Deformation Theories

Let (Υ,{E_α}_α∈T) be a deformation context. Our main goal in this paper is to obtain an algebraic description of the ∞-category Moduli^Υ ⊆Fun(Υ^sm,S) of formal moduli problems. To this end, we would like to have some sort of recognition criterion, which addresses the following question:

(Q) Given an∞-category Ξ, when does there exist an equivalence Moduli^Υ'Ξ?

We take our first cue from Example 1.1.16. To every object A∈ Υ, we can associate a formal moduli problem SpecA ∈Moduli^Υ by the formula (SpecA)(R) = Map_Υ(A, R). Combining this construction with an equivalence Moduli^Υ'Ξ, we obtain a functorD: Υ^op→Ξ. We begin by axiomatizing the properties of this functor:

Definition 1.3.1. Let (Υ,{Eα}_α∈T) be a deformation context. Aweak deformation theoryfor (Υ,{Eα}_α∈T) is a functorD: Υ^op→Ξ satisfying the following axioms:

(D1) The∞-category Ξ is presentable.

(D2) The functorDadmits a left adjoint D⁰ : Ξ→Υ^op.

(D3) There exists a full subcategory Ξ₀⊆Ξ satisfying the following conditions:

(a) For every objectK∈Ξ0, the unit mapK→DD⁰K is an equivalence.

(b) The full subcategory Ξ0 contains the initial object ∅ ∈ Ξ. It then follows from (a) that ∅ ' DD⁰∅ 'D(∗), where ∗denotes the final object of Υ.

(c) For every index α ∈ T and every n ≥ 1, there exists an objectKα,n ∈ Ξ0 and an equivalence Ω^∞−nEα'D⁰Kα,n. It follows that the base point of Ω^∞−nEα determines a map

vα,n:Kα,n'DD⁰Kα,n'D(Ω^∞−nEα)→D(∗)' ∅.

(d) For every pushout diagram

K_α,n //

vα,n

K

∅ //K⁰

whereα∈T andn >0, ifK belongs to to Ξ0 thenK⁰ also belongs to Ξ0.

Definition 1.3.1 might seem a bit complicated at a first glance. We can summarize axioms (D2) and (D3) informally by saying that the functorD: Υ^op→Ξ is not far from being an equivalence. Axiom (D2) requires that there exists an adjointD⁰ toD, and axiom (D3) requires thatD⁰behave as a homotopy inverse toD, at least on a subcategory Ξ0⊆Ξ with good closure properties.

Example 1.3.2. Letk be a field of characteristic zero and let (CAlg^aug_k ,{E}) be the deformation context described in Example 1.1.4. In §2, we will construct a weak deformation theory D : (CAlg^aug_k )^op →Liek, where Lie denotes the ∞-category of differential graded Lie algebras over k (Definition 2.1.14). Here the adjoint functor D⁰ : Lie_k →(CAlg^aug_k )^op assigns to each differential graded Lie algebrag∗ its cohomology Chevalley-Eilenberg complexC^∗(g_∗) (see Construction 2.2.13). In fact, the functorD: (CAlg^aug_k )^op→Liek is even adeformation theory: it satisfies condition (D4) appearing in Definition 1.3.9 below.

Remark 1.3.3. In view of Corollary T.5.5.2.9 and Remark T.5.5.2.10, condition (D2) of Definition 1.3.1 is equivalent to the requirement that the functorD preserves small limits.

Remark 1.3.4. In the situation of Definition 1.3.1, the objects K_α,n∈Ξ₀ are determined up to canonical equivalence: it follows from (a) that they are given byK_α,n'DD⁰K_α,n'D(Ω^∞−nE_α). In particular, the objects Ω^∞−nEα belong to Ξ0.

Our next result summarizes some of the basic features of weak deformation theories.

Proposition 1.3.5. Let (Υ,{E_α}_α∈T) be a deformation context and D : Υ^op → Ξ a weak deformation theory. Let Ξ0 ⊆ Ξ be a full subcategory which is stable under equivalence and satisfies condition (3) of Definition 1.3.1. Then:

(1) The functor D carries final objects ofΥto initial objects ofΞ.

(2) Let A∈Υbe an object having the formD⁰(K), whereK∈Ξ₀. Then the unit map A→D⁰D(A)is an equivalence inΥ.

(3) If A∈Υis small, then D(A)∈Ξ0 and the unit mapA→D⁰DA is an equivalence inΥ.

(4) Suppose we are given a pullback diagram σ: A⁰

//B⁰

φ

A //B

in Υ, whereA andB are small and the morphism φis small. Then D(σ) is a pushout diagram inΞ.

Proof. Let ∅ denote an initial object of Ξ. Then ∅ ∈ Ξ0 so that the adjunction map ∅ → DD⁰∅ is an equivalence. SinceD⁰ : Ξ→Υ^op is left adjoint to D, it carries ∅ to a final object∗ ∈Υ. This proves (1).

To prove (2), suppose thatA=D⁰(K) forK∈Ξ0. Then the unit mapu:A→D⁰DAhas a left homotopy inverse, given by applyingD⁰ to the the mapv : K →DD⁰K in Ξ. Since v is an equivalence (part (a) of Definition 1.3.1), we conclude thatuis an equivalence.

We now prove (3). LetA∈Υ be small, so that there exists a sequence of elementary morphisms A=A0→A1→ · · · →An' ∗.

We will prove thatDAi ∈Ξ0 using descending induction oni. Ifi=n, the desired result follows from (1).

Assume therefore that i < n, so that the inductive hypothesis guarantees that D(Ai+1) ∈ Ξ0. Choose a pullback diagramσ:

Ai //

∗

φ

A_i+1 ^ψ //Ω^∞−nE_α

wheren >0,α∈T, and φis the base point of Ω^∞−nEα. Form a pushout diagramτ : D(Ω^∞−nE_α) ^Dψ //

Dφ

DA_i+1

D(∗) //X

in Ξ. There is an evident transformation ξ : σ → D⁰(τ) of diagrams in Υ. Since both σ and D⁰(τ) are pullback diagrams and the objectsA_i+1, Ω^∞−nE_α, and∗ belong to the essential image ofD⁰|Ξ₀, it follows from assertion (2) thatξis an equivalence, so thatAi'D⁰(X). Assumption (d) of Definition 1.3.1 guarantees thatX ∈Ξ0, so thatAi lies in the essential image ofD⁰|Ξ0.

We now prove (4). The class of morphismsφfor which the conclusion holds (for an arbitrary mapA→B between small objects of Υ) is evidently stable under composition. We may therefore reduce to the case whereφis elementary, and further to the case whereφis the base point map∗ →Ω^∞−nEαfor someα∈T and somen >0. Arguing as above, we deduce that the pullback diagramσ:

A⁰ //

∗

φ

A //Ω^∞−nE_α

is equivalent toD⁰(τ), whereτ is a diagram in Ξ0which is a pushout square in Ξ. ThenD(σ)'DD⁰(τ)'τ is a pushout diagram, by virtue of condition (a) of Definition 1.3.1.

Corollary 1.3.6. Let(Υ,{Eα}α∈T)be a deformation context andD: Υ^op→Ξa weak deformation theory.

Let j: Ξ→Fun(Ξ^op,S)denote the Yoneda embedding. For every objectK∈Ξ, the composition Υ^sm⊆Υ−→^D Ξ^{op j}−→^(K)S

is a formal moduli problem. This construction determines a functorΨ : Ξ→Moduli^Υ⊆Fun(Υ^sm,S).

Remark 1.3.7. Corollary 1.3.6 admits a converse. Let (Υ,{Eα}α∈T) be a deformation context. The functor Spec : Υ^op →Moduli of Example 1.1.16 satisfies conditions (D1), (D2) and (D3) of Definition 1.3.1, and therefore defines a weak deformation theory (we can define the full subcategory Moduli₀ ⊆Moduli whose existence is required by (D3) to be spanned by objects of the form Spec(A), whereA∈Υ^sm).

Combining Corollary 1.3.6 with Proposition 1.2.4, we obtain the following result:

Corollary 1.3.8. Let(Υ,{Eα}_α∈T)be a deformation context andD: Υ^op→Ξa weak deformation theory.

For each α∈T and eachK∈Ξ, the composite map S^fin∗

E_α

−→Υ−→^D Ξ^{op j}−→^(K)S

is strongly excisive, and can therefore be identified with a spectrum which we will denote by e_α(K). This construction determines a functor e_α: Ξ→Sp = Stab(S)⊆Fun(S^fin_∗ ,S).

Definition 1.3.9. Let (Υ,{E_α}_α∈T) be a deformation context. Adeformation theory for (Υ,{E_α}_α∈T) is a weak deformation theoryD: Υ^op→Ξ which satisfies the following additional condition:

(D4) For eachα∈T, leteα: Ξ→Sp be the functor described in Corollary 1.3.8. Then eacheα preserves small sifted colimits. Moreover, a morphism f in Ξ is an equivalence if and only if each eα(f) is an equivalence of spectra.

Warning 1.3.10. Let (Υ,{Xα}_α∈T) be a deformation context, and let Spec : Υ^op → Moduli^Υ be given by the Yoneda embedding (see Remark 1.3.7). The resulting functorseα : Moduli^Υ →Sp are then given by evaluation on the spectrum objects Eα ∈ Stab(Υ), and are therefore jointly conservative (Proposition 1.2.10) and preserve filtered colimits. However, it is not clear that Spec is a deformation theory, since the tangent complex constructionsX 7→X(Eα) do not obviously commute with sifted colimits.

Remark 1.3.11. Let (Υ,{E}) be a deformation context, letD: Υ^op→Ξ be a deformation theory for Υ, and let e: Ξ → Sp be as in Corollary 1.3.8. The functor e preserves small limits, and condition (D4) of Definition 1.3.9 implies thatepreserves sifted colimits. It follows that eadmits a left adjointF : Sp→Ξ.

The composite functore◦F has the structure of a monadA on Sp. Sinceeis conservative and commutes with sifted colimits, Theorem A.6.2.2.5 gives us an equivalence of ∞-categories Ξ'LMod_A(Sp) with the

∞-category of algebras over the monadT. In other words, we can think of Ξ as an∞-category whose objects are spectra which are equipped some additional structure (namely, a left action of the monadA).

More generally, if (Υ,{Eα}_α∈T) is a deformation context equipped with a deformation theoryD: Υ^op→ Ξ, the same argument supplies an equivalence Ξ 'LModA(Sp^T): that is, we can think of objects of Ξ as determines by a collection of spectra (indexed byT), together with some additional structure.

We can now formulate our main result:

Theorem 1.3.12. Let(Υ,{Eα}α∈T)be a deformation context and letD: Υ^op→Ξbe a deformation theory.

Then the functorΨ : Ξ→Moduli^Υ of Corollary 1.3.6 is an equivalence of∞-categories.

The proof of Theorem 1.3.12 will be given in §1.5.

Remark 1.3.13. Let (Υ,{Eα}α∈T) be a deformation context, letD: Υ^op→Ξ a deformation theory, and consider the functor Ψ : Ξ→Moduli of Corollary 1.3.6. The composite functor Υ^op→^D Ξ→^Ψ Moduli^Υcarries an objectA∈Υ to the formal moduli problem defined by the formula

B7→Map_Ξ(DB,DA)'Map_Υ(A,D⁰D(B)).

The unit mapsB→D⁰DB determines a natural transformation of functors β: Spec→Ψ◦D, where Spec : Υ^op→Moduli^Υ is as in Example 1.1.16. It follows from Proposition 1.3.5 that the natural transformationβ is an equivalence. Combining this with Theorem 1.3.12, we conclude thatDis equivalent to weak deformation theory Spec : Υ^op→Moduli^Υ of Remark 1.3.7.

We can summarize the situation as follows: a deformation context (Υ,{E_α}_α∈T} admits a deformation theory if and only if, for each α∈ T, the constructionX 7→ X(Eα) determines a functor Moduli^Υ →Sp which commutes with sifted colimits. If this condition is satisfied, then the deformation theory on Υ is unique (up to canonical equivalence), given by the functor Spec : Υ^op→Moduli^Υ.