary, we begin with the obvious observation that the map Spec(A_red)→ Spec(A) is a formal thickening of Spec(A_red). Hence we have an equivalence 

    𝐓𝐡𝐢𝐜𝐤(Spec(A))≃ ^Spec(A)/𝐓𝐡𝐢𝐜𝐤(Spec(A_red)) .

Then we notice that the fmp A∈𝐅𝐌𝐏_/A_red is the pullback of Spec(A) along 
_/A_red^sm→_^f.t..  
We finally use the above Proposition to get a chain of equivalences 

    FMP_/A≃𝐓𝐡𝐢𝐜𝐤(Spec(A))≃ ^Spec(A)/𝐓𝐡𝐢𝐜𝐤(Spec(A_red))≃ ^A/FMP_/A_red .

For the second part of the statement, we observe that X→ Spec(A) being a formal thickening means that the induced map X_red→ Spec(A_red) is an equivalence. This in turns teaches us that Spec(A_red)≃ X_red→ X is a formal thickening, and thus can be understood as a fmp under Spec(A_red). Hence we have  

    𝐓𝐡𝐢𝐜𝐤(X)≃ ^X/𝐓𝐡𝐢𝐜𝐤(Spec(A_red))≃ ^X/𝐅𝐌𝐏_/A_red ,

As a consequence we get that 𝐓𝐡𝐢𝐜𝐤(X)_/Spec(A)≃ ^X/(𝐅𝐌𝐏_/A_red)_/A. We conclude by observing that (𝐅𝐌𝐏_/A_red)_/A≃𝐅𝐌𝐏_A/A_red (see Subsection <ref>). 




 §.§ DG-Lie algebroids and formal thickenings


The consequences of Corollary <ref> are of particular interest. Let B→ A be a morphism in _^f.t., and assume that the induced morphism B_red→ A_red is an isomorphism. 

We have equivalences: 


  – FMP_/A≃ ^𝕋_A_red/A/_/A_red. 

  – FMP_B/A≃ ^𝕋_B_red/A/_B/B_red. 



This is a direct consequence of (the proof of) Corollary <ref>, Theorem <ref>, and Proposition <ref>. 
Indeed, we have two chains of equivalences: 

    FMP_/A≃ ^A/FMP_/A_red≃ ^𝕋_A_red/A/_/A_red

and

    FMP_B/A≃ ^A/FMP_B/B_red≃ ^𝕋_B_red/A/_B/B_red .


In the rest of this subsection we will provide (sketches of) two alternative proofs that do not rely on formal thickenings and Proposition <ref>. 














We actually prove the following more general result: 

Consider a sequence of -cdgas K→ B→ A, all cohomologically bounded and concentrated in non-positive degree, and with B being small in _/A. Then we have an equivalence 

    FMP_K/B≃ ^B/FMP_K/A .


Observe that B is small in _/A if and only if it is so in _K/A. Also note that if B is almost of finite type and A=B_red, then B is small in _/B_red and thus we recover in particular Proposition <ref>. 

We first observe that there is an adjunction 

    ι_K/B_K/A p ,

where ι(C)=C and p(C)=CA×B. 

Since B is small, and p commutes with limits (being a right adjoint), then p preserves small morphisms and p^*:Fun(_K/B,)→ Fun(_K/A,) sends fmps to fmps. Moreover, for every X∈FMP_K/B, there is a natural morphism B→ p^*X:  

    B=Hom__K/A(B,-)≃ Hom__K/A(B,-)× X(B)→ X(-A×B)=p^*X .

Hence we obtain a functor FMP_K/B→ ^B/FMP_K/A, which we abusively still denote p^*. 

This functor has a right adjoint p_*, which we describe now. 
Since B is small, and ι commutes with pull-backs, then ι preserves small morphisms. 
But it does not preserves fmps, as it doesn't preserve terminal objects. 
To cure this we define, for an X∈ ^B/FMP_K/A, 

    p_*X:=ι^*XX(B)×*∈FMP_K/B ,

where *→ X(B) is given by {id_B}→B(B)→ X(B). 

Let us prove that the adjoint pair (p^*,p_*) is actually an equivalence. 
We start by describing the unit of the adjunction: for every X∈FMP_K/B, we have 

    p_*p^*X	=				ι^*p^*Xp^*X(B)×*
    				≃ 	X(-A×B)X(BA×B)×X(B) .

Hence there is an obvious natural morphism X→ p_*p^*X. 
We claim that it is an equivalence. 
Indeed, for every small C→ B, the square 

    C [r][d]     B [d] 
    
    CA×B [r]     BA×B

is cartesian, and its bottom horizontal morphism is small. 
Therefore X preserves this cartesian square, and thus the unit morphism X→ p_*p^*X is an equivalence. 

We now turn to describing the counit of the adjunction. 
For every X∈ ^B/FMP_K/A, we have 

    p^*p_*X = p_*X(-A×B)=X(-A×B)X(B)* ,

and the counit is the composed morphism p^*p_*X→ X(-A×B)→ X. 
Thus we have, for every C→ A, a composition of pull-back squares 

    p^*p_*X(C) [r][d]    *[d] 
    
    X(CA×B) [r][d]     X(B) [d]
    
    X(C) [r]     X(A)≃*

This shows that the counit morphism p^*p_*X(C)→ X(C) is an equivalence, and ends the proof of the Theorem. 


In the remainder of this subsection, we work under the hypothesis of Theorem <ref>. 
As a consequence of the Theorem we get (using Theorem <ref> and Proposition <ref>) an equivalence

    _K/B≃ ^𝕋_A/B/_K/A .

The functor _K/B→ ^𝕋_A/B/_K/A can be proven to be the derived version of Higgins–Mackenzie pull-back of Lie algebroids (see the end of <ref>). 
The inverse functor is not so easy to describe purely in terms of Lie algebroids, but there is nevertheless a nice way of directly proving that there is an equivalence 

    FMP_K/B≃ ^𝕋_A/B/_K/A

that we explain now. Our strategy is (again) as follows: 


  – we first show that there is a weak Koszul duality context involving _/B and ^𝕋_A/B/_K/A. 

  – we then show that it comes with a weak morphism to a Koszul duality context having the form of Example <ref>. 

  – we finally claim that this weak morphism is a morphism, showing that the original weak Koszul duality context is a Koszul duality context (after Proposition <ref>). 

  – we conclude using Theorem <ref>. 




  
The weak Koszul duality context 

We begin with the following rather evident result: 

We have that 𝔇_K/A(B→ A)≃𝕋_A/B and CE^∙_K/A(𝕋_A/B)≃ B. 
In particular, we have an adjunction 

    𝔇_K/A_K/B(^𝕋_A/B/_K/A)^op CE^∙_K/A .



The first part of the statement is just Proposition <ref>, and the second part follows from the fact that B is small. 
Finally, one concludes using Theorem <ref> and the obvious observation that _K/B≃(_K/A)_/B→ A.  

The deformation context is still the one from Example <ref>: (_K/B,(B⊕ B[n])_n), and the dual deformation context is ^𝕋_A/B/_K/A equipped with the spectrum object 

    𝕋_A/B⊕free (A[-n-1]0→𝕋_A/K) .

for its opposite ∞-category. 

The reader can check directly that this is indeed a spectrum object, but this will also be a consequence of our construction of the weak morphism below (see Lemma <ref>). 

Together, the deformation context and the dual deformation context define a weak Koszul duality context. Indeed: 

    CE^∙_K/A(𝕋_A/B⊕free (A[-n-1]0→𝕋_A/K)) 
    ≃     CE^∙_K/A(𝕋_A/B)A×CE^∙_A(free (A[-n-1])) 
    ≃     BA×(A⊕ A[n]) 
    ≃     B⊕ B[n] .




  
The weak morphism 

We already have the following adjunctions: 

    @C=80pt@R=40pt_K/B@<-2pt>[d]_-(C→ B) ↦ (𝕃_B/K→𝕃_B/C)@<2pt>[r]^-𝔇_K/A   @<2pt>[l]^-CE^∙_K/A(^𝕋_A/B/_K/A)^op
    
    ^𝕃_B/K/_B   @<2pt>[r]^-(-)^∨@<-2pt>[u]_-d↦ Bd⊕M[-1]   @<2pt>[l]^-(-)^∨  (_B/𝕋_B/K)^op


There is an adjunction 

    𝔏_B/𝕋_B/K ^𝕋_A/B/_K/Aℜ

such that 

    𝔏(B[-n-1]0→𝕋_B/K)≃𝕋_A/B⊕free (A[-n-1]0→𝕋_A/K) .



We define 𝔏 as the composition of the following two functors: 


  – _B/𝕋_B/K→ ^𝕋_A/B/_A/𝕋_A/K, sending M→𝕋_B/K to 

    M':=M⊗_BA𝕋_B⊗_BA×𝕋_A/K .

Note that 0'=fib (𝕋_A/K→𝕋_B⊗_BA)≃𝕋_A/B. 

  – ^𝕋_A/B/_A/𝕋_A/K→ ^𝕋_A/B/_K/A, sending 
𝕋_A/B→ N→𝕋_A/K to 

    𝕋_A/Bfree 𝕋_A/B∐free N .


One easily checks that 𝔏(B[-n-1])≃𝕋_A/B⊕free (A[-n-1]0→𝕋_A/K). 

We also define ℜ as the composition of two functors: 


  – first a functor ^𝕋_A/B/_K/A→_𝕋_A/B, where _𝕋_A/B denotes the ∞-category of modules over the (,A)-dg-Lie algebroid 𝕋_A/B (which are just modules over its universal enveloping algebra). Here the functor is defined as cofib (𝕋_A/B→ -), which naturally lands in 𝕋_A/B-modules[To see this, one can use models and compute the cofiber in the case when the map out of 𝕋_A/B is injective. In this case the 𝕋_A/B-module structure is rather classical (see e.g. <cit.>).].  

  – then the functor _𝕋_A/B→_B/𝕋_B, sending a 𝕋_A/B-module E to its Chevalley–Eilenberg complex CE^∙(𝕋_A/B,E) (see e.g. <cit.> for a definition). Here the reader has to check that the 𝕋_A/B-module structure on cofib (𝕋_A/B→𝕋_A/K)≃𝕋_A/K⊗_BA is given by the action of 𝕋_A/B on A, so that 

    ℜ(𝕋_A/K)
    ≃ CE^∙(𝕋_A/B,𝕋_B/K⊗_BA)
    ≃𝕋_B/K⊗_B CE^∙(𝕋_A/B)
    ≃𝕋_B/K .


The details of the construction, and the proof that ℜ is right adjoint to 𝔏, will appear elsewhere. 


The square of right adjoints 

    _K/B   [l]_-CE^∙_K/A(^𝕋_A/B/_K/A)^op
     
    ^𝕃_B/K/_B [u]^-d↦ Bd⊕M[-1]   [u]_-Z:=𝔏^op[l]^-(-)^∨ (_B/𝕋_B/K)^op

commutes. 


The functor CE_K/A is right adjoint, and thus preserves limits. 
In particular, it sends push-outs in _K/A to pull-backs in _K/A. 
Hence we have that 

    CE^∙_K/A𝔏^op(M)≃ (A⊕_d'(M')^∨[-1])A⊕_can𝕃_A/B[-1]×B ,

where d':𝕃_A/K→ (M')^∨ is adjoint to the pullback of M⊗_BA→𝕋_B/K⊗_BA, and  along 𝕋_A/K→𝕋_B/K⊗_BA, and can:𝕃_A/K→𝕃_A/B is the canonical map. 
Hence CE^∙_K/A𝔏^op(M)≃ B⊕_dM^∨[-1], where d:𝕃_B/K→ M^∨ is adjoint to M→𝕋_B/K. 

Together, Lemma <ref> and Lemma <ref> imply that we have a weak morphism of weak Koszul duality contexts from _K/B(^𝕋_A/B/_K/A)^op to ^𝕃_B/K/_B(_B/𝕋_B/K)^op. 

The proof that this weak morphism is indeed a morphism will appear elsewhere. 



 §.§ Formal moduli problems under X


Let X be a formal derived prestack, and let X→ Y be a formal thickening of X.  

For every Spec(A)→ Y, X_A:=X×_YSpec(A)→ Spec(A) is a formal thickening of X_A. 


First of all observe that formal derived prestacks are stable by pullbacks, so that X_A is a formal derived prestack. 
The only thing left to prove is that X_A→ Spec(A) induces an equivalence on reduced prestacks, which is equivalent to require that it induces an equivalence on de Rham prestacks. The functor (-)_DR being a right adjoint, we have that 
(X_A)_DR≃ X_DR×_Y_DRSpec(A)_DR≃ Spec(A)_DR. 

This allows us to build a presheaf of ∞-categories A↦𝐓𝐡𝐢𝐜𝐤(X_A)_/Spec(A) on 𝐝𝐀𝐟𝐟^f.t., and a commuting diagram of functors

    𝐓𝐡𝐢𝐜𝐤(X)_/Y[r][d]    Spec(A)→ Ylim𝐓𝐡𝐢𝐜𝐤(X_A)_/Spec(A)[d] 
    
    ^X/(𝐝𝐏𝐫𝐒𝐭_^f.t.)_/Y[r]    Spec(A)→ Ylim ^X_A/(𝐝𝐏𝐫𝐒𝐭_^f.t.)_/Spec(A)

The bottom horizontal functor has a left adjoint 

    Spec(A)→ Ylim ^X_A/(𝐝𝐏𝐫𝐒𝐭_^f.t.)_/Spec(A)⟶ ^X/(𝐝𝐏𝐫𝐒𝐭_^f.t.)_/Y

that sends a diagram (X_A→ Z(A)→ Spec(A))_Spec(A)→ Y to its colimit 

    X≃Spec(A)→ Ycolim X_A→Spec(A)→ Ycolim Z(A)→Spec(A)→ Ycolim Spec(A)≃ Y .

The counit of the adjunction 

    Spec(A)→ Ycolim(ZY×Spec(A))→ Z

is an equivalence, and thus the bottom horizontal functor is fully faithful. 
Since the two vertical functors are also fully faithful, therefore so is the remaining horizontal one. 
The next Lemma tells us how far it is from being an equivalence. 

The essential image of Spec(A)→ Ylim𝐓𝐡𝐢𝐜𝐤(X_A)_/Spec(A) lies in the essential image of ^X/(𝐝𝐏𝐫𝐒𝐭_^f.t.)_/Y. Moreover, it is included in the sub-∞-category spanned by (images of) those X→ Z→ Y inducing equivalences on associated reduced stacks and such that Z is infinitesimally cohesive. 


The first part of the statement is obvious. 
Let (Z_A)_Spec(A)→ Y be a diagram of formal derived stacks lying over the tautological diagram (Spec(A))_Spec(A)→ Y, and such that (Z_A)_red→ Spec(A_red) is an equivalence. 
The functor (-)_red being left adjoint, it commutes with colimits, and thus Z_red→ Y_red is an equivalence (where 
Z=Spec(A)→ YcolimZ_A). Moreover, filtered colimits do commute with finite limits and thus with pullbacks in particular, so that Z is infinitesimally cohesive because every single Z_A is.  

If we could prove that derived prestacks in the essential image have a procotangent complex, then we would get that the functor 𝐓𝐡𝐢𝐜𝐤(X)_/Y→Spec(A)→ Ylim𝐓𝐡𝐢𝐜𝐤(X_A)_/Spec(A) is an equivalence. 
This is probably not true in full generality, but we think that it is for several examples: 

If the functor IndCoh(Y)→QCoh(Y) is an equivalence, then the functor 

    𝐓𝐡𝐢𝐜𝐤(X)_/Y→Spec(A)→ Ylim𝐓𝐡𝐢𝐜𝐤(X_A)_/Spec(A)

is an equivalence as well. 

Combining the above Lemma for Y=X_DR, together with the results of the previous Subsections, we get the following Theorem: 

There is a fully faithful functor 

    𝐓𝐡𝐢𝐜𝐤(X) →_X_DR/X:=Spec(A)→ X_DRlim ^𝕋_Spec(A_red)/X_A/_A/A_red .



We simply have to prove that 𝐓𝐡𝐢𝐜𝐤(X)≃𝐓𝐡𝐢𝐜𝐤(X)_/X_DR. 
This is true as X_DR happens to be the terminal formal thickening of X. 

We strongly believe that the functor appearing in the Theorem is actually an equivalence. This is in fact a consequence of our conjecture, as we know that IndCoh(X_DR)→QCoh(X_DR) is an equivalence (see <cit.>). 



  
Comparison with formal localization 

Following Remark <ref>, let us pretend that our results do hold in the more general graded mixed setting from <cit.> (see <ref> for a recollection), and consider the following situation: 
𝐊=𝔻_X_DR is the mixed crystalline structure sheaf of a derived Artin stack X and 𝐀=ℬ_X is the mixed principal parts of X (after <cit.>). These are prestacks over X_DR with values in graded mixed cdgas, and they are defined as follows: 


  – (Spec(B)→ X_DR)𝐊⟼𝔻(B):=DR(B_red/B)≃ CE^ϵ(𝕋_Bred/B), with |𝔻(B)|≃ B; 

  – (Spec(B)→ X_DR)𝐀⟼𝔻(X_B):=DR(X_B/B). 

We would thus have a prestack of ∞-categories _𝐊/𝐀 over X_DR, of which the global sections are denoted ^f_X_DR/X. Presumably, this would be an ind-coherent version of _X_DR/X defined above, which has also been defined as global sections of a prestack of ∞-categories. Let us explain why do believe so. 


  – First observe that, according to our results, we shall have an equivalence 

    _𝔻(B)/𝔻(X_B)≃ ^𝕋_B_red/𝔻(X_B)/_𝔻(B)/B_red .


  – Then, the realization functor |·| and its left adjoint shall produce two adjunctions, related by a forgetful functor as follows: 

    @C=75pt@R=50pt ^𝕋_Spec(B_red)/X_B/_B/B_red@<-2pt>[d] @<2pt>[r]    @<2pt>[l]  ^𝕋_B_red/𝔻(X_B)/_𝔻(B)/B_red@<-2pt>[d] 
    _B@<2pt>[r]     _𝔻(B)@<2pt>[l]


  – Finally, it has been proven in <cit.> that, restricted to suitable subcategories of perfect modules, the adjunction 

    𝔻(X_B)⊗_B_red-_B _𝔻(B)|·|

restricts to an equivalence. 
Moreover, it is conjectured that _𝔻(X_B) is equivalent to 𝐈𝐧𝐝𝐂𝐨𝐡(X_B) (see e.g. <cit.>). 	

Let us conclude with the following observation. Let X→ Y be a formal thickening for which Y is also a derived geometric stack. 
In particular we know that X_DR→ Y_DR is an equivalence, and we thus get a sequence of graded mixed cdgas 

    𝐊→𝐁:=ℬ_Y→𝐀 ,

and hence graded mixed (𝐊,𝐀)-Lie algebroid 𝕋_𝐀/𝐁. 
We therefore get a functor 𝐓𝐡𝐢𝐜𝐤𝐆𝐞𝐨𝐦(X)→^f_X_DR/X, which shall fit into the following conjectural commuting diagram of functors: 

    𝐓𝐡𝐢𝐜𝐤𝐆𝐞𝐨𝐦(X) [r][d]    𝐓𝐡𝐢𝐜𝐤(X) [d] 
    _X_DR/X^f[r][d]    _X_DR/X[d]
    𝐈𝐧𝐝𝐂𝐨𝐡(X) [r]    𝐐𝐂𝐨𝐡(X)

where ThickGeom(X) denotes the subcategory of Thick(X) consisting of geometric thickenings. 


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Let us end this paper with several observations. 

  – the functor 

    ℜ: ^𝕋_Spec(A_red)/X_A/_A/A_red→_A/𝕋_A

gives rise to a functor 

    _X_DR/X→Spec(A)→ X_DRlim_A/𝕋_A≃𝐐𝐂𝐨𝐡(X_DR)_/q_*𝕋_X ,

where q:X→ X_DR is the canonical morphism. 

  – the composed functor 

    𝐓𝐡𝐢𝐜𝐤(X)→_X_DR/X→𝐐𝐂𝐨𝐡(X_DR)_/q_*𝕋_X

coincides with the the functor sending a formal thickening X→ Y to q_*𝕋_X/Y.[Note that since X and Y only admits a procotangent complex, 𝕋_X and 𝕋_X/Y are a priori an ind-objects, but then q_*𝕋_X and q_*𝕋_X/Y lie in 𝐈𝐧𝐝𝐂𝐨𝐡(X_DR)≃𝐐𝐂𝐨𝐡(X_DR). 

  – TALK ABOUT THE DIAGRAMX→ X× X→ X× Y→ Y→ *. 


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§ APPENDIX


In this appendix our aim is to compare the Chevalley-Eilenberg functor with the de Rham functor that is used in the formal localization procedure of <cit.>. 
This is a very first step for a future comparison between the various approaches to formal geometry in the derived setting. 







 §.§ Recollection on graded mixed stuff


We fix a base cdgaA(not assuming anything at the moment), and start with Definitions. 
 
  – A graded module is a sequence (M_n)_n∈ℤ of A-modules. We refer to the index n as the weight. 

  – We denote by _A^gr the category of graded modules. It comes equipped with the weight shifting functor M↦ M(1), defined by M(1)_n=M_n+1. 

  – A graded mixed module is a pair (M,ϵ), with M a graded module and ϵ:M→ M[1](1) a morphism of graded modules such that ϵ[1](1)∘ϵ=0 (which we will abusively write ϵ^2=0). We call ϵ the mixed differential. 

  – We denote by _A^ the category of graded mixed modules. The weight shifting functor obviously lifts to _A^. 

  – Both categories (_A^gr and _A^) are symmetric monoidal, so all categories of algebraic structures on modules that we considered for _A can be considered as well within them. We will add superscripts gr and  to the already introduce notation when emphasize which underlying symmetric monoidal category we are working with. For instance, _A^ denotes the category of commutative monoids in _A^. 

We equip_A^grwith the objectwise model structure from the one of_A. We can transport this model structure along the forgetful functor_A^(-)^♯⟶_A^grand get a model structure[The injective one (though we will soon use the projective one, with which it is easier to compute mapping spaces, but for which the monoidal unit is not cofibrant). ] on_A^for which the monoidal unitAis cofibrant. 
This model structure is nice enough so that all model category theoretic constructions based on_Astill make sense on_A^, see e.g.<cit.> and <cit.>. Thus we have for instance an∞-category_A^of graded mixedA-cdgas. 

Then recall from <cit.> that the Quillen adjunction_A _A^HOM__A^(A,-)induces an adjunction_A _A^|-|for which the right adjoint|-|, called (standard) realization, is lax symmetric monoidal. This implies in particular that this realization functor is also well-defined on the other aforementioned categories carrying the superscript. It has a very explicit description: 
[<cit.>]
For a graded mixed module (M,ϵ), |M|≃ (∏_n≥0M_n,d_M+ϵ), where d_M:M→ M[1] is the (weight preserving) differential. 

Let us (temporarily) use the projective model structure on _A^. For this model structure, every object is fibrant. Moreover, one has the following explicit cofibrant replacement A for A, which is quasi-free (as a graded mixed A-module): 

  – let us consider the free A-module generated by x_i's for i≥ 0 and y_j's for j≥1. 

  – assign to x_i cohomological degree 0 and weight i, and to y_j cohomological degree 1 and weight j. 

  – define ϵ(x_i)=y_i+1. 

  – modify the differential[Before doing that, our graded mixed module was free. ] by imposing that d(x_i)=y_i (by convention, y_0=0). 

Finally observe that HOM__A^(Ã,M)=(∏_n≥0M_n,d_M+ϵ). 


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Let us observe that there is a rather weird phenomenon appearing in the ∞-category _A^: 

There is a canonical non-zero morphism A(1)→ A in _A^. 

Let us (temporarily) use the projective model structure on _A^. For this model structure, every object is fibrant. Moreover, one has the following explicit cofibrant replacement A(1) for A(1), which is quasi-free (as a graded mixed A-module): 

  – let us consider the free A-module generated by x_i's for i≥ -1 and y_j's for j≥0. 

  – assign to x_i cohomological degree 0 and weight i, and to y_j cohomological degree 1 and weight j. 

  – define ϵ(x_i)=y_i+1. 

  – modify the differential[Before doing that, our graded mixed module was free. ] by imposing that d(x_i)=y_i. 

Thus we see that a morphism from A(1) to A in _A^ is determined by an actual morphism A(1)→ A in _A^. There is a canonical such morphism, that sends x_0 to 1∈ A. 

Several observations are in order: 

  –One sees from the proof that one actually has that 
HOM__A^(A(n),A(m))≃ Aδ_n≥ m
(meaning that it is A if n≥ m, 0 else). More generally, if M and N are A-modules, then 
HOM__A^(N(n),M(m))≃HOM__Aδ_n≥ m .

  –The fiber of the above morphism A(1)→ A in _A^ is the actual fiber of A(1)→ A in _A^. 
More precisely: fib (A(1)→ A)≃ A{x_-1,y_0|ϵ(x_-1)=y_0}=:A(1,0). 
More generally, the fiber of the composed map M(1)→ M→ N can be described as follows: in weight -1 we have M, in weight 0 we have N[-1], and the mixed differential is the original map M→ N=N[-1][1](1). 


Note that the realization functor exists if one replaces _A^ with a reasonable stable A-linear symmetric monoidal ∞-category 𝐌 (we refer to <cit.> for the detailed assumptions on 𝐌), and get an adjunction 

-⊗1_𝐌_A 𝐌|-| .

There is another lax symmetric monoidal functor |-|^t:_A^→_A, called Tate realization, which is defined as the composition 
_A^⟶𝐈𝐧𝐝(_A^) |-|⟶_A ,

where the first arrow is the functor -⊗_A(A(0)→ A(-1)→ A(-2)→⋯) sending a graded mixed A-module M to the 
ind-object M(∞):=M→ M(-1)→ M(-2)→⋯. More explicitely:

|M|^t≃ colim_i≥0(∏_n≥-iM_n,d_M+ϵ) .

It is lax symmetric monoidal because the standard realization functor |-| is, and the ind-object 

A(∞)="colim"_n≥0A(-n)=A(0)→ A(-1)→ A(-2)→⋯
is a commutative monoid, with product given by A(i)⊗ A(j)→̃A(i+j). 

Actually, the product on A(∞) provides an equivalence between A(∞)⊗_AA(∞) and A(∞). Indeed, one first observes that the morphism A(1)→ A(0) leads to an equivalence 
A(1)(∞)="colim"_nA(1-n)→̃ "colim"_nA(-n)=A(∞). Then one gets, 

A(∞)⊗_AA(∞)="colim"_n,mA(-n)⊗ A(-m)→̃"colim"_n,mA(-n-m)≃ colim_nA(∞)=A(∞) .

Let M=A[t,t^-1], with t^± n having cohomological degree 0 and weight ± n: in other words, M=⊕_n∈ℤA(n). 
The standard realization of M is 

|M|≃∏_n≥0 A=A[[x]]

while its Tate realization is 

|M|^t≃ colim_i≥ 0∏_n≥ -i A=A[[x,x^-1] .

Let us finally state a very useful, though rather obvious, result: 

Within the sequence of lax symmetric monoidal functors 
_A⟶_A^(∞)⟶𝐈𝐧𝐝(_A^)=:_A^∞ ,

the first functor is fully faithful, the second one is not, and their composition is fully faithful. 

The fact that the functor _A→_A^ is fully faithful is obvious. 
Let us then prove that tensoring with A(∞) is not fully faithfull. 
We already know that HOM__A^(A,A(1))=0. 
Nevertheless, 

    HOM__A^∞(A(∞),A(1)(∞)) 
        =    HOM__A^∞("colim"_nA(-n),"colim"_mA(1-m)) 
        =     lim_ncolim_mHOM__A^(A(-n),A(1-m)) 
        =     lim_ncolim_m Aδ_-n≥ 1-m = lim_ncolim_m Aδ_n≤ m-1
        =     lim_nA=A .

Let us finally observe that tensoring with A(∞) becomes fully faithful when restricted to weight zero graded mixed A-modules (i.e. on A-modules). Indeed, if M and N sit in weight zero, then we have the very same calculation: 

    HOM__A^∞(N(∞),M(∞)) 
        =     lim_ncolim_mHOM__A^(N(-n),M(-m)) 
        =     lim_ncolim_m HOM__A(N,M)δ_m≤ n = HOM__A(N,M) .


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 §.§ Graded mixed cdgas and dg-Lie algebroids


LetB→ Abe a morphism of cdgas which, as usual, we assume to be cofibrant. 

The Chevalley–Eilenberg functor CE^∙_B/A factors through a commuting diagram 
   _B/A^[rd]^-|-|   
_B/A^op[rr]^-CE^∙_B/A[ru]^-CE^ϵ_B/A      _B/A
For a dg-(B,A)-algebroid L we set CE^ϵ_B/A(L)_n:=S^n(L[1])^∨, together with 
ϵ(ω)(ℓ_0,…,ℓ_n):=±∑_i=0^nρ(ℓ_i)(ω(ℓ_0,…ℓ̂_̂î,…,ℓ_n))±∑_0≤ i<j≤ nω([ℓ_i,ℓ_j],ℓ_0,…ℓ̂_̂î,…,ℓ̂_̂ĵ,…,ℓ_n) .

We let the reader check that we indeed have that |CE^ϵ_B/A(L)|≃CE^∙_B/A(L) using Proposition<ref>.[We simply observe that the discrepancy between d_CE and ϵ is resolved precisely thanks to the specific form of the realization functor. ]

Hence, composing with the forgetful functor_B/A^→_B^, we get a functor_B/A^op→_B^ ,which lands in the full sub-∞-category consisting of graded mixed cdgasCsuch thatC_0≃ A. 
It even factors through the∞-category consisting of graded mixed cdgasCtogether with a the data of an equivalenceC_0≃ A, which has an initial object, denotedDR(A/B)(an explicit description of which is provided in <cit.>). 
Actually, the functorDR(-/B):_B→_B^is left adjoint to the functor sending a graded mixedB-cdgaCto its weight zero componentC_0. 


The morphism DR(A/B)→ CE^ϵ_B/A(𝕋_A/B) is an equivalence whenever 𝕃_A/B is coherent and (cohomologically) bounded above. 

Equivalences between graded mixed objects are checked componentwise. 
On weight components the map is given by 
DR(A/B)_n≃ S^n_A(𝕃_A/B[-1])→ S^n_A(𝕋_A/B[1])^∨≃ CE^ϵ_B/A(𝕋_A/B)_n , 

which is an equivalence if 𝕃_A/B is coherent and (cohomologically) bounded above. 


IfBis a non-positively graded-cdga (being a field, for simplicity) that is almost finitely presented, then we know that|𝐃𝐑(B_red/B)|≃ Bafter <cit.>. 
Hence we get that|CE_/B_red^ϵ(𝕋_B_red/B)|≃|CE_B/B_red^ϵ(𝕋_B_red/B)|≃ B ,which we already knew forBalmost of finite type. 



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From the universal property of DR and naturality, B↦𝔻(B):=𝐃𝐑(B_red/B) defines a functor _→_^. 
We now state a series of crucial results about the functor 𝔻. 
[<cit.>, Proposition 2.2.3, Lemma 2.2.4 & Corollary 2.2.5]
When restricted to the full sub-∞-category _^f.p. of almost finitely presented -cdgas, 𝔻 is fully faithfull and its image lands in the full sub-∞-category of graded mixed cdgas C such that 

  –C_0 is concentraded in degre 0, and is a reduced -algebra of finite type. 

  – As a C_0 module, C_1 is almost finitely presented and cohomologicall sits in non-positive degree.  

  – The natural morphism S_C_0(C_1)→ C^♯ is an equivalence of graded cdgas. 

In particular, if B is almost finitely presented then the natural morphism B→|𝔻(B)| is an equivalence, and the sequence of symmetric monoidal stable functors 
_B [r]^-𝔻(B)⊗_B-   _𝔻(B)^[r]^-(-)(∞)   _𝔻(B)^∞
moreover induces equivalences 
𝐏𝐞𝐫𝐟_B≃𝐏𝐞𝐫𝐟_𝔻(B)^≃𝐏𝐞𝐫𝐟_𝔻(B)^∞
between their perfect sub-∞-categories, where a 𝔻(B)-module (resp. a 𝔻(B)(∞)-module) M is perfect if M^♯≃𝔻(B)^♯⊗_B_redM_0 (resp. M^♯≃𝔻(B)(∞)^♯⊗_B_redM_0) with M_0 in 𝐏𝐞𝐫𝐟_B_red. 

The only difficult statement to prove in the first part of the Proposition is fully faithfulness. We have 

    Hom__^(𝔻(B),𝔻(A))    ≃   Hom__^(𝐃𝐑(B_red),𝔻(A))Hom__^(𝐃𝐑(B),𝔻(A))×Hom__^(B,𝔻(A)) 
       ≃   Hom__(B_red,A)Hom__(B,A)×Hom__^(B,𝔻(A)) 
        =    Hom__^(B,𝔻(A))  ≃ Hom__(B,|𝔻(A)|) .

Hence it is sufficient to prove that the natural morphism A→|𝔻(A)| is an equivalence, which is precisely the content of <cit.>. 

In the second part of the Proposition, the only difficult statement is that the symmetric monoidal stable functors involved are fully faithful when restricted to perfect modules. 
By stability, it is sufficient to prove fully faithfulness for B itself:   
Hom__𝔻(B)^(𝔻(B),𝔻(B)) ≃Hom__B^(B,𝔻(B)) ≃Hom__B(B,|𝔻(B)|)≃Hom__B(B,B)) ,

and similarly, 
Hom__𝔻(B)^∞(𝔻(B)(∞),𝔻(B)(∞) ≃Hom__B(B,|𝔻(B)(∞)|)≃Hom__B(B,B)) .

Indeed, since 𝔻(B) is concentrated in non-negative weight, then |𝔻(B)(∞)|≃|𝔻(B)|^t≃|𝔻(B)|≃ B. 

As a consequence, if B is almost finitely presented then 𝕃_B_red/𝔻(B)≃𝕃_B_red/B(-1): 

  – The first part of the above Proposition tells us this is true as graded complexes: 
𝕃_B_red/𝔻(B)^♯≃𝕃_B_red/B(-1)^♯ .

  –𝕃_B_red/B(-1) being concentrated in weight one, its mixed differential can't be anything else than trivial. 

One can even further prove that the natural map 𝕃_B_red/B→𝕃_B_red/𝔻(B) is id⊗_B_red(B_red→ B_red(-1)). 

TO DO NOW : 

LIE_B/B_red↪ LIE_B/B_red^∞←̃(LIE_B/B_red^∞)_/𝕋_B_red/𝔻(B)(∞) =LIE_𝔻(B)/B_red^∞
This is because 𝕋_B_red/𝔻(B)(∞)≃ 0. 

Combining the above Proposition together with the considerations in the end of Section <ref>, we get the following Theorem: 

There is a functor 
_/A⟶_/𝔻(A)^
that leads to natural equivalences of functors
FMP_/A^perf[r]^-∼[rd]_-𝕋_Spec(A)/-   _/A^perf[r]^-∼[d]^-forget   _/𝔻(A)^,perf[d]^-forget
   Perf_A [r]^-∼   Perf_𝔻(A)^
where C^perf:=C_A×𝐏𝐞𝐫𝐟_A, for a category C over _A. 


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plain[ 

D.C.: IMAG, Univ Montpellier, CNRS, Montpellier, France & Institut Universitaire de France 

Email: <damien.calaque@umontpellier.fr>
 

J.G.: CNRS, I2M (Marseille) & IHÉS 

Email: <jgrivaux@math.cnrs.fr>]]