Contents DEFORMATIONS OF CATEGORIES ANTHONY BLANC, LUDMIL KATZARKOV, PRANAV PANDIT Abstract. Let k be a eld. In this paper we use the theory of formal moduli problems developped by

FORMAL DEFORMATIONS OF CATEGORIES

ANTHONY BLANC, LUDMIL KATZARKOV, PRANAV PANDIT

Abstract. Let k be a field. In this paper we use the theory of formal moduli problems

developped by Lurie in order to study the space of formal deformations of a k-linear ∞-

category. Our main result states that if C is a k-linear ∞-category which has a compactgenerator whose groups of self extensions vanish for higher positive degrees, then every

formal deformation of C has a compact generator. To establish this result, we prove

a general formula for the associated formal moduli problem functor in terms of formalgroup actions.

Contents

1. Introduction 11.1. Main results 41.2. Outline of this paper 51.3. Conventions 52. Formal moduli and approximation after Lurie 62.1. Formal moduli problems 62.2. Axiomatic deformation theory 92.3. Classification of En-formal moduli problems 132.4. Proximate formal moduli problems and loop spaces 163. Deformations of linear ∞-categories 223.1. Linear ∞-categories 223.2. Deformations of objects 243.3. Deformations of categories 273.4. Deformations as Ind-coherent loop actions 333.5. Deformations of associative algebras 383.6. Formal deformations 40References 45

1. Introduction

Let k be a field of characteristic 0 and let B be an associative algebra over k. It is knownsince the work of Gerstenhaber that the deformation theory of B up to isomorphism has aclose relation with the Hochschild cohomology of B. First order infinitesimal deformations(=over k[t]/t2) of B are classified by the group HH2(B). If µ0 denotes the multiplica-tion of B, a 2-cocycle φ defines a first order deformation of µ0 given by µ = µ0 + εφ.Obstructions to extend infinitesimal deformations live in the group HH3(B). Moreover ifL = HH∗(B)[1] denote the dg-Lie algebra given by the shifted Hochschild complex togetherwith the Gerstenhaber bracket, formal deformations (=over k[[t]]) of B are classified by theset MC(L⊗k tk[[t]]) of solutions of the Maurer–Cartan equation in (L⊗k tk[[t]])1 up to gaugeequivalence.

Date: May 4, 2016.

1

2 ANTHONY BLANC, LUDMIL KATZARKOV, PRANAV PANDIT

In [LVdB05] and [LVdB06], W. Lowen and M. Van den Bergh study deformations ofan abelian category up to equivalence of categories and its relation to a suitably definedHochschild cohomology, as well as obstructions for infinitesimal deformations. However, asin the case of associative algebras, there is no mathematical statement which exhibits theprecise relation between the corresponding deformation functor and the Hochschild coho-mology complex.

Deformations of dg-categories. In geometry we study derived categories attached tomanifolds and varieties (e.g. local systems, coherent sheaves...). Moreover many invariantswhich play a central role in mirror symmetry like cyclic homology are invariant under Moritaequivalence. This forces us, in order to have a well behaved Morita theory (see [Toe07]),to work in the setting of differential graded categories (or linear ∞-categories). Given adg-category C, it is known that the group HH2(C) parametrizes more than the dg-categorydeformations of C, namely every 2-cocycle corresponds to a curved A∞-deformation of C(see [Low08] and [KL09]). From the purpose of homotopy theory, curved A∞-categories areconsidered as pathological objects since no they do not possess a good notion of equivalencenor derived categories. It is therefore useful from this point of view to have another descrip-tion of the space of deformations controlled by the Hochschild cohomology complex. Wewill see that such a new description as well as its relation with the space of deformations ofa dg-category was given by J. Lurie using his theory of formal moduli problems and Koszulduality.

Derived deformation theory and formal moduli problems. Adopting the fonc-teur des points point of view, deformation theory was first formalized through the notionof functors from artinian local algebras to sets satisfying the Schlessinger conditions. Afterthe work of Drinfeld, Kontsevitch, and more recently of Hinich, Manetti, Kapranov andCiocan-Fontanine, and finally Pridham and Lurie, it is now understood that deformationtheory is in essence derived, and that it is naturally formalized using certain derived func-tors from artinian dg-algebras to simplicial sets satisfying derived analogs of the Schlessingerconditions. In [Lurb], J. Lurie makes a systematic study of these functors under the nameof formal moduli problems, and through a spectacular use of the theory of ∞-categories andtheir monoidal structures, achieves the classification of formal moduli problems by dg-Lie al-gebras, making a mathematical statement out of what was then a principle. This result wasalso proven independently by J. Pridham [Pri10] using different techniques. More precisely,in the setting of [Lurb], there exists an equivalence of ∞-categories

dg-Lie algebras/k ∼−→ formal moduli problems/kThis equivalence is defined in terms of the Koszul duality functor between augmented E∞-algebras and dg-Lie algebras, which gives a different model for the space of Maurer–Cartansolutions. The inverse functor is given by the sifted tangent complex F 7→ TF [−1] which hastherefore the structure of a dg-Lie algebra. Moreover, Lurie develops an axiomatic approachfor proving this equivalence, which therefore applies in other contexts. In particular, if n ≥ 0is an integer, he defines formal moduli problems over En-algebras and obtains an equivalence

augmented En-algebras/k ∼−→ En-formal moduli problems/kwhere En is the ∞-operad of little n-cubes. Here the equivalence is valid for k a field ofarbitrary characteristic. The functor above is defined using the self duality functor of theEn-operad and gives therefore a model of a less commutative version of the space Maurer–Cartan solutions. It is often useful to find the minimum n for which a given deformationproblem is defined, precisely because it gives a finer structure on the tangent complex, asfor example in the case of deformations of a category.

FORMAL DEFORMATIONS OF CATEGORIES 3

Formal moduli problem of a linear ∞-category. Let k be a field. In this paper wehave choosen to work in the setting of k-linear∞-categories instead of k-linear dg-categoriesmainly to fit with [Lurb] whose constructions and results are used in this paper. The relationbetween these two classes of categories will be mentionned in §3.1. A k-linear∞-category isroughtly a presentable∞-category which is enriched over the∞-category Modk of k-modulespectra. Any dg-category gives a linear ∞-category by applying the dg-nerve construction.

If A is an E1-algebra, the ∞-category LModA of left A-module spectra has in generalno natural monoidal structure. However if A is an E2-algebra, LModA ihnerits a naturalmonoidal structure and we can speak of A-linear ∞-categories. For this reason the de-formation functor of a k-linear ∞-category is defined over artinian E2-algebras. If C is a(compactly generateda) k-linear ∞-category, we can define an ∞-functor CatDefcC from ar-tinian E2-algebras to spaces such that for each artinian E2-algebra A, the space CatDefcC(A)is a classifying space of (compactly generated) deformations of C over A. In general, the∞-functor CatDefcC does not satisfy the derived Schlessinger conditions and is therefore nota formal moduli problem. The moral reason for this is that compact objects of C do notalways deform. The typical example is the ∞-category ModB where B = k[u±1] is the freegraded commutative algebra with u of degree 2 and zero differential where any B-modulehas a non zero obstruction with respect to the 2-cocycle given by u (see Example 3.34).

This situation has been explained clearly by J. Lurie in the following way. The Hochschildcohomology complex HH∗(C) of C admits the structure of an E2-algebra by [Lurc, §5.3].Therefore the augmented E2-algebra k ⊕ HH∗(C) corresponds to a formal moduli problemΨHH∗(C) defined over artinian E2-algebras through the equivalence of the previous paragraph.Using Koszul duality, Lurie defines a natural transformation

θ : CatDefcC −→ ΨHH∗(C)

and shows that it is in general a 0-truncated map, meaning that its fibers are discrete spaces.This map is a higher analog of the map π0CatDefcC(k[t]/t2) → HH2(C) studied in [KL09].Moreover Lurie shows that the formal moduli problem ΨHH∗(C) is the best approximation ofCatDefcC by a formal moduli problem (see also [Pre12, Lem 5.3.3.6]), and it should thereforebe regarded as the correct space of deformations of C, as well as a substitute of the space ofcurved A∞-deformations. To emphasize this, we set the notation CatDef∧C = ΨHH∗(C). Bythe universal property of Hochschild cohomology, for each artinian E2-algebra A, the spaceCatDef∧C (A) has the following interpretation (see [Lurc, §5.3]):

CatDef∧C (A) ' D(2)(A)− linear structures on Cwhere D(2)(A) is the E2-Koszul dual of A. The interesting question becomes: does thereexists a good class of categories C for which θ is an equivalence?

Killing curvatures. The example B = k[u±1] above shows that there might existHochschild cocycles which do not correspond to any uncurved deformations of C throughthe map θ. However because the map θ is 0-truncated, it is not far from being an equivalence,and therefore CatDefcC is not far from being a formal moduli problem. In [Lurb, Prop 5.3.21],Lurie shows that if C is such that its spaces of morphisms have bounded above cohomology,then the fibers of θ are either empty or contractible. We will use this result below. This lattercondition on C is rather reasonable and is satisfied by the derived category of quasi-coherentsheaves on a finite type scheme over k, but also excludes the example above. Moreover heshows ([Lurb, Thm 5.3.33]) that if in addition C is generated by a set of unobstructiblecompact generators, the map θ is an equivalence, or in other words CatDefcC is a formal

aThis condition means that C is generated under filtrant colimits by its subcategory of compact objects;condition which is always satisfied for most categories of geometric origin like the derived category of quasi-

coherent sheaves on a finite type scheme.


moduli problem. However this latter condition is not satisfied by most derived categories ofvarieties of interest.

1.1. Main results. In this paper we study the space of formal deformation of a k-linear∞-category C defined as

CatDefcC(k[[t]]) = limiCatDefcC(k[t]/ti).

In this case the (formal) E2-Koszul dual of k[[t]] is equivalent to the underlying E2-algebraof the commutative graded alegbra k[β] with β of cohomological degree 2. ThereforeCatDef∧C (k[[t]]) is equivalent to the space of k[β]-linear structures on C (see [Toe14, Thm5.1]). We obtain the following.

Theorem 1.1. (See Theorem 3.63). — Let k be a field and C a k-linear ∞-category.Suppose that C admits a single compact generator E such that ExtmC (E,E) = 0 for m 0.Then the natural transformation θ induces an equivalence

CatDefcC(k[[t]]) ' CatDef∧C (k[[t]]) ' k[β]− linear structures on C.Moreover, for any formal deformation Cii≥0 of C, each Ci has a compact generator whichsatisfies the same condition as E.

The assumptions on C in this theorem are rather reasonable. For example if C is thederived category of quasi-coherent sheaves on a finite type scheme over k, we know by atheorem of Bondal–Van-den-Bergh that C admits a compact generator E which satisfies theassumption of Theorem 1.1.

To prove this theorem we use a new description of the space CatDef∧C (k[[t]]) in terms of

actions of the loop group ΩA1 on C where A1 = Spf(k[[t]]) is the formal completion of theaffine line at 0. A similar description appears in [Pre12, Lem 5.3.3.6].

Theorem 1.2. (See Corollary 3.52). — Let k be field and C a compactly generated k-linear∞-category whose spaces of morphisms are cohomologically bounded above. Then there existsa natural equivalence

CatDef∧C (k[[t]]) ' ΩA1 −Actions on C.

Moreover we obtain a similar formula not only for k[[t]] but for deformations over anypro-artinian E2-algebra. We derive this result from a fundamental fact concerning the ∞-category of formal moduli problems over any reasonable deformation context A in the senseof Lurie: we show that the loop functor

Ω : formal moduli problems over A → Group objects in formal moduli problems over Ais an equivalence of∞-categories (see Proposition 2.44). This fact implies a general formulafor the best approximation to a deformation functor (see Theorem 2.48). Consider forexample the deformation functor AlgDefB of an E1-algebra B over k. It can be shown thatif B is connective, AlgDefB is a formal moduli problem (see Proposition 3.56) whose tangentcomplex is given by the shifted derived derivations Der(B,B)[1]. Moreover this fact aboutthe loop functor above implies that there exists an equivalence

AlgDefB(k[[t]]) ' ΩA1 −Actions on B.

Sketch of proof of Theorem 1.1. Under these assumptions on C, we know by [Lurb,Prop 5.3.21] that the map θ induces an isomorphism on πi for i > 0 and an injection on π0.Therefore for the first part, it only remains to prove the surjectivity on π0. If E ∈ C is anobject, we can define an pre-formal moduli ObjDefE encoding deformations of E in C. Wecan define as well a pre-formal moduli Defc(C,E) encoding simulnateous deformations C and


E. A point in Def(C,E)(A) corresponds to a compactly generated deformation CA of C andof a deformation EA of E in CA. There exists a map Defc(C,E) → CatDefcC which forget thedeformation of the object. This latter map induces a map on the loops Aut(C,E) → AutCwhich takes an infinitesimal autoequivalence of C fixing E to the underlying autoequivalenceof C (forgetting that it fixes E). Let α ∈ π0CatDef∧C (k[[t]]) which corresponds to an action

ρ : ΩA1 → AutC via Theorem 1.2. If E is a compact generator of C, it might not be fixedunder ρ. However because ρ corresponds to a D(2)(k[[t]]) ' k[β]-linear structure on C viaTheorem 1.1, it is possible to find another compact generator of C fixed under ρ. For thiswe can show that it suffices to take the cofiber E′ of the map β : E[−2] → E given by the

multiplication by β. Therefore there exists a lift ρ : ΩA1 → Aut(C,E′) to autoequivalencesof C fixing E′. But such autoequivalences restrict to algebra autoequivalences of the E1-

algebra B = EndC(E′). Hence we have an action ρ′ : ΩA1 → Autalg

B which corresponds to aformal deformation Bt of B through the equivalence above. By construction, left modulesover Bt are sent to α through the map θ, hence we have surjectivity on π0. It also provesthat any formal deformation of C is equivalent to left modules over a formal deformation ofan E1-algebra, and has therefore a compact generator.

This argument is deeply connected to the fact that D(2)(k[[t]]) ' k[β] is a graded poly-nomial algebra and cannot be adapted to the case of infinitesimal deformations whereD(2)(k[t]/t2) ' k ⊕ k[−2].

1.2. Outline of this paper. Section 2.1 contains some motivations and a first definitionof formal moduli problems over artinian commutative dg-algebras.

Section 2.2, 2.3, as well as 3.1, 3.2 and most of 3.3 are expository and devoted to makerecalls from Lurie’s fundamental work in [Lurb], in a form that is suitable for our use. Thecorresponding references are written. We apologize for the formal aspect of the exposition.

Section 2.4 contains some reminders about proximate formal moduli problems but alsocontains new materials concerning the loop functor on formal moduli as well as our firstmain results concerning the explicit formula for the associated formal moduli in terms ofgroup actions.

Section 3.4 contains new material concerning the description of deformations of a linear∞-categories in terms of Ind-coherent group actions.

Section 3.5 contains folklore facts about deformations of associative algebras which weuse in the proof of Theorem 3.63. However they are put in a modern form.

Finally Section 3.6 contains our main result concerning the compact generation of formaldeformations as well as its proof.

1.3. Conventions.

• Let U be a Grothendieck universe, with U satisfying the axiom of infinity. TheU-small mathematical objects will be called only small. We assume the axiomof Universes. Some arguments in this article will require to enlarge the universeU, which is always possible by assuming the axiom of Universes. If V is such anenlargement in which U is small, the V-small mathematical objects will be callednot necessarily small.

• We work within the theory of ∞-categories in the sense of Lurie [Lur09], a.k.aquasicategories. We follow the terminology and the conventions of loc.cit. regardingthe theory of ∞-categories.


• The only exception is for morphisms between ∞-categories, which are called ∞-functors (instead of just functors in loc.cit.). For two ∞-categories C and C′, wedenote by Fun(C, C′) the ∞-category of ∞-functors from C to C′.

• If C is an ∞-category admitting a final object ∗. The ∞-category C∗ of pointed ob-jects in C is by definition the full subcategory of Fun(∆1, C) consisting of morphismsC → C ′ such that C is a final object of C.

• We denote by S the∞-category of small spaces in the sense of [Lur09, Def 1.2.16.1].It is therefore defined as the simplicial nerve of the simplicial category of smallKan complexes. We denote by Sbig the ∞-category of not necessarily small spaces.The word space means an object of the ∞-category S. The expression notnecessarily small space means an object of Sbig for some appropriate universe V.We denote by ∗ the final object ∆0 of S.

• We follow the terminology of [Lurc] concerning n-connective spaces. Let n ≥ 0 bean integer. A space X is n-connected if πi(X,x) = 0 for every i < n and everyvertex x ∈ X. Every space is declared to be (−1)-connective.

• We follow the terminology of [Lurc] concerning n-truncated spaces. Let n ≥ −1be an integer. A space X is n-truncated if πi(X,x) = 0 for every i > n and everyvertex x ∈ X. By convention a space is (−2)-truncated if it is contractible. A mapof pointed spaces X → Y is called n-truncated if its fiber is n-truncated.

• We denote by Sp the∞-category of spectra (see [Lurc, Def 1.4.3.1]). We will reviewin §2.2 the definition of spectra objects in an ∞-category admitting finite limits.

• Let n ∈ Z be an integer. We consider the usual t-structure on the ∞-category Sp.A spectrum X is n-connective (resp. n-truncated) if πiX = 0 for every i < n (resp.i > n).

2. Formal moduli and approximation after Lurie

2.1. Formal moduli problems. We start by some motivations toward the notion of for-mal moduli problems, and we define them and their tangent complex in the context ofcommutative dg-algebras. The introduction of [Lurb] is highly recommended.

Let k be a field of characteristic 0 and B an associative k-algebra. Let artk denotethe ordinary category of local artinian commutative k-algebras and Gpd the category ofgroupoids. The ordinary deformation theory of B is encoded in a functor

defB : artk −→ Gpd

such that for each A ∈ artk, the set defB(A) is the set of isomorphism classes of deformationsof B over A and functoriality is given by base change. In the introduction we saw that thetangent space of defB is given by

π0defB(k[t]/t2) ' HH2(B).

Moreover it can be shown that the group of automorphisms of any first order deformationB1 of B is given by

π1(defB(k[t]/t2), B1) ' HH1(B).

The natural question in deformation theory is to ask whether a first order deformation B1

of B can be extended to a second order deformation. It is possible to prove by direct com-putation that for every first order deformation B1 of B, there exists a class o(B1) ∈ HH3(B)such that there exists a lift of B1 to a second order deformation of B if and only if o(A1) = 0.This latter fact is much worse than the facts above giving an identification of Hochschildcohomology with a space of deformations. Indeed it does not give a natural way to obtainthe obstruction and it does not give a deformation theoretic interpretation of the wholegroup HH3(B).


At the cost of passing from deformations over artinian algebras to deformations over ar-tinian differential graded algebras, we can give a satisfactory answer to the above questions.This is the point of view of derived deformation theory. Moreover, derived deformationtheory gives an answer to another related important question (which was perhaps its origi-nal motivation): the problem of classifying deformation problems by differential graded Liealgebras.

Recall that the category of commutative dg-algebras cdgak admits a simplicial combi-natorial model structure with equivalences being quasi-isomorphisms and fibrations beinglevelwise surjective maps. Recall as well that for any model category M with subclass ofweak equivalences W , we have an associated ∞-category N(M)[W−1] to M which is bydefinition the localization (in the Joyal model structure on simplicial sets) of the nerve of

M along W (see [Rob14, 2.2.0.1]). We denote by CAlgdgk the ∞-category associated to the

model category cdgak.

If n ≥ 0 is an integer, we denote by k ⊕ k[n] the shifted trivial square zero extension ofk by an element in degree −n. If n = 0, k ⊕ k[n] = k[t]/t2 and if n > 0, the underlyingcomplex of k ⊕ k[n] consists of k in degree 0 and k in degree −n, with zero differential. Itis well known that there exists a diagram

k[t]/t3 //

k[t]/t2

k // k ⊕ k[1]

in the∞-category CAlgdgk , which is a cartesian diagram. This important fact provides a way

to analyze more carefully the behavior of the projection map defB(k[t]/t3)→ defB(k[t]/t2).Deformation theory was at first formalized through the notion of functors defined on lo-cal artinian algebras satisfying some exactness properties called the Schlessinger conditions.Nevertheless to make the above idea concrete, we need to allow our functors to be defined onartinian dg-algebras. In practise, most if not all the deformation functors naturally extendto dg-algebras.

A commutative k-dg-algebra A is artinian if H0(A) is a local artinian ring with residuefield k, Hi(A) = 0 for i > 0 and i 0 and if for every i, the vector space Hi(A) is finitedimensional over k. The trivial square zero extensions k ⊕ k[n] are examples of such. We

denote by dgartk the full subcategory of CAlgdgk consisting of artinian dg-algebras. The

deformation functor DefB extends naturally on artinian dg-algebras. If A ∈ dgartk, a defor-mation of B over A is given by an associative dg-algebra BA together with an equivalence

BA ⊗A k ' B in CAlgdgk . It is possible to define an ∞-functor

DefB : dgartk → S

such that for each A ∈ dgartk, the space DefB(A) is the classifying space of deformation ofB over A up to equivalence. It can be shown (see section §3.5) that the induced diagram

DefB(k[t]/t3) //

DefB(k[t]/t2)

DefB(k) // DefB(k ⊕ k[1])


is cartesian in S and that there exists a bijection π0DefB(k ⊕ k[1]) ' HH3(B). MoreoverDefB(k) ' ∗ so that we obtain a fiber sequence of pointed sets

π0DefB(k[t]/t3)→ π0DefB(k[t]/t2)→ HH3(B).

The map on the right associates to each (equivalence class of) first order deformation theobstruction of lifting it to a second order deformation. This motivates the following centraldefinition introduced in [Lurb, Def 0.0.8].

Definition 2.1. An ∞-functor F : dgartk → S is called a formal moduli problem if itsatisfies the following two conditions:

(1) The space F (k) is contractible.(2) For every cartesian diagram of the form

A //

A′

k // k ⊕ k[n]

in dgartk, the induced diagram

F (A) //

F (A′)

F (k) // F (k ⊕ k[n])

is cartesian in S.

The first condition means that we are looking at the deformation of one object (up toequivalence). Indeed we should think about a formal moduli problem as a formal neibourg-hood of a point in a derived moduli space. The second condition, which is a derived analog ofthe Schlessinger conditions, provides a way to relate the spaces of deformations over variousrings with the spaces of obstructions, and provides as well the tangent complex associatedto the formal moduli problem. Indeed for each m ≥ 1 the diagram

k ⊕ k[m− 1] //

k

k // k ⊕ k[m]

is cartesian in dgartk. Therefore if F is a formal moduli problem, the induced map

F (k ⊕ k[m− 1])→ ∗×F (k⊕k[m]) ∗ ' ΩF (k ⊕ k[m])

is a homotopy equivalence, and provides the bonding maps of a spectrum TF with (TF )m =F (k⊕k[m]). Here the loop space is taken with respect to the point ∗ ' F (k)→ F (k⊕k[m]).The zero level (TF )0 = F (k[t]/t2) is the usual Zariski tangent space. The level 1 is theobstruction space for lifting infinitesimal deformations and the higher positive levels arehigher obstruction spaces. As expected, it can be shown that a map of formal moduliproblems f : F → G is an equivalence if and only if the induced map of spectra TF → TG isan equivalence (see [Lurb, Prop 1.2.10]). It can be shown that the tangent complex functorcommutes with finite limits so for any formal moduli problem we obtain an equivalence

TΩF ' ΩTF = TF [−1].

It is therefore not suprising that the shifted tangent complex TF [−1] has the structure of adg-Lie algebra being the tangent complex of a formal group. Moreover the central theoremin the theory of formal moduli problems is the following classification result, and which


shows the power of this derived point of view. If FMPk denotes the ∞-category of formalmoduli problems over k, Lurie defines an ∞-functor

dgLiek −→ FMPk

which is proven to be an equivalence of∞-categories, and whose inverse equivalence is givenon objects by F 7→ TF [−1] (see [Lurb, Thm 0.0.13]). We will review in §2.3 this classificationand as well as in less commutative deformation contexts.

2.2. Axiomatic deformation theory. As explained in the introduction, some deforma-tion functors are often defined on En-algebras for some n < ∞, and it is therefore usefulto have a theory of such formal moduli problems at our disposal. In [Lurb], an axiomaticapproach to formal moduli problems is given, and allows the author to obtain results in allpossible deformation context at the same time, in particular the context of En-algebras forevery 0 ≤ n ≤ ∞. In this section we recall definitions and results from [Lurb, §1] related tothe axiomatic approach to deformation theory. The main result is the classification of formalmoduli problems relative to a deformation context which admits a deformation theory. Westart by recalling definitions and terminology related to spectra objects.

Notation 2.2. Following [Lurc, Rem 1.1.2.9] we can define the loop functor for any pointed∞-category C. Consider the subcategoryMΩ of the∞-category of square diagrams Fun(∆1×∆1, C) consisting of pullback diagrams

Y //

0

0′ // X

where 0 and 0′ are zero objects of C. Applying [Lur09, Prop 4.3.2.15] we have that the∞-functor e :MΩ → C given by evaluation at the final vertex is a trivial fibration. Let s bea section of e and let e′ :MΩ → C be the evaluation at the initial vertex. The loop functorof C is by definition the composite Ω = e′ s : C → C.

Notation 2.3. We recall the definition of spectra objects in an∞-category admitting finitelimits. Let Sfin denote the full subcategory of S which contains ∗ and is stable under finitecolimits, or in other words the ∞-category of finite spaces. Let Sfin

∗ be the ∞-categoryof pointed finite spaces and let A be an ∞-category admitting finite limits. Recall (see[Lurc, Def 1.4.2.1]) that an ∞-functor X : Sfin

∗ → A is pointed excisive if it carries pushoutdiagrams to pullback diagrams in A and if X(∗) is a final object of A. The ∞-category ofspectra objects in A is defined to be the full subcategory Sp(A) of Fun(Sfin

∗ ,A) consistingof pointed excisive ∞-functors. If X : Sfin

∗ → A is a spectrum in A and n ≥ 0, we denoteby Ω∞−nX the evaluation of X on the pointed n-sphere Sn (for which we choose a modelin Sfin

∗ ).

Remark 2.4. In the situation of Notation 2.3, denote by Ω : A∗ → A∗ the loop functorof A∗. Unwinding the definition, a spectrum object X : Sfin

∗ → A is essentially given by asequence of finite pointed spaces Ω∞−nX, n ≥ 0, together with equivalences ΩΩ∞−nX 'Ω∞−n−1X. Indeed by [Lurc, Prop 1.4.2.24], there exists an equivalence of ∞-categories

Sp(A) ' lim(. . .Ω−→ A∗

Ω−→ A∗Ω−→ A∗).

Definition 2.5. • A deformation context is a pair (A, (Eα)α∈T ) where A is a pre-sentable ∞-category and (Eα)α∈T is a sequence of objects of the ∞-category ofspectra objects Sp(A) in A.


• A map f : A→ A′ in A is called elementary if it can be written as a pullback

A //

f

∗

A′ // Eα[n]

for an index α ∈ T and an integer n.• A map f : A −→ A′ in A is called small if it can be written as a finite composition

of elementary maps A0 = A→ A1 → . . .→ Am = A′ in A.• An object A ∈ A is called artinian if the map A → ∗ is small. We denote by Aart

the full subcategory of A spanned by artinian objects.

Example 2.6. Commutative dg-algebras. — Let k be a commutative ring. Consider thesimplicial model category of unbounded dg-modules over k (we consider differential Z-gradedmodules) dgmodk with class of equivalences W1 being the class of quasi-isomorphisms andfibrations are levelwise surjective morphisms. Let cdgak = AlgComm(dgmodk) denote thecategory of commutative algebra objects in dgmodk, its objects are commutative dg-algebrasover k. We endow cdgak with its usual simplicial model structure with equivalences W2 beingquasi-isomorphisms and fibrations are levelwise surjectives morphisms.

Let CAlgdgk := N(cdgack)[W−1

2 ] be the associated∞-category. Let CAlgdg,augk := (CAlgdg

k )/kbe the ∞-category of augmented commutative dg-algebras over k. The ∞-category of spec-

tra in CAlgdgk and in CAlgdg,aug

k is equivalent to the∞-category Moddgk := N(dgmodck)[W−1

1 ]of dg-modules over k.

The presentable ∞-category CAlgdg,augk together with T = ∗ a one point set and E =

k ∈ Moddgk concentrated in degree zero form a deformation context (CAlgdg,aug

k , k). Thedeloopings E[n] of E are the square zero extensions k ⊕ k[n] and the elementary maps aregiven by the inclusion maps k → k⊕ k[n] for n ≥ 0. The artinian objects are given by localartinian commutative dg-algebras (similar to [Lurb, Prop 1.1.11]), which are by definitioncommutative dg-algebras A satisfying the following conditions:

• Hi(A) = 0 for i < 0 and for i 0• For every i, the k-vector space Hi(A) is finite dimensional• The commutative ring H0(A) is local with residue field k.

We denote by CAlgdg,artk the subcategory of artinian commutative dg-algebras over k. A

map A → B in CAlgdg,artk is small if and only if the induced ring map H0(A) → H0(B) is

surjective (similar to [Lurb, Lem 1.1.20]).

Definition 2.7. (Lurie [Lurb, Def 1.1.14]). — Let (A, (Eα)α∈T ) be a deformation contextand let ∗A denote a final object of A.

• An ∞-functor F : Aart −→ S is called a pre-formal moduli problem if the spaceF (∗A) is contractible.

• An ∞-functor F : Aart −→ S is called a formal moduli problem if it is a pre-formalmoduli problem and if in addition for every pullback square

A //

A0

A1// A01


in Aart such that the map A0 → A01 is small, the induced diagram

F (A) //

F (A0)

F (A1) // F (A01)

is a pullback in S.

The following result allows us to test the second condition in the previous definition on asmaller class of pullback diagrams of artinian objects.

Proposition 2.8. [Lurb, Prop 1.1.15]. — Let (A, (Eα)α∈T ) be a deformation context. An∞-functor F : Aart −→ S is a formal moduli problem if and only if the two conditions aresatisfied:

1) The space F (∗A) is contractible.2) For every pullback diagram of the form

A //

∗A

A′ // Ω∞−mEα

in Aart, for some α ∈ T , the induced diagram

F (A) //

∗

F (A′) // F (Ω∞−mEα)

is a pullback in S.

Example 2.9. Let (CAlgdg,augk , k) be the deformation context of Example 2.6 formed

by augmented commutative dg-algebras. By Proposition 2.8, a pre-formal moduli problem

F : CAlgdg,artk → S is a formal moduli problem in the sense of Definition 2.7 in and only if

it is a formal moduli problem in the sense of Definition 2.1.

Notation 2.10. Let (A, (Eα)α∈T ) be a deformation context. We denote by PFMPA the∞-category of pre-formal moduli problem and by FMPA the ∞-category of formal moduliproblems. Let iA : FMPA → PFMPA denote the natural inclusion ∞-functor. BothFMPA and PFMPA are presentable ∞-categories, and iA preserves small limits. Hence bythe adjoint ∞-functor theorem [Lur09, Cor 5.5.2.9] iA has a left adjoint denoted by

FMPA

iA// PFMPA

LA

vv

.

For every pre-formal moduli problem F we therefore have a natural map F → iA(LA(F ))in PFMPA given by the unit map of this adjunction, which is universal among maps fromF to formal moduli problems. The formal moduli problem LA(F ) is therefore the best ap-proximation we were refering to in the introduction. For convenience, when the deformationcontext is clear, we denote LA simply by L.

Notation 2.11. Let (A, (Eα)α∈T ) be a deformation context. For each α, the object Eα isa spectrum in the ∞-category A, in particular we have Eα(∗) ' ∗A. By [Lurb, Prop 1.2.3],for every α ∈ T and every pointed finite set S, the object Eα(S) is artinian. Hence for eachα we have a spectrum object

Eα : Sfin∗ −→ Aart.


Proposition 2.12. [Lurb, Prop 1.2.4]. — Let (A, (Eα)α∈T ) be a deformation context andlet F : Aart −→ S be a formal moduli problem. For every α ∈ T the composite ∞-functor

Sfin∗

Eα // Aart F // S

is pointed excisive in the sense of Notation 2.3 and therefore defines a spectrum in S denotedby F (Eα).

Remark 2.13. The idea behind Proposition 2.12 is to consider for each α and each integerm ≥ 0 the pullback diagram

Ω∞−mEα //

∗A

∗A // Ω∞−m−1Eα

in Aart. If F is a formal moduli problem the induced diagram of spaces

F (Ω∞−mEα) //

∗

∗ // F (Ω∞−m−1Eα)

is a pullback, and therefore gives the bonding maps of the tangent spectrum of F at α:

ΩF (Ω∞−mEα) ' F (Ω∞−m−1Eα).

Definition 2.14. [Lurb, Def 1.2.5]. — Let (A, (Eα)α∈T ) be a deformation context and letF : Aart −→ S be a formal moduli problem. The tangent spectrum of F at α is by definitionthe spectrum F (Eα).

Proposition 2.15. [Lurb, Prop 1.2.10]. — Let (A, (Eα)α∈T ) be a deformation context andlet f : F → G be a map in FMPA. Then f is an equivalence if and only if for each α ∈ Tthe induced map on tangent spectra F (Eα)→ G(Eα) is an equivalence.

Theorem 2.16. (Lurie [Lurb, Thm 1.3.12]). — Let (A, (Eα)α∈T ) be a deformation contextwhich admits a deformation theory D : Aop → B in the sense of [Lurb, Def 1.3.9]. Thenthere exists an ∞-functor

Ψ : B −→ FMPA

which to an object B ∈ B assigns the formal moduli problem ΨB defined by ΨB(A) =MapB(D(A), B). Moreover Ψ is an equivalence of ∞-categories.

We now set some notations which will be useful in the sequel of the paper.

Notation 2.17. Let (A, (Eα)α∈T ) be a deformation context. Let A ∈ Aart be an artinianobject. We denote by Spf(A) : Aart −→ S the ∞-functor j′(A) where j′ the co-Yoneda em-bedding. For every R ∈ Aart we have therefore an equivalence Spf(A)(R) ' MapA(A,R).By the ∞-categorical co-Yoneda lemma, the pre-formal moduli problem Spf(A) is a formalmoduli problem. By the ∞-categorical Yoneda lemma ([Lur09, Prop 5.1.3.1]) this construc-tion defines a fully faithful ∞-functor

Spf : (Aart)op −→ FMPA.

Remark 2.18. In the deformation context (CAlgdg,augk , k) of example 2.6, suppose A is

a local artinian discrete commutative k-algebra with residue field k viewed naturally as anartinian commutative dg-algebra. Then the formal moduli problem Spf(A) parametrizesdeformations of the point corresponding to the augmentation A→ k.


Notation 2.19. Let C be a small ∞-category. We denote by Ind(C) the ∞-category ofInd-objects in C (see [Lur09, Def 5.3.5.1] for a definition). By [Lur09, Cor 5.3.5.4], Ind(C)is equivalent to the full subcategory of P(C) formed by functors F : Cop −→ S which areleft exact. By [Lur09, Prop 5.3.5.10] the∞-category Ind(C) is equivalent to the∞-categoryfreely generated by C under filtered colimits. Namely if D is a small ∞-category whichadmits filtered colimits, then the Yoneda embedding C −→ Ind(C) induces an equivalenceof ∞-categories

Funω(Ind(C),D) −→ Fun(C,D)

where the left handside is the ∞-category of ∞-functors which commute with filtered col-imits. The dual notion to Ind-objects is that of Pro-objects. The ∞-category of pro-objectsassociated to C is the ∞-category Pro(C) := Ind(Cop)op. If D is a small ∞-category whichadmits cofiltered limits, then the Yoneda embedding C −→ Ind(C) induces an equivalenceof ∞-categories

Funω(Pro(C),D) −→ Fun(C,D)

where the left handside is the ∞-category of ∞-functors which commute with cofilteredlimits.

Notation 2.20. Let (A, (Eα)α∈T ) be a deformation context. We consider the ∞-categoryPro(Aart) of pro-artinian objects. The ∞-functor

Spf : (Aart)op −→ FMPA

gives an essentially unique ∞-functor which commutes with filtered colimits

Ind(Spf) : Ind((Aart)op) ' Pro(Aart)op −→ FMPA

By abusing notations, we denote again the functor Ind(Spf) by Spf. If A = limiAi is apro-object in Aart, then there is an equivalence

Spf(A) ' colimiSpf(Ai)

in FMPA.

Notation 2.21. Let (A, (Eα)α∈T ) be a deformation context and F : Aart −→ S a pre-formal moduli problem. Then we denote again by F the natural extension of F to pro-artinian objects

F : Pro(Aart) −→ Swhose existence is ensured by the universal property of the Pro completion (see Notation2.19). Its value on a pro-artinian object A = limiAi is given informally by

F (A) = limiF (Ai).

For example if we work in the deformation context (CAlgdg,augk , k) of augmented commu-

tative dg-algebras over k, the discrete algebra of formal power series over k gives a pro-objectk[[t]] := limik[t]/ti. If F is a formal moduli problem, the space F (k[[t]]) := limiF (k[t]/ti)is often called the space of formal arcs in F .

2.3. Classification of En-formal moduli problems. We recall from [Lurb, §2 and §4]the classification of formal moduli problems related to the deformation contexts formed byE∞-algebras and by En-algebras (see examples 2.22 and 2.27).

Example 2.22. E∞-algebras. — Let k be a field. We denote by Modk the ∞-categoryof k-modules spectra whose objects are decribed by spectra together with an action of thering spectrum Hk associated to k. Let CAlgk := AlgE∞(Modk) be the ∞-category ofE∞-algebras over k and CAlgaug

k := (CAlgk)/k the ∞-category of augmented E∞-algebrasover k. By [Lurc, Thm 4.5.4.7], if k is of characteristic zero, there exists an equivalence

of ∞-categories CAlgk ' CAlgdgk and therefore an equivalence CAlgaug

k ' CAlgdg,augk . The

stabilization Stab(CAlgaugk ) is equivalent to the ∞-category Modk of k-module spectra.


Together with T = ∗ a one point set and E = k as discrete module, the presentable ∞-category CAlgaug

k form a deformation context (CAlgaugk , k). The deloopings E[n] of E are

given by the square zero extensions k ⊕ k[n]. The artinian objects (see [Lurb, Prop 1.1.11])are given by E∞-algebras A which satisfy the following conditions:

• πiA = 0 for i < 0 and for i 0.• For every i, the k-vector space πiA is finite dimensional.• The commutative ring π0A is local with residue field k.

We denote by CAlgartk ⊂ CAlgaug

k the subcategory of artinian E∞-algebras. A map A→ B

in CAlgartk is small if and only if the induced ring map π0A→ π0B is surjective (see [Lurb,

Lem 1.1.20]).

Definition 2.23. An E∞-formal moduli problem over k is a formal moduli problem relativeto the deformation context (CAlgaug

k , k) formed by augmented E∞-algebras (example2.22). We call them only formal moduli problems when the context is clear. We denote byFMPE∞(k) the ∞-category of E∞-formal moduli problems over k.

Notation 2.24. Let k be a field of characteristic zero and let dgliek denote the category ofdifferential Z-graded Lie algebras over k. There exists a combinatorial model structure ondgliek with equivalences W being quasi-isomorphisms and fibrations are given by levelwisesurjective morphisms (see [Lurb, Prop 2.1.10]). We denote by AlgLiek := N(dglieck)[W−1]the ∞-category associated to dgliek.

Notation 2.25. Koszul duality between E∞ and Lie. — Let k be a field of characteristiczero and let

D : (CAlgaugk )op −→ AlgLiek

denote the ∞-functor left adjoint to the ∞-functor

C∗ : AlgLiek −→ (CAlgaugk )op

given by Lie algebra cochains (see [Lurb, §2.3]). We call D the Koszul duality functorbetween E∞ and Lie-algebras. For a differential graded Lie algebra g over k and an ar-tinian E∞-algebra A, the space MapAlgLiek

(D(A), g) is a model for the space MC(g⊗k mA)

of solutions of the Maurer–Cartan equation in the Lie algebra g ⊗k mA where mA is theaugmentation ideal of A.

Theorem 2.26. (Lurie [Lurb, Thm 2.0.2]). — Let k be a field of characteristic zero. The

∞-functor D : (CAlgaugk )op −→ AlgLiek defined above is a deformation theory in the sense of

[Lurb, Def 1.3.9]. As a corollary to Theorem 2.16 we obtain an equivalence of ∞-categories

Ψ : AlgLiek −→ FMPE∞(k).

Moreover the diagram of ∞-categories

AlgLiekΨ //

U&&

FMPE∞(k)

T [−1]

Sp

where U is the forgetful functor is commutative.

We now recall the classification of formal moduli problems defined over En-algebras. Inthe sequel, k is a field of any characteristic and n ≥ 0 is an integer.

Example 2.27. En-algebras. — Let k be a field. Let n ≥ 0 be an integer and let Alg(n)k :=

AlgEn(Modk) denote the ∞-category of En-algebras over k. Let moreover Alg(n),augk be

the ∞-category of augmented En-algebras over k. The stabilization Stab(Alg(n),augk ) is


equivalent to the ∞-category Modk of k-module spectra. Together with T = ∗ and E = k

as a discrete module, the presentable ∞-category Alg(n),augk form a deformation context

(Alg(n),augk , k). The deloopings E[n] of E are given by the square zero extensions k⊕k[n].

The artinian objects (see [Lurb, Prop 4.5.1] are given by En-algebras which satisfy thefollowing conditions:

• πiA = 0 for i < 0 and for i 0.• For every i, the k-vector space πiA is finite dimensional.• Let r denote the radical of the algebra π0A, then the natural map (π0A)/r → k is

an isomorphism.

We denote by Alg(n),artk ⊂ Alg

(n),augk the full subcategory of artinian En-algebras. A map

A→ B in Alg(n),artk is small if and only if the induced ring map π0A→ π0B is surjective (see

[Lurb, Prop 4.5.3]). In this paper we will study the deformation functor of a fixed k-linear∞-category which is naturally defined in the deformation context formed by augmentedE2-algebras.

Definition 2.28. Let n ≥ 1. An En-formal moduli problem over k is a formal moduli

problem relative to the deformation context (Alg(n),augk , k) formed by augmented En-

algebras over k (exemple 2.27). We call them only formal moduli problems when the contextis clear. We denote by FMPEn(k) the∞-category of En-formal moduli problems over k andby PFMPEn(k) the ∞-category of pre-En-formal moduli problems over k.

Notation 2.29. Let A ∈ Alg(n)k be an En-algebra over k. We denote by Aug(A) =

MapAlg

(n)k

(A, k) the space of augmentations of A. Let now A,B be two augmented En-

algebras over k with augmentations ε : A → k and η : B → k respectively. Note that thenatural maps of En-algebras A→ A⊗k B ← B induce a map of spaces

Aug(A⊗k B)→ Aug(A)×Aug(B).

We recall that the tensor product A ⊗k B differs a priori from the coproduct in Alg(n)k .

We denote by Pair(A,B) the space of pairings between A and B; it is by definition thehomotopy fiber of Aug(A ⊗k B) → Aug(A) × Aug(B) over the point (ε, η). The spacePair(A,B) have points corresponding to augmentations A ⊗k B → k which extends thegiven augmentations ε : A→ k and η : B → k.

Proposition 2.30. [Lurb, Prop 4.4.1]. — Let A ∈ Alg(n),augk be an augmented En-algebra

over k. The construction B 7→ Pair(A,B) defines an ∞-functor (Alg(n),augk )op → S. More-

over this ∞-functor is representable. In other words, there exists an augmented En-algebraD(n)(A) and a universal pairing ν : A ⊗k D(n)(A) → k such that for every augmentedEn-algebra B the map ν induces an equivalence of spaces

MapAlg

(n),augk

(B,D(n)(A)) −→ Pair(A,B).

Notation 2.31. In the situation of Proposition 2.30 we refer to D(n)(A) as the En-Koszuldual to A. Proposition 2.30 moreover implies that the construction A 7→ D(n)(A) determinesan ∞-functor

D(n) : (Alg(n),augk )op −→ Alg

(n),augk

called the Koszul duality functor. Because the construction (A,B) 7→ Pair(A,B) is symmet-ric in A and B, the ∞-functor D(n) is self adjoint, meaning that there exists an equivalenceof spaces

MapAlg

(n),augk

(B,D(n)(A)) ' MapAlg

(n),augk

(A,D(n)(B)).

Theorem 2.32. (Lurie [Lurb, Thm 4.5.5, Thm 4.0.8]). — Let n ≥ 0 be an integer. The

∞-functor D(n) : (Alg(n),augk )op −→ Alg

(n),augk defined above is a deformation theory in the


sense of [Lurb, Def 1.3.9]. As a corollary to Theorem 2.16 we obtain an equivalence of∞-categories

Ψ : Alg(n),augk −→ FMPEn(k).

Moreover there exists a commutative diagram of ∞-categories

Alg(n),augk

Ψ //

m

FMPEn(k)

T [−n]

Modk // Sp

where mB is the augmentation ideal of an augmented En-algebra B.

The following corollary will be useful in the sequel and follows from the fact that theKoszul duality functor D(n) is a deformation theory.

Corollary 2.33. Let n ≥ 1 be an integer and denote by t : FMPEn(k) → Alg(n),augk the

equivalence given by Theorem 2.32. Let B be any pro-artinian En-algebra over k. Then there

exists a natural equivalence tSpf(B) ' D(n)f (B) in Alg

(n),augk or equivalently an equivalence

of formal moduli problems Spf(B) ' ΨD

(n)f (B)

. In particular if B is artinian we have an

equivalence of augmented En-algebras tSpf(B) ' D(n)(B).

Proof. Because t commutes with colimits, it suffices to prove the statement for B artinian.In this case, for each artinian En-algebra A, we have a natural map

Spf(B)(A) = MapAlg

(n),augk

(B,A) −→ MapAlg

(n),augk

(D(n)(A),D(n)(B)) = ΨD(n)(B)(A).

By [Lurb, Prop 4.4.21], this map is an equivalence when A is n-coconnective (the localfiniteness assumption is satisfied whenever A is artinian). The En-algebra k ⊕ k[m] is n-coconnective as long as m > n. This implies that the map above is an equivalence whenA = k ⊕ k[m] for every m > n, which in turn implies that the natural transformationSpf(B) → ΨD(n)(B) is an equivalence on tangent spaces and therefore an equivalence offormal moduli problems.

2.4. Proximate formal moduli problems and loop spaces. We recall from the notionof n-proximate formal moduli problem from [Lurb, §5.1], which permits to study deformationfunctors which do not satisfy the derived Schlessinger conditions, but are very close to. Anexample is provided by the deformation functor CatDefC of a k-linear ∞-category C. Wethen describe a general formula for the approximation of some formal moduli problems withrespect to a general deformation context.

Definition 2.34. [Lurb, Def 5.1.5]. — Let (A, (Eα)α∈T ) be a deformation context. Apre-formal moduli problem F : Aart −→ S is an n-proximate formal moduli problem if forevery pullback square

A //

A0

A1// A01

in Aart such that the map A0 → A01 is small, the induced map of spaces

F (A) −→ F (A0)×F (A01) F (A1)

is (n− 2)-truncated.


Remark 2.35. By [Lurb, Prop 5.1.4] the condition of Definition 2.34 for being n-proximatecan be tested on the smaller class of pullback diagram of artinian objects the form

A //

∗A

B // Ω∞−mEα

for α ∈ T and m ≥ 0.

Notation 2.36. We denote by FMP(n)A ⊂ PFMPA the full subcategory of n-proximate

formal moduli problems. There is a tower of ∞-categories

FMPA = FMP(0)A ⊂ FMP

(1)A ⊂ . . . ⊂ FMP

(n)A ⊂ FMP

(n+1)A ⊂ . . . ⊂ PFMPA

For example, we will see below that the deformation functor of an object in a stable k-linear∞-category is in general 1-proximate and that the deformation functor of a fixed stablek-linear ∞-category itself is in general 2-proximate.

Theorem 2.37. [Lurb, Thm 5.1.9]. — Let (A, (Eα)α∈T ) be a deformation context whichadmits a deformation theory and let F : Aart −→ S be a pre-formal moduli problem. Thenthe following conditions are equivalent:

1) F is an n-proximate formal moduli problem.2) There exists a formal moduli problem F ′ and a (n− 2)-truncated map F → F ′.3) The natural map F → LA(F ) is (n− 2)-truncated.

For all the sequel of this subsection we work in a fixed deformation context (A, (Eα)α∈T )and we suppose that the∞-category A is pointed. This will be the case in all our applicationswhere A will be an ∞-category of augmented En-algebras for some n.

Remark 2.38. Let A be an ∞-category having an initial object ∅. Let Fun∗(A,S) denotethe subcategory of Fun(A,S) consisting of ∞-functors F : A → S such that F (∅) is acontractible space. It can be shown that the ∞-category Fun∗(A,S) is pointed, or in otherwords has an object which is both initial and final. This object is described informally bythe ∞-functor sending every object of A to a fixed final object ∗ of S.

Notation 2.39. Let (A, (Eα)α∈T ) be a deformation context such that A is a pointed ∞-category. Fix a final object ∗A of A which is therefore an initial object. Then the∞-categoryPFMPA of pre-formal moduli problem is by definition the ∞-category Fun∗(A

art,S) of∞-functors sending ∗A to a contractible space. In this situation, by Remark 2.38 the ∞-category PFMPA is pointed and we denote by

Ω : PFMPA −→ PFMPA

its loop functor in the sense of Notation 2.2. Because the inclusion FMPA → PFMPA andthe loop functor above commute with limits, for every formal moduli problem F then ΩFis again a formal moduli problem. For an integer n ≥ 0, we denote by Ωk = Ω . . . Ω (ktimes) the iterated loop functor.

Remark 2.40. In the situtation of Notation 2.39, if F : Aart → S is a pre-formal moduliproblem, the loop object ΩF is the pre-formal moduli problem described on objects by(ΩF )(A) = Ω∗F (A) where the loop space Ω∗ is the loop space taken at the base point∗ → F (A) corresponding to the map ∗ ' F (∗A)→ F (A) induced by the essentially uniquemap ∗A → A. If we denote by E the element in the one point set π0F (∗A) = E, which weimagine being an object of an ∞-category C. Informally, the point ∗ → F (A) correspondsto the trivial deformation of E over A. Moreover the ∞-functor ΩF parametrizes theinfinitesimal automorphisms of E in C, or in other words the deformations of idE in C.


Proposition 2.41. Let (A, (Eα)α∈T ) be a deformation context such that A is pointed. LetF : Aart −→ S be a pre-formal moduli problem and let n ≥ 0. Then F is an n-proximateformal moduli problem if and only if ΩnF is a formal moduli problem.

Proof. It is a direct consequence of the fact that the loop space functor commutes withsmall limits and that if X is an m-truncated pointed space then the space ΩX is (m − 1)-truncated.

Notation 2.42. Let C be an ∞-category admitting finite products. We consider the carte-sian symmetric monoidal structure C× on C (see [Lurc, §2.4.1]). The ∞-category MonEk(C)of Ek-monoids objects in C is the ∞-category AlgEk(C×) of Ek-algebra objects with respectto the cartesian symmetric monoidal structure on C. An Ek-monoid A with multiplicationmap m : A×A→ A in C is called grouplike if the maps

(m, p1) : A×A→ A×A, (m, p2) : A×A→ A×A

are equivalences in C. We denote by GpEk(C) ⊂ MonEk(C) the full subcategory of grouplikeEk-monoids in C, whose objects are also called Ek-groups in the sequel.

Notation 2.43. Let (A, (Eα)α∈T ) be a deformation context. The ∞-category PFMPA isan∞-topos. We consider the cartesian symmetric monoidal structure PFMP×A on PFMPA.Because the inclusion FMPA → PFMPA commutes with small products, the ∞-categoryFMPA inherits this cartesian symmetric monoidal structure and we have a symmetricmonoidal ∞-category FMP×A with a symmetric monoidal inclusion FMP×A → PFMP×A.

Let k ≥ 0, we denote by GpEk(A) the ∞-category GpEk(FMPA) and we called its objectsformal Ek-groups relative to A or just formal Ek-groups when the context is clear.

A consequence of the construction of Bar/Cobar functors (see [Lurc, Not 5.2.6.11]) is thatthe loop functor PFMPA −→ PFMPA given by Ωk factorizes through the ∞-category ofEk-group objects GpEk(PFMPA) giving an ∞-functor still denoted by

Ωk : PFMPA −→ GpEk(PFMPA).

Remark that if F : Aart −→ S is a k-proximate formal moduli problem then ΩkF is a formalmoduli problem which is moreover a formal Ek-group. We obtain a commutative diagramof ∞-categories

PFMPAΩk // GpEk(PFMPA)

FMP(k)A

OO

Ωk|55

FMPA

OO Ωk|FMPA

;;

We adopt the notation ΩkA = Ωk|FMPA.

The following result gives conditions on A under which ΩkA is an equivalence and is thefirst key result of this paper.

Proposition 2.44. Let (A, (Eα)α∈T ) be a deformation context which admits a deformationtheory D : Aop −→ B and such that A is pointed. We suppose moreover that there existsan ∞-functor U : B −→ A such that

• A is a stable ∞-category.• U is conservative, commutes with small limits and with sifted colimits.


Then for every k ≥ 0, the functor

ΩkA : FMPA −→ GpEk(A)

is an equivalence of ∞-categories.

Proof. By iteration it suffices to prove it for k = 1. By Theorem 2.16, there exists anequivalence of ∞-categories B

∼−→ FMPA. We therefore have a diagram of ∞-categories

FMPAΩA //

Ψ

GpE1(A)

GpE1(Ψ)

BΩB //

U

GpE1(B)

U

A ΩA // GpE1(A)

This diagram is commutative because both Ψ and U commutes with small limits. BecauseΨ and therefore GpE1

(Ψ) are equivalences, for proving the proposition it suffices to provethat ΩB is an equivalence. The ∞-functor ΩA is an equivalence because A is stable. Wedenote by BB and BA the left adjoints to ΩB and ΩA respectively. These are given bythe Bar construction which is given by a sifted colimit (see [Lurc, §5.2.2]). Because U isconservative and ΩA is an equivalence, in order to prove that ΩB is an equivalence it sufficesto prove that U BB ' BA U , but this is ensured by the condition that U commutes withsifted colimits.

Construction 2.45. Let (A, (Eα)α∈T ) be a deformation context and k ≥ 0 an integer.Consider the ∞-functor

Spf : Aart −→ (FMPA)op

from Notation 2.17 and denote by

Spf ′ : Aart → (FMP(k)A )op

its composition with the inclusion FMPA ⊂ FMP(k)A into k-proximate formal moduli prob-

lems. Denote by G the composite functor

Aart × FMP(k)A

Spf′×id// (FMP

(k)A )op × FMP

(k)A

Ωk| ×Ωk|// (GpEk(A))op × GpEk(A)

MapGpEk

(A)

S

where MapGpEk

(A)is the ∞-functor given by the mapping space in GpEk(A). For each

A ∈ Aart and F ∈ FMP(k)A , we have

G(A,F ) = MapGpEk

(A)(ΩkSpf(A),ΩkF ).

Because G(k, F ) ' F (∗A) is a contractible space, the∞-functor G determines an∞-functor

L′k : FMP(k)A → PFMPA well defined up to equivalence. For each A ∈ Aart and F ∈ FMP

(k)A

we have L′k(F )(A) = MapGpEk


Lemma 2.46. Let (A, (Eα)α∈T ) be a deformation context satifiying all the conditions ofProposition 2.44 and n ≥ 0 an integer. For each k-proximate formal moduli problem F , thepre-formal moduli problem L′k(F ) is a formal moduli problem.


Proof. For every pullback square

A //

A0

A1// A01

in Aart with A0 → A01 small, the induced diagram

Spf(A) Spf(A0)oo

Spf(A1)

OO

Spf(A01)

OO

oo

is a pushout in FMPA. By Proposition 2.44, for any integer k ≥ 0 the ∞-functor ΩkA :

FMPA → GpEk(A) is an equivalence. Hence the induced diagram

ΩkASpf(A) ΩkASpf(A0)oo

ΩkASpf(A1)

OO

ΩkASpf(A01)

OO

oo

is a pushout in GpEk(A). Let F ∈ FMP(k)A and we apply the Yoneda∞-functor Map

GpEk(A)

(−,ΩkF )

to obtain a pullback square

L′k(F )(A) //

L′k(F )(A0)

L′k(F )(A1) // L′k(F )(A01)

in S, and therefore L′k(F ) is a formal moduli problem.

Notation 2.47. Let (A, (Eα)α∈T ) be a deformation context satifiying all the conditions ofProposition 2.44 and n ≥ 0 an integer. It follows from Lemma 2.46 that the ∞-functor L′nform Construction 2.45 factors as an ∞-functor

L′n : FMP(n)A −→ FMPA.

Theorem 2.48. Let (A, (Eα)α∈T ) be a deformation context satifiying all the conditions ofProposition 2.44 and let k ≥ 0 be an integer. Consider the restriction

L| : FMP(k)A → FMPA

of the associated formal moduli problem functor from Notation 2.10. Then L| is equivalentto the ∞-functor L′k from Construction 2.45. In other words, for each k-proximate formalmoduli problem F , the associated formal moduli problem to F is given by

L(F )(A) ' MapGpEk


Proof. For convenience we denote the ∞-functor L′k simply by L′. It suffices to prove that

L′ : FMP(k)A → FMPA is a localization functor in the sense of [Lur09, Def 5.2.7.2]. Indeed it

implies that L′ is left adjoint to the inclusion i : FMPA → FMP(k)A and therefore that there

exists an equivalence

L′ ' L|FMP(k)A

.


By [Lur09, Prop 5.2.7.4] we only need to prove that there exists a natural transformation

αF : F → L′(F ) for each F ∈ FMP(k)A such that the two maps

αL′(F ), L′(αF ) : L′(F ) −→ L′ L′(F )

are equivalences in FMPA. For F ∈ FMP(k)A and A ∈ Aart, we have a natural map

F (A) = MapFMPA(Spf(A), F )

Ωk−→ MapGpEn (A)

(ΩkSpf(A),ΩkF ) = L′(F )(A)

which defines a natural transformation αF : F → L′(F ). The map αL′(F ) is given by

αL′(F ) : L′(F )(A) = MapFMPA(Spf(A), L′(F ))

Ωk−→ MapGpEn (A)

(ΩkSpf(A),ΩkL′(F )) = L′L′(F )(A).

which is an equivalence by Proposition 2.44. The map L′(αF ) gives for A ∈ Aart a map ofspaces

L′(F )(A) = MapGpEn (A)

(ΩkSpf(A),ΩkF )L′(αF )(A)−→ Map

GpEn (A)(ΩkSpf(A),ΩkL′(F )).

which by definition is equivalent to composing on the right with

ΩkαF : ΩkF −→ ΩkL′(F ).

But through the equivalence

ΩkL′(F )(R) ' MapGpEn (A)

(ΩkSpf(R),Ωk(ΩkF ))

for R ∈ Aart, the composition on the right by ΩkαF is homotopic to

Ωk : (ΩkF )(R) −→ MapGpEn (A)

(ΩkSpf(R),Ωk(ΩkF ))

which is an equivalence by Proposition 2.44 and because ΩkF is a formal moduli problem.Hence L′(αF ) is an equivalence and this finishes the proof.

Corollary 2.49. Let (A, (Eα)α∈T ) be a deformation context satifiying all the conditions ofProposition 2.44 and let n ≥ 0 be an integer. Then the restricted ∞-functor

L|FMP(k)A

: FMP(k)A −→ FMPA

commutes with small limits.

Proof. Let I be a small simplicial set and F : I → FMP(k)A an I-diagram of n-proximate

formal moduli problems. We need to show that for each A ∈ Aart the natural mapL(limIF )(A)→ limIL(F )(A) is an equivalence in S. But through the equivalence

L(F )(A) ' MapGpEn (A)

(ΩkSpf(A),ΩkF )

provided by Theorem 2.48, this map is equivalent to a map

MapGpEn (A)

(ΩSpf(A),Ωk(limIF ))→ limIMapGpEn (A)

(ΩSpf(A),ΩkF ).

Through the equivalence

limIMapGpEn (A)

(ΩSpf(A),ΩkF ) ' MapGpEn (A)

(ΩSpf(A), limIΩkF )

the latter map is induced by the natural map Ωk(limIF ) → limIΩkF in GpEn(A) which

is an equivalence. Hence the map L(limIF )(A) → limIL(F )(A) is an equivalence for eachA ∈ Aart.


3. Deformations of linear ∞-categories

We begin with notations and terminology related to presentable, compactly generatedand linear ∞-categories, appropriate ∞-functors between them, their monoidal structures,as well as the relation with differential graded categories. We follow the terminology of[Lur09] and [Lurc]. We also point out the reference [Rob14] where we can find a quick andefficient overview about this subject.

3.1. Linear ∞-categories. We denote by PrL the ∞-category of not necessarily smallpresentable∞-categories together with∞-functors which preserve small colimits (see [Lur09,Def 5.5.3.1] for a construction of PrL). By [Lur09, Thm 5.5.1.1] every object of PrL is anaccessible reflexive localization of an ∞-category of presheaves P(C0) = Fun(C0,S) for somesmall ∞-category C0 and is in particular cocomplete.

An ∞-category C is said to be compactly generated if it is presentable and if in additionthere exists a small∞-category C0 which admits finite colimits and an equivalence Ind(C0) 'C. In this situation, by [Lur09, Thm 5.4.2.2] the small∞-category C0 which generates C canbe choosen to be the subcategory Cc ⊆ C consisting of compact objects of C. We denote byPrLω the non full subcategory of PrL spanned by compactly generated∞-categories togetherwith ∞-functors which commutes with small colimits and preserve compact objects.

By [Lurc, Prop 4.8.1.14], the ∞-category PrL admits a symmetric monoidal structure inthe sense of [Lurc, Def 2.1.2.13] and we denote the induced symmetric monoidal product by⊗. If C, C′ are presentable ∞-categories of the form C = P(C0) and C′ = P(C′0) with C0, C′0small ∞-categories, then we have

C⊗C′ = P(C0 × C′0)

where × is the cartesian symmetric monoidal structure on Cat∞. The unit of PrL,⊗ is the∞-category S of small spaces. Moreover this symmetric monoidal structure is closed andfor two presentable∞-categories C, C′, the∞-category FunL(C, C′) of colimit preserving∞-functors is presentable. A presentable∞-category is dualizable with respect to the monoidalstructure ⊗ if and only if it is compactly generated and the dual is given by the opposite∞-category. This implies that for presentable ∞-categories C and C′, with C′ compactlygenerated, there exists equivalences of ∞-categories

FunL(C, C′) ' FunL(C⊗(C′)op,S) ' Cop⊗C′.The symmetric monoidal structures PrL,⊗ restricts to a symmetric monoidal structure

PrL,⊗ω on PrLω (see [Lurc, Lem 5.3.2.11]) such that the natural inclusion PrL,⊗ω → PrL,⊗ is asymmetric monoidal∞-functor. In other words, if C and C′ are compactly generated k-linear∞-categories, the tensor product C⊗C′ is again compactly generated and the subcategory(C⊗C′)c is generated under finite colimits by Cc × C′c.

Let k be an E∞-ring. The∞-category Modk of k-module spectra is a compactly generated∞-category and admits a symmetric monoidal structure Mod⊗k given by the smash productover k. It can therefore be seen as an E∞-algebra in the symmetric monoidal ∞-categoryPrL,⊗ω and in PrL,⊗.

We define the ∞-category of k-linear ∞-categories to be

Catk := ModMod⊗k(PrL,⊗)

the ∞-category of presentable ∞-categories which are tensored over Mod⊗k . We call itsobjects k-linear ∞-categories and its morphisms are called k-linear ∞-functors. We definethe ∞-category of compactly generated k-linear ∞-categories to be

Catck := ModMod⊗k(PrL,⊗ω ).

The inclusion PrLω → PrL induces a fully faithfull embedding Catck → Catk whose essentialimage consists of k-linear ∞-categories which are moreover compactly generated. Because


Modk is a stable ∞-category in the sense of [Lurc, Def 1.1.1.9], the underlying ∞-categoryof any k-linear ∞-category is stable.

The symmetric monoidal structure on PrL induces a symmetric monoidal structure onCatk. We denote the corresponding tensor product by ⊗k. Because there exists a symmetricmonoidal inclusion PrLω → PrL, the symmetric monoidal structure on Catk restricts toa symmetric monoidal structure on Catck denoted again by ⊗k. Moreover the symmetricmonoidal structure on Catk is closed. For k-linear ∞-categories C, C′ and C′′ there exists ak-linear ∞-category FunLk (C, C′) of k-linear colimit preserving ∞-functors, such that thereexists an equivalence

MapCatk(C⊗kC′, C′′) ' MapCatk

(C,FunLk (C′, C′′)).For every C, C′ ∈ Catk such that C′ is compactly generated, there exists equivalences

FunLk (C, C′) ' FunLk (C⊗kC′op,Modk) ' Cop⊗kC′.

In the case of C = C′, we denote by End(C) = FunLk (C, C) the k-linear∞-category of k-linearendofunctors of C.

Any k-linear ∞-category C ∈ Catk is enriched over Modk. Indeed if C,D ∈ C are objects,the ∞-functor Modk → S given on objects by M 7→ MapC(M ⊗C,D) commutes with smallcolimits and by presentability of C is therefore representable by an object MapC(C,D) ∈Modk satisfying the universal property

MapC(M ⊗ C,D) ' MapModk(M,MapC(C,D)).

We denote by

h− : C −→ FunLk (Cc,Modk)

the restricted Yoneda embedding, it is an equivalence because C is presentable. For E ∈ Cwe have hE(F ) ' MapC(F,E). We denote by

h− : Cop −→ FunLk ((Cc)op,Modk)

the restricted coYoneda embedding.We now recall the relation of the above with the homotopy theory of differential graded

categories. Let k be a discrete commutative ring. We consider two ∞-categories of k-lineardg-categories (we use the terminology of [Rob14, §6.1.1]):

• Dg(k) : the ∞-category encoding the homotopy theory of presentable k-linear dg-categories up to quasi-equivalence. Morphisms are dg-functors which commute withsmall sums. The dg-nerve provides an ∞-functor

NLdg : Dg(k) −→ PrL.

• Dgc(k) : the ∞-category encoding the homotopy theory of presentable compactlygenerated k-linear dg-categories up to quasi-equivalence. Morphisms are dg-functorswhich commute with small sums and preserve compact objects. The dg-nerve re-stricts to an ∞-functor

Ncdg : Dgc(k) −→ PrLω .

By the work of Cohn [Coh13] the dg-nerve functor factorize through Catk and induces anequivalence. More precisely there exists a commutative diagram of ∞-categories

Dg(k)NLdg// Catk

Dgc(k)Ncdg//

?

OO

Catck?

OO

where the horizontal arrows are equivalences of ∞-categories.


Notation 3.1. Let k be an E∞-ring and C a k-linear ∞-category. If E is a collection ofobjects of C, we denote by E⊥ the full subcategory of C consisting of objects C such thatMapC(E,C) ' 0 for every E ∈ E .

Definition 3.2. Let k be an E∞-ring. Let C be a k-linear ∞-category.

(1) C is smooth if the identity functor idC is a compact object of the k-linear∞-categoryEnd(C).

(2) A collection E of objects of C generates C if E⊥ consists of zero objects of C.(3) A collection E of objects of C is a family of compact generators of C if each object

E ∈ E is compact and if the collection E generates C.(4) C has a compact generator if there exists a family of compact generators E = E

of C consisting of a single object.

Lemma 3.3. Let k be an E∞-ring and C a smooth compactly generated k-linear∞-category.Then C has a compact generator.

Proof. Because (Cop⊗kC)c is generated under finite colimits by (Cc)op⊗kCc, the identity idCcan be written as a finite colimit idC ' colimi∈IFi ⊗ Gi, where I is a simplicial set withfinitely many vertices, and the Fi and Gi are compact objects of C. We identity Fi⊗Gi withthe corresponding representable bimodule hFi⊗Gi : (Cop⊗kC)c → Modk. Set G =

⊕iGi.

Because G is a finite sum of compact objects of C, G is a compact object of C. Suppose nowthat E ∈ C is such that MapC(G,E) ' 0. This implies that MapC(Gi, E) ' 0 for every i.Then we have equivalences in C,

hFi⊗Gi(E) ' (hFi ⊗ hGi)(E) ' hFi ⊗ hGi(E) ' 0.

Taking the colimit over I we get equivalences

0 ' colimi∈IhFi⊗Gi(E) ' idC(E) ' E.

Hence G is a compact generator of C.

3.2. Deformations of objects.

In the rest of this article, k denotes a field of arbitrary characteristic.

We recall from [Lurb, §5.2] the fundamental results about the deformation functor of anobject in a k-linear ∞-category.

Construction 3.4. ([Lurb, Const 5.2.1]). — We consider the∞-category RMod(C) of rightmodules objects in C. It is the ∞-category of pairs (B,EB) with B an E1-algebra over kand EB ∈ RModB(C) a right B-module in C. The projection ∞-functor

q : RMod(C) −→ Algk

is a cocartesian fibration of∞-categories. Let RMod(C)cocart ⊆ RMod(C) be the subcategoryof q-cocartesian morphisms such that q restricts to a left fibration of ∞-categories

RMod(C)cocart −→ Algk.

Morally, the passage from RMod(C) to RMod(C)cocart consists in keeping only the morphisms(B,EB) → (B′, EB′) given by a morphism of E1-algebras B → B′ and an equivalence ofright B-modules EB ⊗B B′ ' EB′ .

Let C be a k-linear ∞-category and E ∈ C an object. The pair (k,E) defines an objectof RMod(C)cocart. We define the ∞-category of deformations of E in C to be

Def[E] := RMod(C)cocart/(k,E).


The induced left fibration Def[E] −→ Algaugk given by (B,EB) 7→ B is classified by an

∞-functor Algaugk −→ Sbig whose restriction to artinian E1-algebras is a pre-formal moduli

problem denoted by

ObjDefE : Algartk −→ S.

It follows from [Lurb, Cor 5.2.3] that whenB is an artinian E1-algebra, the space ObjDefE(B)is essentially small. For an artinian E1-algebra B, the space ObjDefE(B) classifies defor-mations of E over B or in other words it classifies pairs (EB , v) with EB ∈ RModB(C) andv : EB ⊗B k ' E an equivalence in C.

Proposition 3.5. [Lurb, Cor 5.2.5]. — Let C a k-linear ∞-category and E ∈ C an object.Then the ∞-functor ObjDefE defined in Construction 3.4 is a 1-proximate formal moduliproblem.

Notation 3.6. Recall from Notation 2.10, the notion of associated formal moduli prob-lem. In the deformation context formed by E1-algebras (Algaug

k , k) we therefore have an∞-functor by L : PFMPE1(k) → FMPE1(k) left adjoint to the inclusion FMPE1(k) →PFMPE1

(k). If k is a field, C is a k-linear ∞-category and E ∈ C an object, we denoteby ObjDef∧E the E1-formal moduli problem L(ObjDefE). Recall the classification of formalmoduli problems in the deformation context formed by augmented E1-algebras (see Theorem2.32) which asserts the existence of an equivalence

Ψ : Algaugk −→ FMPE1

(k).

Notation 3.7. Let C be a k-linear∞-category and E ∈ C an object. By [Lurc, Thm 4.8.5.11]the k-module EndC(E) := MapC(E,E) of endomorphisms of E in C is naturally an E1-algebra, with multiplication induced by the opposite of the composition of endomorphisms.We can be more precise. Consider the ∞-category (Catk)Modk/ of k-linear ∞-categories Cendowed with a colimit preserving k-linear ∞-functor Modk → C. The data of an objectof (Catk)Modk/ is equivalent to a couple (C, E) where E is the image of the object k byModk → C. By loc.cit. there exists an ∞-functor

Algk → (Catk)Modk/

given on objects by A 7→ (LModA, A) which has a right adjoint

E : (Catk)Modk/ → Algk

given on objects by E(C, E) = EndC(E). Moreover, by construction, the diagram of ∞-categories

Alg(Catk)E⊗ //

Alg(2)k

(Catk)Modk/E // Algk

is commutative. Here the left vertical map is given on objects by A⊗ 7→ (A,1A) where 1Ais the unit, and the right vertical map is the forgetful functor.

Notation 3.8. Let n ≥ 0 be an integer. The forgetful ∞-functor Alg(n),augk → Alg

(n)k

which forgets the augmentation commutes with small colimits and by presentability of the

∞-categories Alg(n),augk and Alg

(n)k , it admits a right adjoint Alg

(n)k → Alg

(n),augk given on

objects by A 7→ k × A, where × is the product in the ∞-category Alg(n)k , and k × A is

endowed with the obvious augmentation given by the projection on k. The product k × Ais equivalent as a k-module to the sum k ⊕A in k-modules. We will denote the augmentedEn-algebra k ×A by k ⊕A.


Theorem 3.9. [Lurb, Thm 5.2.8]. — Let C a k-linear ∞-category and E ∈ C an ob-ject. Then there exists a natural transformation ObjDefE → Ψk⊕EndC(E) which induces an

equivalence ObjDef∧E ' Ψk⊕EndC(E). In other words, there is a natural equivalence

ObjDef∧E(B) ' MapAlgk(D(1)(B),EndC(E)).

Corollary 3.10. Let C a k-linear ∞-category and E ∈ C an object. Then there exists anequivalence of k-module spectra TObjDef∧E

' EndC(E)[1].

Proof. By Theorem 2.32, the shifted tangent spectrum TObjDef∧E[−1] is equivalent to the

augmentation ideal mB of the E1-algebra B associated to ObjDef∧E . But by Theorem 3.9there is an equivalence B ' k ⊕ EndC(E).

Proposition 3.11. [Lurb, Prop 5.2.14]. — Let C a k-linear ∞-category and E ∈ C≥0 aconnective object of C. Then the ∞-functor ObjDefE is a formal moduli problem.

Definition 3.12. [Lurb, Def 3.4.1, Def 3.4.4]. — Let B be an artinian E1-algebra over k.

• A right or left B-module M is said to be artinian if it is perfect when viewed as ak-module.

• The ∞-category of Ind-coherent right B-modules is by definition the ∞-categoryRMod!

R := Ind(RModartB ). Similarly LMod!

B := Ind(LModartB ) is by definition the

∞-category of Ind-coherent left B-modules.• The ∞-category RMod!

B is a k-linear ∞-category (**Ref**). Let C be a k-linear∞-category. The ∞-category of Ind-coherent right B-modules in C is by definitionthe ∞-category RMod!

B(C) := RMod!B ⊗k C. Similarly LMod!

B(C) := LMod!B ⊗k C.

Remark 3.13. The terminology of Definition 3.12 comes from the fact that when theE1-algebra B is artinian, then every coherent B-module (in the sense of [?]) is artinian.Therefore the Ind-completion of the∞-category of coherent right B-module is equivalent tothe Ind-completion of RModart

B .

The following result shows the interaction between Ind-coherent modules and Koszulduality of E1-algebras and is known as Kozsul duality for modules. See Proposition 2.30 forKoszul duality of E1-algebras.

Theorem 3.14. [Lurb, Thm 3.5.1]. — Let B be an artinian E1-algebra over k. Then there

exists a natural equivalence of ∞-categories RMod!B ' LModD(1)(B).

Remark 3.15. [Lurb, Rem 5.2.16]. — Let C be a k-linear ∞-category. It can be shown

that the construction B 7→ RMod!B(C) can be promoted to an ∞-functor from Algk to Catk

(see [Lurb, §3.4]). Let E ∈ C be an object. The description of ObjDef∧E given in Theorem3.9 can be interpreted in terms of Ind-coherent modules by the following. We have a chainof natural equivalences

ObjDef∧E(B) = MapAlgk(D(1)(B),EndC(E))

' LModD(1)(B)(C)' ×C' E

' (RMod!B(C))' ×C' E.

We deduce that the space ObjDef∧E(B) classifies pairs (EB , v) where EB is an Ind-coherentright B-module in C and v is an equivalence EB ⊗B k ' E in C. Moreover the fact that theembedding RModB → RMod!

B induces an equivalence on connective objects ([Lurb, Prop3.4.18] reflects in Proposition 3.11.


3.3. Deformations of categories. We still work over a base field k of arbitrary charac-teristic.

Construction 3.16. ([Lurb, Not 5.3.1, Const 5.3.2]). — By [Lurc, Thm 4.8.5.16] theconstruction

Algk −→ Catk

A 7−→ LModA

can be promoted to a symmetric monoidal ∞-functor. We then pass to associative algebraobjects and obtain an ∞-functor

Alg(2)k ' Alg(Algk) −→ Alg(Catk)

B 7−→ LMod⊗B

which to an E2-algebra B associates the monoidal ∞-category of left B-modules. For anE2-algebra B, we define a right B-linear ∞-category (resp. left B-linear ∞-category) tobe a right LMod⊗B-module in Catk (resp. a left LMod⊗B-module in Catk). If B → B′ isa morphism of E2-algebras, and CB a B-linear ∞-category, we denote by CB ⊗B B′ theextension of scalars CB⊗LModBLModB′ .

On the other hand we consider the cocartesian fibration

RMod(Catk) −→ Alg(Catk)

whose fiber over a monoidal k-linear ∞-category A⊗ is the ∞-category of k-linear ∞-categories which are right tensored over A⊗. We set

RCat(k) := Alg(2)k ×Alg(Catk) RMod(Catk)

The objects of RCat(k) are described informally by pairs (B, CB) where B is an E2-algebraover k and CB is a right B-linear ∞-category. The projection ∞-functor

p : RCat(k) −→ Alg(2)k

is a cocartesian fibration of ∞-categories. Let RCat(k)cocart ⊆ RCat(k) be the subcategoryof p-cocartesian morphisms, such that p restricts to a left fibration of ∞-categories

RCat(k)cocart −→ Alg(2)k .

Morally, the passage from RCat(k) to RCat(k)cocart consists in keeping only the morphisms(B, CB)→ (B′, CB′) given by a morphism of E2-algebras B → B′ and a B-linear equivalenceCB ⊗B′ ' CB′ .

Let C ∈ Catk be a fixed k-linear∞-category. The pair (k, C) defines an object of RCat(k).We define the ∞-category of deformations of C to be

Def[C] := (RCat(k)cocart)/(k,C).

The projection ∞-functor

Def[C] −→ Alg(2),augk

described informally by (B, CB) 7→ B, is a left fibration of ∞-categories and is classified by

an ∞-functor Alg(2),augk −→ Sbig whose restriction to artinian E2-algebras is a pre-formal

moduli problem denoted by

CatDefC : Alg(2),artk −→ Sbig

For an E2-algebra B, the space CatDefC(B) classifies deformations of C over B or in otherwords it classifies pairs (CB , u) where CB is a B-linear ∞-category and u : CB ⊗B k ' C is ak-linear equivalence.


Notation 3.17. Let C be a k-linear ∞-category. We denote by CatDef∧C := L(CatDefC)the E2-formal moduli problem associated to CatDefC where L : PFMPE2

(k) → FMPE2(k)

is left adjoint to the inclusion. By [Lurb, Cor 5.3.8] for every artinian E2-algebra B thespace CatDef∧C (B) is essentially small and we can therefore see CatDef∧C as an ∞-functor

CatDef∧C : Alg(2),artk −→ S

to small spaces. We denote by

θC : CatDefC −→ CatDef∧C

the natural map of pre-formal moduli problems (in a larger universe) given by the unit ofthe adjunction i : FMPE2(k) PFMPE2(k) : L.

Proposition 3.18. [Lurb, Cor 5.3.8]. — Let C be a k-linear ∞-category. Then CatDefC :

Alg(2),artk → Sbig is a 2-proximate formal moduli problem (after enlargement of the universe).

Notation 3.19. Let C be a compactly generated k-linear ∞-category. For each B ∈Alg

(2),augk , we let CatDefcC(B) denote the summand of CatDefC(B) consisting of pairs (CB , u)

where CB is compactly generated. By [Lurb, Variant 5.3.6], for each B the space CatDefcC(B)is essentially small. We therefore obtain an ∞-functor

CatDefcC : Alg(2),augk −→ S.

encoding compactly generated deformations of C.

Remark 3.20. [Lurb, Rem 5.3.9]. — Let C be a compactly generated k-linear ∞-category.By [Lurb, Variant 5.3.4], the ∞-functor CatDefcC is a 2-proximate formal moduli problem.Moreover the composite map

CatDefcC −→ CatDefC −→ CatDef∧C

has 0-truncated fibers. Theorem 2.37 then implies that L(CatDefcC) ' CatDef∧C .

Definition 3.21. A k-linear ∞-category C is said to be tamely compactly generated if C iscompactly generated and if for any compact objects E,E′ of C we have ExtnC(E,E

′) = 0 forn 0.

Theorem 3.22. [Lurb, Prop 5.3.21]. — Let C be a tamely compactly generated k-linear∞-category. Then CatDefcC is a 1-proximate formal moduli problem.

Notation 3.23. Center of a k-linear ∞-category (see [Lurb, Def 5.3.10]) — Let C be ak-linear ∞-category. We set

LCat(k) := Alg(2)k ×Alg(Catk) LMod(Catk).

An object of LCat(k) is a pair (A, CA) with A an E2-algebra over k and CA a left A-linear∞-category. We consider the fiber product LCat(k) ×Catk C of pairs (A, CA) ∈ LCat(k)such that the image of CA in Catk via the restriction of scalars is equivalent to C. Bydefinition a pair (A, CA) exhibits A as the k-linear center of C if (A, CA) is a final object ofLCat(k)×Catk C. By [Lurb, Rem 5.3.11], a k-linear center A of C is characterized by theuniversal property

MapAlg

(2)k

(B,A) ' RModLModA(Catk)' ×Cat'kC,

or in other words the space MapAlg

(2)k

(B,A) classifies right actions of LMod⊗A on C. By

[Lurb, Prop 5.3.12], there exists a k-linear center of C and it is given by the following. The∞-functor

Alg(2)k −→ Alg(Catk)


given on objects by B 7→ LMod⊗B (see Construction 3.16) admits a right adjoint

E⊗ : Alg(Catk) −→ Alg(2)k

given on objects by D⊗ 7→ EndD(1D). Let End(C) = FunLk (C, C) denote the k-linear ∞-category of colimit preserving k-linear ∞-functors from C to itself. It can be promoted to amonoidal ∞-category via the composition of endomorphisms. Let

A = E⊗(End(C)⊗) ' EndEnd(C)(idC)

be the E2-algebra of endomorphisms of the identity of C. By adjunction we have a naturalequivalence

MapAlg

(2)k

(B,A) ' MapAlg(Catk)(LMod⊗B ,End(C)).

which implies that A is a k-linear center of C. We denote by ξ(C) a k-linear center of C. Wedenote by HH∗(C) the underlying k-module of the E2-algebra ξ(C) and call it the Hochschildcomplex of C. The Hochschild cohomology groups of C are defined by HHi(C) = π−iHH∗(C)for every i ∈ Z.

Notation 3.24. Let C be a k-linear ∞-category and consider the E2-algebra ξ(C) given bya k-linear center of C. The commutative diagram of Notation 3.7 implies that the underlyingE1-algebra of ξ(C) is equivalent to the E1-algebra given by E(End(C), idC). We will denotethis E1-algebra by ξ1(C).

Notation 3.25. Let C be a k-linear ∞-category and E ∈ C an object. Consider thepairs (End(C), idC) and (C, E) as objects of the ∞-category (Catk)Modk/. The ∞-functorevE : End(C) → C given by evaluation at E induces a morphism (End(C), idC) → (C, E) in(Catk)Modk/. By applying the ∞-functor E from Notation 3.7 we have an induced map ofE1-algebras

χE : ξ1(C) ' E(End(C), idC) −→ E(C, E) ' EndC(E)

which we call the obstruction map associated to (C, E).

Theorem 3.26. [Lurb, Thm 3.5.1]. — Let C be a k-linear ∞-category. There exists a nat-ural transformation CatDefC → Ψk⊕ξ(C) which induces an equivalence CatDef∧C ' Ψk⊕ξ(C).

In other words, the formal moduli problem CatDef∧C is given by

CatDef∧C (B) ' MapAlg

(2)k

(D(2)(B), ξ(C)).

Notation 3.27. If B ∈ Alg(n),artk is an artinian En-algebra, the Koszul dual D(n)(B) is not

necessarily artinian. However, by universal property of the pro-completion, there exists anessentially unique extension

D(n)f : Pro(Alg

(n),artk ) −→ (Alg

(n),augk )op

of D(n) to pro-artinian En-algebras, which commutes with cofiltered limits. If B = limiBiis a pro-artinian En-algebra, we have D

(n)f (B) ' colimiD

(n)(Bi).

Corollary 3.28. Let C be a k-linear ∞-category. Then the ∞-functor

CatDef∧C : Pro(Alg(2),artk )→ S

(see Notation 2.21) is given by

CatDef∧C (B) ' MapAlg

(2)k

(D(2)f (B), ξ(C)).

Corollary 3.29. Let C be a k-linear ∞-category. Then there exists an equivalence of k-module spectra TCatDef∧C

' ξ(C)[2].


Proof. By Theorem 2.32, the shifted tangent spectrum TCatDef∧C[−2] is equivalent as a k-

module to the augmentation ideal mB of the E2-algebra B associated to CatDef∧C . But byTheorem 3.26 there is an equivalence B ' k⊕ξ(C), so we obtain the desired equivalence.

Following the same lines as in Construction 3.16, we define a formal moduli of deforma-tions of a couple (C, E) with C a k-linear ∞-category and E ∈ C an object.

Construction 3.30. Consider the symmetric monoidal ∞-functor

Algk → (Catk)Modk/

which is given on objects by A 7→ (LModA, A). By applying associative algebras objects weobtain an ∞-functor

Alg(2)k → Alg((Catk)Modk/).

Consider the cocartesian fibration

RMod((Catk)Modk/)→ Alg((Catk)Modk/)

whose fiber over a monoidal ∞-category A⊗ is the ∞-category of pairs (C, E) where C is ank-linear ∞-category right tensored over A⊗ and E ∈ C is an object. We set

RCat(k)∗ = Alg(2)k ×Alg((Catk)Modk/

) RMod((Catk)Modk/).

An object of RCat(k)∗ is given by a triple (A, CA, EA) with A an E2-algebra over k, CA aright A-linear ∞-category and EA ∈ CA an object. The projection ∞-functor

u : RCat(k)∗ → Alg(2)k

is a cocartesian fibration. Let RCat(k)cocart∗ ⊆ RCat(k)∗ be the subcategory consisting of

u-cocartesian morphisms such that we obtain a left fibration RCat(k)cocart∗ → Alg(2)k .

Let C be a k-linear ∞-category and E ∈ C an object. The triple (k, C, E) defines anobject of RCat(k)cocart∗ . The ∞-category of deformations of (C, E) is the ∞-category

Def[(C, E)] := (RCat(k)cocart∗ )/(k,C,E).

The induced left fibration Def[(C, E)]→ Alg(2),augk is classified by an∞-functor Alg

(2),augk →

Sbig whose restriction to artinian E2-algebras is denoted by

Def(C,E) : Alg(2),artk −→ Sbig.

For each E2-algebra A, the space Def(C,E)(A) classifies uples (CA, EA, u, v) where CA isa right A-linear ∞-category, u : CA ⊗A k ' C is an equivalence, EA ∈ CA is an object andv : EA⊗A k ' A is an equivalence in C. The ∞-functor Def(C,E) is an E2-pre-formal moduli

problem. We denote by Def∧(C,E) := L(Def(C,E)) the associated formal moduli problem.

The forgetful∞-functor (Catk)Modk/ → Catk induces an∞-functor RCat(k)∗ → RCat(k)which in turn induces an∞-functor Def[(C, E)]→ Def[C] commuting with the projection to

Alg(2),augk . This ∞-functor induces a natural transformation

Def(C,E) −→ CatDefC .

Informally this map associates to a triple (A, CA, EA) the couple (A, CA).Moreover the ∞-functor C → (Catk)Modk/ given by C 7→ (C, C) induces an ∞-functor

RMod(C)→ RCat(k)∗ which in turn induces an∞-functor Def[E]→ Def[(C, E)] commuting

with the projection to Alg(2),augk . This ∞-functor induces a natural transformation

ObjDefE −→ Def(C,E)

where we consider ObjDefE as an E2-pre-formal moduli problem by restricting along the

natural map Alg(2),artk → Algart

k . Informally, this map is given on objects by (A,EA) 7→(A, CtA, EA) where CtA = C ⊗k A denotes the trivial deformation of C over A.


Proposition 3.31. Let C be a k-linear ∞-category and E ∈ C an object. The sequence ofnatural transformations

ObjDefE −→ Def(C,E) −→ CatDefC

from Construction 3.30 is a fiber sequence in PFMPE2(k). As a consequence we get a

commutative diagram

ObjDefE //

Def(C,E)

// CatDefC

ObjDef∧E // Def∧(C,E)// CatDef∧C

whose horizontal lines are fiber sequences.

Proof. The fiber of the ∞-functor (Catk)Modk/ → Catk at C is equivalent to the mappingspace MapCatk

(Modk, C) ' C'. This implies that the fiber of the map Def(C,E) → CatDefCis given by ObjDefE . The second part follows from the fact that the three ∞-functorsinvolded are n-proximate formal moduli problems for some n by Propositions 3.5, 3.18 and3.36 and from the fact that L : PFMPE2

(k) → FMPE2(k) commutes with small limits of

n-proximate formal moduli problems by Corollary 2.49.

Remark 3.32. Let C be a k-linear ∞-category and E ∈ C an object. The commutative

diagram of Proposition 3.31 gives for each B ∈ Alg(2),artk a commutative diagram of spaces

(1) ΩDef(C,E)(B) //

ΩCatDefC(B) //

ObjDefE(B)

ΩDef∧(C,E)(B) // ΩCatDef∧C (B) // ObjDef∧E(B)

where the lines are fiber sequences. The upper right map in this diagram associates toevery autoequivalence fB : CB → CB of the trivial deformation CB = C ⊗k B the evaluationfB(E) ∈ CB at E, which is a deformation of E over B. By applying the tangent spectrumon the bottom line, we obtain a fiber sequence of k-modules

TΩDef∧(C,E)

−→ TΩCatDef∧C−→ TObjDef∧E

which by Corollaries 3.29 and 3.10 is equivalent to a fibration sequence of k-modules

TΩDef∧(C,E)

−→ HH∗(C)[1] −→ EndC(E)[1].

Hence we obtain a map of k-modules HH∗(C) −→ EndC(E). Recall from Notation 3.25 theobstruction map of E1-algebras

χE : ξ1(C) −→ EndC(E)

where the underlying k-module of ξ1(C) is HH∗(C). By construction, the underlying map ofk-modules of χE coincides with the map HH∗(C)→ EndC(E) we have just defined.

Remark 3.33. Let C be a k-linear∞-category and E ∈ C an object. Consider the diagram(1) of Remark 3.32 for the E2-algebra B = k ⊕ k[m + 1] for an integer m ≥ 0. Because

there is an equivalence k⊕ k[m] ' k×k⊕k[m+1] k in Alg(2),artk , we get commutative diagram

of spaces


CatDefC(k ⊕ k[m]) //

θC

ΩCatDefC(k ⊕ k[m+ 1])

// ObjDefE(k ⊕ k[m+ 1])

CatDef∧C (k ⊕ k[m])∼ // ΩCatDef∧C (k ⊕ k[m+ 1]) // ObjDef∧E(k ⊕ k[m+ 1])

The bottom left map is an equivalence because CatDef∧C is a formal moduli problem. Infact by Corollary 3.29, there is an isomorphism of groups π0CatDef∧C (k⊕k[m]) ' HHm+2(C).Moreover by Corollary 3.10 there is an isomorphism of groups

π0ObjDef∧E(k ⊕ k[m+ 1]) ' π−m−1(End(E)[1]) ' Extm+2C (E,E).

The previous commutative diagram then induces a commutative diagram of sets

π0Def(C,E)(k ⊕ k[m])α //

π0CatDefC(k ⊕ k[m]) //

θC

π0ObjDefE(k ⊕ k[m+ 1])

π0Def∧(C,E)(k ⊕ k[m]) // HHm+2(C)χE // Extm+2

C (E,E)

where only the bottom line is an exact sequence of groups. The right vertical map is injectivebecause by Proposition 3.5 the∞-functor ObjDefE is a 1-proximate formal moduli problem.

Now let C1 be a deformation of C over k ⊕ k[m] and let φ = θC(C1). Suppose E1 ∈ C1deforms E or in other words verifies E1 ⊗B k ' E in C1 ⊗B k ' C. In this case, thepair (C1, E1) is sent to C1 via the map α and because the bottom line is a exact we haveχE(φ) = 0.

Example 3.34. We can now explain a simple example in which CatDefC fails to be aformal moduli problem. It is taken from [KL09, Ex 3.14]. Let k be a field and A = k[u, u−1]the free graded commutative k-algebra generated by a degree 2 variable u and its inverseover k. We regard A as an E∞-algebra and let C = ModA be the k-linear ∞-category ofA-module spectra. A computation gives HH2(C) ' k generated by the multiplication byu whose corresponding cocycle is denoted by φu. Then for every A-module M , we haveχM (φu) is non zero in Ext2

A(M,M) because multiplication by u on M is an equivalence.It follows from the observation made in Remark 3.33 that φu is not in the image of themap θC : π0CatDefC(k[t]/t2) → HH2(C), otherwise there would exist an M ∈ C such thatχM (φu) = 0.

Remark 3.35. Let A = k[u, u−1] be the commutative graded algebra of Example 3.34.The k-linear ∞-category C = ModA is not tamely compactly generated and by loc.cit. wesee that CatDefcC is not a formal moduli problem by default of surjectivity of the mapπ0CatDefcC(k[t]/t2)→ HH2(C).

Proposition 3.36. Let C be a k-linear ∞-category and E ∈ C an object. Then Def(C,E)

is a 2-proximate formal moduli problem. Moreover TDef∧(C,E)

[−2] is equivalent to the fiber of

the obstruction map χE : HH∗(C)→ EndC(E).

Proof. It suffices to prove that the map Def(C,E) → Def∧(C,E) is 0-truncated. But it followsfrom the fact that this map fits in the commutative diagram of Proposition 3.31 where themap ObjDefE → ObjDef∧E is (−1)-truncated by Proposition 3.5 and the map CatDefC →CatDef∧C is 0-truncated by Proposition 3.18. The formula for the tangent is proven inRemark 3.32.


Notation 3.37. Let C be a compactly generated k-linear ∞-category and E ∈ C an object.

For each B ∈ Alg(2),augk we denote by DefcC,E(B) the summand of Def(C,E)(B) consisting of

uplets (CB , u, EB , v) with CB compactly generated. We obtain an ∞-functor

Defc(C,E) : Alg(2),artk → Sbig.

By the same argument as for Proposition 3.36 the map Defc(C,E) → Def∧(C,E) is 0-truncated

and we have an equivalence L(Defc(C,E)) ' Def∧(C,E). By the same argument used in Propo-sition 3.31 we get a commutative diagram

ObjDefE //

Defc(C,E)

// CatDefcC

ObjDef∧E // Def∧(C,E)// CatDef∧C

Corollary 3.38. Let C be a tamely compactly generated k-linear ∞-category and E ∈ C anobject. Then DefcC,E is a 1-proximate formal moduli problem.

Proof. It follows from the same argument as in the proof of Proposition 3.36 using the factthat, because C is tamely compactly generated, by Theorem 3.22 we have that CatDefcC is1-proximate.

Remark 3.39. Let C be a tamely compactly generated k-linear ∞-category and E ∈ C anobject. Using the same arguments as in Remark 3.33 we have for each integer m ≥ 0 acommutative diagram of sets

π0Defc(C,E)(k ⊕ k[m]) //

π0CatDefcC(k ⊕ k[m]) //

θC

π0ObjDefE(k ⊕ k[m+ 1])

π0Def∧(C,E)(k ⊕ k[m]) // HHm+2(C)χE // Hm+2(End(E))

where the bottom line is an exact sequence of groups. This time all vertical maps areinjective because when C is tamely compactly generated, then Defc(C,E) and CatDefcC are1-proximate and moreover ObjDefE is 1-proximate. This implies that the upper line isan exact sequence of pointed sets. We deduce from this that given a compactly generateddeformation C1 of C over k ⊕ k[m], there exists an object E1 ∈ C1 deforming E if and onlyif χE(θC(C1)) = 0.

3.4. Deformations as Ind-coherent loop actions.

Construction 3.40. Let C be a compactly generated k-linear ∞-category. We denote byEnd(C) = FunLk (C, C) the k-linear∞-category of k-linear endofunctors of C. Let RMod(End(C))be the ∞-category of pairs (A,M) where A is an E1-algebra over k and M is a right A-module in the k-linear∞-category End(C). In other words M is an object in the∞-category

FunLA(C ⊗k A, C ⊗k A) ' End(C)⊗ModkLModA

of colimit preservingA-linear∞-functors. Consider the cocartesian fibration RMod(End(C))→Algk. Consider the full subcategory RMod(End(C))inv of RMod(End(C)) consisting of pairs(A,M) where M is an autoequivalence of C⊗kA. Because autoequivalences are stable by ex-tension of scalars, the cocartesian fibration RMod(End(C))→ Algk restricts to a cocartesianfibration s : RMod(End(C))inv → Algk. Denote by RMod(End(C))inv,cocart the subcategoryof s-cocartesian morphisms so that s restricts to a left fibration

RMod(End(C))inv,cocart −→ Algk.


Consider the pair (k, idC) as an object of RMod(End(C))inv,cocart. The∞-category of defor-mations of idC is the ∞-category

Def[idC ] := RMod(End(C))inv,cocart/(k,idC) .

The induced left fibration Def[idC ] → Algaugk is classified by an ∞-functor Algaug

k −→ Sbigwhose restriction to artinian E1-algebra is denoted by

AutC : Algartk −→ S

For each A ∈ Algaugk , the space AutC(A) classifies pairs (M,u) with M : C ⊗k A → C ⊗k A

a colimit preserving A-linear autoequivalence and u : M⊗Ak ' idC is an equivalence. Bydefinition we have AutC = ObjDefidC where idC ∈ End(C) is the identity functor of C. Henceby [Lurb, Cor 5.2.3] for each artinian E1-algebra A, the space AutC(A) is essentially small.

Remark 3.41. Let A be an ∞-category and A ∈ A an object. Then there exists a naturalmap of E1-groups in spaces AutA(A) → ΩAA' which is an equivalence. We deduce fromthis that for C a compactly generated k-linear ∞-category, there exists a natural map ofE1-groups in pre-formal moduli problems

AutC → ΩCatDefC

which is an equivalence, where we restrict AutC to artinian E2-algebras via the natural map

Alg(2),artk → Algart

k . Moreover if E ∈ C is an object, the map

AutC ' ΩCatDefC → ObjDefE

defined in Remark 3.32 coincide with the map induced by the ∞-functor evE : Aut(C) →C evalutation at E. Because equivalences of ∞-categories preserve compact objects, theinclusion CatDefcC → CatDefC induces an equivalence ΩCatDefcC ' ΩCatDefC .

Remark 3.42. By Proposition 3.18 and Remark 3.41, AutC is a 1-proximate formal moduliproblem for any compactly generated ∞-category C.

Remark 3.43. Let C be a compactly generated k-linear ∞-category and E ∈ C an object.The∞-functor evE : End(C)→ C given on objects by evaluation at E induces an∞-functorRMod(End(C))→ RMod(C) which in turn induces a natural transformation

AutC → ObjDefE .

For an A ∈ Algartk , the map AutC(A) → ObjDefE(A) associates to each autoequivalence

fA : C ⊗k A → C ⊗k A the object fA(EtA), where EtA is the trivial deformation of Eover A. Hence we see that AutC → ObjDefE coincides with the natural transformationΩCatDefC → ObjDefE defined in Remark 3.32 through the equivalence ΩCatDefC ' AutC ,though we will not need this fact.

Lemma 3.44. Let C be a compactly generated k-linear ∞-category. Let A ∈ Alg(2),artk be

an artinian E2-algebra and let MA : C ⊗k A→ C ⊗k A be an A-linear ∞-functor. Then MA

is an equivalence if and only if M = MA ⊗A k : C → C is an equivalence.

Proof. The only if direction is immediate. Conversely suppose that M is an equivalence.Because A is artinian there exists a finite sequence

A = A0 → A1 → . . .→ An = k

and pullback diagrams

Ai //

Ai+1

k // k ⊕ k[mi]


in Alg(2),artk . In particular we have fiber sequences of A-modules

Ai → Ai+1 → k[mi].

We have induced fiber sequences of A-modules in End(C)

MA ⊗A Ai →MA ⊗A Ai+1 →M [mi].

Hence we deduce by descending induction on i that MA is an equivalence.

Remark 3.45. Let C be a compactly generated k-linear∞-category. Let EndC : Algartk → S

denote the pre-formal moduli problem obtained by following the same construction as forAutC but omitting taking invertible endofunctors. For each A ∈ Algart

k , the space EndC(A)classifies pairs (MA, v) with MA : C ⊗k A → C ⊗k A an A-linear endofunctor and v :MA ⊗A k ' idC an equivalence in End(C). We have EndC = ObjDefidC with the notationof §3.2. The inclusion Aut(C) ⊆ End(C) induces a natural transformation AutC → EndC .Morever this natural transformation is an equivalence by Lemma 3.44. We have thereforean equivalence of pre-formal moduli problems AutC ' ObjDefidC .

Construction 3.46. Let C be a compactly generated k-linear ∞-category. Consider the∞-category RMod!(End(C)) of pairs (A,M) where A is an E1-algebra over k and M is anInd-coherent right A-module in End(C) in the sense of Definition 3.12. In other words M is

a filtered colimit of artinian objects in FunLA(C ⊗k A, C ⊗k A). There is an evident cocarte-

sian fibration RMod!(End(C))→ Algk. Consider the full subcategory RMod!(End(C))inv of

RMod!(End(C)) consisting of autoequivalences. Because Ind-coherent autoequivalences are

stable by extension of scalars, the cocartesian fibration RMod!(End(C))→ Algk restricts to

a cocartesian fibration s! : RMod!(End(C))inv → Algk. Denote by RMod!(End(C))inv,cocart

the subcategory of s!-cocartesian morphisms so that s! restricts to a left fibration

RMod!(End(C))inv,cocart −→ Algk.

Consider the pair (k, idC) as an object of RMod!(End(C))inv,cocart and define the∞-category

of deformations of idC in RMod!(End(C))inv,cocart to be the ∞-category

Def[idC ]! := RMod!(End(C))inv,cocart

/(k,idC) .

The induced left fibration Def[idC ]! → Algaug

k is classified by an ∞-functor Algaugk → Sbig

whose restriction to artinian E1-algebras is denoted by

Aut!C : Algart

k −→ Sbig.

For each A ∈ Algartk , the space Aut!

C(A) classifies pairs (M,u) with M : C ⊗k A → C ⊗k Aan Ind-coherent colimit preserving A-linear autoequivalence and u : M⊗Ak ' idC is anequivalence. For each A ∈ Algaug

k , there is a natural ∞-functor RModA → RMod!A which

induces a natural transformation AutC → Aut!C .

Remark 3.47. Let C be a compactly generated k-linear∞-category. Let End!C : Algart

k → Sdenote the pre-formal moduli problem obtained by following the same construction as forAut!

C but omitting taking invertible Ind-coherent endofunctors. For each A ∈ Algartk , the

space End!C(A) classifies pairs (MA, v) with MA : C ⊗kA→ C⊗kA an A-linear Ind-coherent

endofunctor and v : MA⊗Ak ' idC an equivalence in End(C). A similar proof as the proof of

Lemma 3.44 shows that the inclusion Aut!C → End!

C is an equivalence. Moreover in Remark3.15 we saw that the associated formal moduli problem L(AutC) ' ObjDef∧C is given onobjects by

ObjDef∧idC (A) ' (RMod!A(End(C))cocart)/(k,idC).


More precisely the sequence of equivalences in Remark 3.15 gives a natural transformationObjDef∧idC → End!

C which is an equivalence. Hence we have a diagram

ObjDefidC// ObjDef∧idC

∼ // End!C

AutC

o

OO

// Aut!C

o

OO

where the decorated arrows are equivalences.

Proposition 3.48. Let C be a compactly generated k-linear ∞-category. The natural trans-formation AutC → Aut!

C induces an equivalence L(AutC) ' Aut!C. In particular Aut!

C is aformal moduli problem which takes values in essentially small spaces.

Proof. See Remark 3.47.

Corollary 3.49. Let C be a tamely compactly generated k-linear ∞-category. Then thenatural map AutC → Aut!

C is an equivalence.

Proof. Consider the sequence ΩCatDefcC → ΩCatDefC → ΩCatDef∧C . By Remark 3.41, themap ΩCatDefcC → ΩCatDefC is an equivalence. By Theorem 3.22, under the assumptionthat C is tamely compactly generated, the map ΩCatDefcC → ΩCatDef∧C is an equivalence.This implies that the map ΩCatDefC → ΩCatDef∧C is an equivalence as well. We have acommutative diagram

ΩCatDef∧C

ΩCatDefC

∼77

//

o

L(ΩCatDefC)

OO

o

AutC //

''

L(AutC)

Aut!C

By Corollary 2.49, the map L(ΩCatDefC) → ΩCatDef∧C is an equivalence. By corollary

3.48, the map L(AutC)→ Aut!C is an equivalence. Hence the map Aut!

C → ΩCatDef∧C is anequivalence.

Notation 3.50. We denote by GpE1the ∞-category Gp(FMPE2(k)) of E1-group objects

in the ∞-category FMPE2(k) of E2-formal moduli problems and by GpE2

the ∞-categoryof E2-groups objects in FMPE2

(k) (see Notation 2.42 for group objects).

Corollary 3.51. Let C be a compactly generated k-linear ∞-category. Then the formalmoduli problem CatDef∧C is given by

CatDef∧C (B) ' MapGpE1

(ΩSpf(B),Aut!C).

Moreover the above formula also holds as an ∞-functor CatDef∧C : Pro(Alg(2),artk ) → S on

pro-artinian E2-algebras.

Proof. By Theorem 2.48, the formal moduli problem CatDef∧C is given by


(Ω2Spf(B),Ω2CatDefC)


where GpE2is the ∞-category of formal E2-groups. By Remark 3.41 we obtain a natural

equivalence


(Ω2Spf(B),ΩAutC).

By Proposition 3.48 and Corollary 2.49, the map ΩAutC → ΩAut!C is an equivalence and we

obtain a natural equivalence


(Ω2Spf(B),ΩAut!C).

Proposition 2.44 then gives a natural equivalence


(ΩSpf(B),Aut!C).

The formula on pro-artinian algebras follows from Notations 2.20 and 2.21 and from the factthat filtered colimits commute with finite limits.

Corollary 3.52. Let C be a tamely compactly generated k-linear ∞-category. Then theformal moduli problem CatDef∧C is given by


(ΩSpf(B),AutC).

Moreover the formula holds on pro-artinian E2-algebras.

Proof. Combine Corollary 3.51 and Corollary 3.49.

Following the same lines as Construction 3.40, but replacing the ∞-category Catk by(Catk)Modk/, we give a construction of ΩDef(C,E) in terms of cocartesian fibrations whichwill be useful below.

Construction 3.53. Let C be a compactly generated k-linear ∞-category and E ∈ C anobject. We denote by End(C, E) = Fun(Catk)Modk/

((C, E), (C, E)) the k-linear ∞-category

of endofunctors of C fixing E up to equivalence. Let RMod(End(C, E)) be the ∞-categoryof pairs (A,M) where A is an E1-algebra over k and M is a right A-module in the k-linear ∞-category End(C, E). In other words M is a colimit preserving A-linear ∞-functorsC ⊗k A→ C ⊗k A fixing E up to equivalence.

Consider the cocartesian fibration RMod(End(C, E)) → Algk. Consider the full sub-category RMod(End(C, E))inv of RMod(End(C)) consisting of pairs (A,M) where M is anautoequivalence of C ⊗k A. Because autoequivalences are stable by extension of scalars,the cocartesian fibration RMod(End(C)) → Algk restricts to a cocartesian fibration s :RMod(End(C, E))inv → Algk. Denote by RMod(End(C, E))inv,cocart the subcategory of s-cocartesian morphisms so that s restricts to a left fibration

RMod(End(C, E))inv,cocart −→ Algk.

Consider the pair (k, idC) as an object of RMod(End(C, E))inv,cocart. The ∞-category ofdeformations of idC in End(C, E) is the ∞-category

Def[idC ]E := RMod(End(C, E))inv,cocart/(k,idC) .

The induced left fibration Def[idC ]E → Algaugk is classified by an ∞-functor Algaug

k → Sbigwhose restriction to artinian E1-algebras is denoted by

Aut(C,E) : Algartk −→ S

For each A ∈ Algartk , the space Aut(C,E)(A) classifies pairs (M,u) with M : C⊗kA→ C⊗kA

a colimit preserving A-linear autoequivalence fixing E and u : M⊗Ak ' idC is an equivalencefixing E up to equivalence.


Remark 3.54. Let C be a compactly generated k-linear ∞-category and E ∈ C an object.For the same reasons as in Remark 3.41, there exists a natural map

Aut(C,E) → ΩDefc(C,E) ' ΩDef(C,E)

in Gp(PFMPE2(k)) which is an equivalence.

3.5. Deformations of associative algebras.

Construction 3.55. Consider the∞-category Alg∗(Modk) of pairs (A1, A2) where A2 is anE2-algebra over k andA1 is an E1-algebra over A2. The projection∞-functor Alg∗(Modk)→Alg

(2)k is a cocartesian fibration. Let Alg∗(Modk)cocart ⊆ Alg∗(Modk) be the subcategory

consisting of cocartesian morphisms so that we have a left fibration Alg∗(Modk)cocart →Alg

(2)k . Let B ∈ Algk be an E1-algebra over k. The ∞-category of deformations of B is

Def[B] = (Alg∗(Modk)cocart)/(B,k)

where we see the pair (B, k) as an object of Alg∗(Modk)cocart. The induced left fibration

Def[B] → Alg(2),augk is classfied by an ∞-functor Alg

(2),augk → Sbig whose restriction to

artinian E2-algebras is denoted by

AlgDefB : Alg(2),artk −→ S.

For each A ∈ Alg(2),artk , the space AlgDefB(A) classifies pairs (BA, u) with BA an E1-algebra

over A and u an equivalence BA ⊗A k ' B of E1-algebras over k.

Proposition 3.56. Let n ∈ Z be an integer and B ∈ Algk an n-connective E1-algebra (i.e.the underlying k-module of B is n-connective). Then the ∞-functor AlgDefB is a formalmoduli problem.

Proof. By a variant of [Lura, Prop 7.6], for every pullback diagram

A //

A0

A1// A01

in Alg(2),artk such that the maps π0A0 → π0A01 ← π0A1 are surjective, the induced ∞-

functor

RMod≥nA −→ RMod≥nA0×

RMod≥nA01

RMod≥nA1

is an equivalence, where RMod≥nA is the ∞-category of n-connective right A-modules. Be-cause the ∞-functor Alg : Alg(PrL)→ PrL commutes with small limits, passing to algebraobjects we obtain that for every pullback diagram as above, the induced map

Alg≥nA −→ Alg≥nA0×

Alg≥nA01

Alg≥nA1

is an equivalence, where Alg≥nA is the∞-category of n-connective A-algebras. Remark that ifM is an n-connective k-module and if MA is a deformation of M over an artinian E1-algebraA, the A-module MA is n-connective by the proof of [Lurb, Prop 5.2.14]. This implies thatthe induced map

AlgDefB(A) −→ AlgDefB(A0)×AlgDefB(A01) AlgDefB(A1)

is an equivalence and therefore that AlgDefB is a formal moduli problem.

Notation 3.57. Let B ∈ Algk be an E1-algebra over k. We denote by AutalgB := ΩAlgDefB

the ∞-functor encoding infinitesimal autoequivalences of B, or in other words deformationsof the identity idB .


Corollary 3.58. Let n ∈ Z be an integer and B ∈ Algk an n-connective E1-algebra. Thenthe formal moduli problem AlgDefB is given by

AlgDefB(A) ' MapGpE2

(ΩSpf(A),AutalgB ).

Moreover the formula holds for A a pro-artinian E2-algebra.

Notation 3.59. Let A ∈ Alg(2)k and BA ∈ Alg(RModB) be a E1-algebra over A. Passing

to left modules, the action map BA ⊗k A → BA gives an ∞-functor LModBA⊗LModA →LModBA . Using this we can regard LModBA as a right A-linear∞-category. This construc-tion defines an ∞-functor Alg(RModA)→ RModLModA(PrL), and an ∞-functor

Alg∗(Modk) −→ RCat(k)

commuting with the projections to Alg(2)k and given on objects by (A,BA) 7→ (A,LModBA).

Let n ∈ Z and B an n-connective E1-algebra over k. The latter ∞-functor defines a naturaltransformation

mB : AlgDefB −→ CatDefcLModB .

Under this assumption onB, the k-linear∞-category LModB is tamely compactly generated.Passing to the associated formal moduli problems, we obtain a commutative diagram

AlgDefBθalgB //

mB

AlgDef∧B

m∧B

CatDefcLModB

θC // CatDef∧LModB

in PFMPE2(k), where θalg

B is an equivalence by Proposition 3.56. Moreover this commutativediagram extends naturally on pro-artinian E2-algebras. We can be more precise with respect

to the descriptions in terms of formal group actions. For each A ∈ Pro(Alg(2),artk ) we have

a commutative diagram

AlgDefB(A)θalgB //

mB

σB

**

AlgDef∧B(A)

m∧B

σ∧B // MapGpE2

(ΩSpf(A),AutalgB )

ν

CatDefcLModB (A)θC //

ϕC

33

CatDef∧LModB (A)ϕ∧C // Map

GpE2

(ΩSpf(A),AutLModB )

where σB and ϕC are the map induced by the loop space ∞-functor

FMP(1)E2

(k)→ FMPE2(k)

from 1-proximate formal moduli problems to formal moduli problems, σ∧B and ϕ∧C are thenaturally induced maps providing the equivalences of Corollary 3.58 and 3.52 and ν is the

map induced by the natural map AutalgB → AutC itself induced by the ∞-functor Algk →

Catk given on objects by C 7→ LModC . The left square in this diagram is commutative bythe above. By construction of mB and ν, the outer figure in this diagram is commutative,and therefore by definition of σ∧B and ϕ∧C , the right square is commutative as well.

Remark 3.60. Let C be a k-linear ∞-category and E ∈ C an object. The ∞-functorE : (Catk)Modk/ → Algk from Notation 3.7 induces an ∞-functor

Def[(C, E)]→ Def[EndC(E)]


given on objects by (CA, EA, u, v) 7→ (EndCA(EA), w) where w : EndCA ⊗A k ' EndC(E)is the equivalence induced by u : CA ⊗A k ' C and v : EA ⊗A k ' E. Here Def[(C, E)] is∞-category of deformations of the pair (C, E) from Construction 3.30. This latter∞-functorinduces in turn a natural transformation

Def(C,E) −→ AlgDefEndC(E).

Remark 3.61. Let C be a compactly generated k-linear ∞-category and E ∈ C an object.We have a diagram

Aut(C,E)//

AutC // ObjDefE

AutalgEndC(E)

in PFMPE2(k), where the left horizontal map is induced by the forgetful ∞-functor

(Catk)Modk/ → Catk,

the map AutC → ObjDefE is induced by the ∞-functor evE : End(C) → C evaluation atE (see Remark 3.43). By construction, the horizontal line is equivalent to the fibrationsequence from Remark 3.32. The vertical map is induced by the natural transformationDef(C,E) → AlgDefEndC(E) from Remark 3.60. This diagram will be useful to study the

compact generation of formal deformations.

3.6. Formal deformations. The letter k still denote a base field of arbitrary characteristic.

Notation 3.62. Recall from Notations 2.20 and 2.21 the conventions for extending formalmoduli problems to pro-artinian algebras. We will now consider the case of the discretecommutative pro-artinian algebra k[[t]] = limik[t]/ti of formal power series considered as a

pro-artinian E2-algebra over k. By definition, if F : Alg(2),artk → S is a pre-formal moduli

problem, we have F (k[[t]]) = limiF (k[t]/ti).If C be a compactly generated k-linear ∞-category and F = CatDefcC , a compacly gener-

ated formal deformation of C is by definition a representative of an element in π0CatDefcC(k[[t]]).The functor π0 does not commute with limits. In general there is only a map

π0CatDefcC(k[[t]])→ limiπ0CatDefcC(k[t]/ti)

If Ct is a compacly generated formal deformation of C, we will write Cii≥0 for the imageof the connected component of Ct by the above map.

Theorem 3.63. Let C be a tamely compactly generated k-linear ∞-category which has acompact generator. Then the map of spaces CatDefcC(k[[t]])→ CatDef∧C (k[[t]]) is an equiva-lence. Moreover for every compactly generated formal deformation Cii≥0 of C, each Ci istamely compactly generated and has a compact generator.

Remark 3.64. In the situation of Theorem 3.63, the fact that each Ci is tamely compactlygenerated for every compactly generated formal deformation Cii≥0 of C is already a con-sequence of [Lurb, Lem 5.3.31] and does not need the assumption that C has a compactgenerator.

Remark 3.65. By Theorem 3.22 we know that when C is tamely compactly generated,

CatDefcC is 1-proximate. This implies that for each artinian E2-algebra B ∈ Alg(2),artk , the

map

CatDefcC(B)→ CatDef∧C (B)

is (−1)-truncated or in other words induces an isomorphism on πk for k > 0 (based at thetrivial deformation) and an injection on π0.


Proposition 3.66. Let C be a tamely compactly generated k-linear ∞-category. Then foreach pro-artinian E2-algebra B = limiBi, the map CatDefcC(B) → CatDef∧C (B) is (−1)-truncated.

Proof. The proof of [Hir15, Thm 2.2] proves that if

. . . // Xn//

Xn−1//

. . . // X0

. . . // Yn // Yn−1// . . . // Y0

is a morphism of towers of fibrations in the category of topological spaces such that each mapXn → Yn is (−1)-truncated, then the induced map limnXn → limnYn is (−1)-truncated.Our statement follows by taking Xn = CatDefC(Bn), Yn = CatDef∧C (Bn) and the mapsXn → Yn are induced by θC in the ∞-category S of spaces, and from Remark 3.65.

Therefore to prove the first part of Theorem 3.63 it suffices to prove that when C istamely compactly generated and has a compact generator, the map π0CatDefcC(k[[t]]) →π0CatDef∧C (k[[t]]) is surjective. We will deduce this fact below from a more precise assertion.For this we need some preliminaries.

Remark 3.67. Let k[β] denote the free graded commutative k-algebra on one generator βof cohomological degree 2. We regard k[β] as an augmented E2-algebra over k. By [Pre11,

Prop 3.1.4] there exists an equivalence of augmented E2-algebras D(2)f (k[[t]]) ' k[β].

Remark 3.68. Let C be a k-linear ∞-category. By applying Corollary 3.28 and Remark3.67 we obtain equivalences

CatDef∧C (k[[t]]) ' MapAlg

(2)k

(D(2)f (k[[t]]), ξ(C)) ' Map

Alg(2)k

(k[β], ξ(C)).

In other words, CatDef∧C (k[[t]]) is the space of left k[β]-linear structures on C.

Remark 3.69. Let C be a tamely compactly generated k-linear ∞-category. By applyingCorollary 3.52 for the pro-artinian algebra B = k[[t]], we get an equivalence

CatDef∧C (k[[t]]) ' MapGpE1

(ΩSpf(k[[t]]),AutC).

In other words, CatDef∧C (k[[t]]) is the space of actions of the formal group G = ΩSpf(k[[t]])on C.

The following lemma will be useful below.

Lemma 3.70. Let n ≥ 1 be an integer and denote by t : FMPEn(k) → Alg(n),augk the

equivalence given by Theorem 2.32. Let A be a pro-artinian En+1-algebra over k. Then

there exists a natural equivalence tΩSpf(A) ' D(n+1)f (A) of augmented En-algebras over k,

where we regard D(n+1)f (A) as an En-algebra via the natural map Alg

(n+1),augk → Alg

(n),augk .

Proof. Because t, Ω and the Koszul duality commute with filtrant colimits, it suffices toshow the statement for A artinian. In this case, we have an equivalence of En-formal moduli

problems ΩSpf(A) ' Spf(k⊗Ak) = Spf(Bar(A)) where Bar : Alg(n+1),augk → Alg

(n),augk is the

Bar construction in the sense of [Lurb, §4.3] and is given on objects by the formula Bar(A) =k ⊗A k. By Corollary 2.33 we obtain a natural equivalence tΩSpf(A) ' D(n)(Bar(A)). But

the proof of [Lurb, Prop. 4.4.21] shows that D(n) Bar ' D(n+1) for every n ≥ 1, so weobtain the desired equivalence.

We need now to set up some notations concerning formal group actions on categories andtheir fixed objects.


Notation 3.71. Let G be a group object in FMPE1(k) and let C be a compactly generatedk-linear ∞-category. The space MapGp(PFMPE1

(k))(G,AutC) is the space of formal actions

of G on C. If E ∈ C is an object, we say that a formal action ρ : G → AutC fixes E up toequivalence if there exists a lift

Gρ

yy

ρ

Aut(C,E)// AutC

of ρ through the natural map Aut(C,E) → AutC in the ∞-category Gp(PFMPE1(k)).

Remark 3.72. Let G be a group object in FMPE1(k) and C a tamely compactly generated

k-linear ∞-category with an object E ∈ C. By Theorem 3.22 and Corollary 3.38, AutC andAut(C,E) are formal moduli problems. Recall the fibration sequence

Aut(C,E) −→ AutC −→ ObjDefE

from Remark 3.61. Let ρ : G → AutC be a formal action of G on C. We then form thepullback diagram

GE //

G

ρ

AutEC //

AutC

∗ // ObjDefE

in PFMPE1(k). The formal group GE is the stabilizer group of E with respect to ρ, it is aformal moduli problem the ∞-category of formal moduli problems FMPE1(k) is complete.The formal action ρ fixes E up to equivalence in the sense of Notation 3.71 if and only ifthe map GE → G is an equivalence in Gp(FMPE1

(k)). Because the forgetful ∞-functorGp(FMPE1

(k)) → FMPE1(k) is conservative, ρ fixes E if and only if the map GE → G is

an equivalence in FMPE1(k), which is equivalent to the condition that the map of spectra

TGE → TG is an equivalence.By Corollary 2.49 and because ObjDefE is 1-proximate by Proposition 3.5, the fiber

sequence

GE −→ G −→ ObjDefE

in PFMPE1(k) induces a fiber sequence

GE −→ G −→ ObjDef∧E

in Gp(FMPE1(k)). Because the tangent spectrum functor T : FMPE1

(k)→ Modk commuteswith finite limits it induces a fiber sequence

TGE [−1] −→ TG[−1] −→ TObjDef∧E[−1] ' EndC(E).

Hence we have proved the following.

Lemma 3.73. Let G be a group object in FMPE1(k) and C a tamely compactly generatedk-linear ∞-category with an object E ∈ C. Then a formal action ρ : G → AutC fixes E inthe sense of Notation 3.71 if and only if the induced map of k-modules TG[−1]→ EndC(E)is null homotopic.


Remark 3.74. Consider the diagram of ∞-categories

Alg(2)k

GpE1

U

Alg(2),augk

k⊕−

OO

Ψ //

m&&

FMPE2(k)

too

Ω

88

//UΩ //

T [−2]

FMPE2(k)

T [−1]xx

Modk

where U is the ∞-functor which forget the group structure, Ω is the equivalence given byProposition 2.44, the adjunction (t,Ψ) is given by Theorem 2.32, m is the ∞-functor givenon objects by the augmentation ideal and T is the tangent complex functor. By Theorem2.32, the left triangle is commutative. Moreover because the tangent complex commuteswith finite limits, the right triangle is commutative.

Let G be a group object in FMPE2(k) and C a tamely compactly generated k-linear ∞-

category with an object E ∈ C. Suppose G is of the form G = ΩSpf(A) for some pro-artinian

E2-algebra A ∈ Alg(2),artk . The previous commutative diagram of ∞-categories induces a

commutative diagram of spaces

MapAlg

(2)k

(D(2)f (A), ξ(C))

o

MapAlg

(2),augk

(D(2)f (A), k ⊕ ξ(C))

o

MapGpE1

(G,AutC)

U

MapFMPE2(k)(Spf(A),CatDef∧C )

UΩ //

Ω∼

33

T [−2]

MapFMPE2(k)(ΩSpf(A),AutC)

T [−1]

MapModk(m

D(2)f (A)

,HH∗(C)) ∼ //

r

MapModk(TG[−1], TAutC [−1])

s

MapModk(m

D(2)f (A)

,EndC(E))∼ // MapModk

(TG[−1], TObjDef∧E)[−1]

where the map r is induced by composition on the right with the obstruction map χE :HH∗(C)→ EndC(E), the map s is induced by the natural map AutC → ObjDef∧E and wherethe last two bottom lines are equivalences.

Let now ρ : G → AutC be a formal action of G on C and let S : D(2)f (A) → ξ(C)

be the corresponding left D(2)f (A)-linear structure on C. We then see from the previous

commutative diagram and from Lemma 3.73 that ρ fixes E if and only if the map of k-module m

D(2)f (A)

→ EndC(E) induced by S is null homotopic.

Proposition 3.75. Let C be tamely compactly generated k-linear ∞-category which hasa compact generator, and let α ∈ π0CatDef∧C (k[[t]]). Then there exists an integer n, ann-connective E1-algebra B over k, and an n-connective formal deformation Bt = Bii≥0

of B such that there exists equivalences C ' LModB and moreover we have the equalityθC(mB(Bt)) = α in π0CatDef∧C (k[[t]]). In particular the map

π0θC : π0CatDefcC(k[[t]])→ π0CatDef∧C (k[[t]])


is surjective.

Proof. Let α ∈ π0CatDef∧C (k[[t]]). Let ρ : G → AutC be the corresponding formal actionof G = ΩSpf(k[[t]]) on C through the equivalence of Remark 3.69. Let S : k[β] → ξ(C) bethe corresponding left k[β]-linear structure on C through the equivalence of Remark 3.68.The map S gives a Hochschild 2-cocycle φ = π2S(β) ∈ HH2(C). For each object M ∈ C,we denote by φM = π2χM (φ) ∈ Ext2

C(M,M) where χM : HH∗(C) → EndC(M) is theobstruction map from Notation 3.25.

Let E denote a compact generator of C. Consider the morphism φE : E → E[2] in C. LetE′ = Cofib(φE) so we have a cofiber sequence E → E[2] → E′ in C. First we observe thatE′ is another compact generator of C. Indeed E′ is compact because it is a finite colimitof compact objects. Suppose F ∈ C is such that MapC(E

′, F ) ' 0. This implies that the

map MapC(E,F [−2]) ' MapC(E[2], F ) → MapC(E,F ) induced by φE is an equivalence.

By iteration, the map MapC(E,F [−2n]) → MapC(E,F ) induced by φnE : E → E[2n] isan equivalence for every integer n. But because C is tamely compactly generated we haveMapC(E,F [N ]) ' 0 for N 0. Hence we obtain MapC(E,F ) ' 0. But E is a generator of

C so this implies F ' 0. Therefore E′ is a compact generator of C.The object E′ is fixed by the formal action ρ : G→ AutC in the sense of Notation 3.71. To

see this, by Lemma 3.73 we need to show that the corresponding map TG[−1]→ EndC(E′)

is null homotopic. By Remark 3.74, it is equivalent to show that the map of k-modulesmk[β] → EndC(E

′) induced by S is null homotopic. We know that this map is obtained fromS : k[β] → ξ(C) by composing with the obstruction map χE′ : ξ(C) → EndC(E

′) which is amap of E1-algebras, therefore mk[β] → EndC(E

′) is a map of nonunital E1-algebras, and to

show that it is null homotopic it suffices to show that the image of β is zero in Ext2C(E

′, E′)or in other words that φE′ = 0. Denote by k[β]/β the cofiber of the map β : k[β][−2]→ k[β]given by the multiplication by β. Because β is a non zero divisor in the ordinary commutativealgebra k[β], we have an isomorphism of groups π2(k[β]/β) ' π2(k[β])/β. However we havean equivalence E′ ' E ⊗k k[β]/β in C, and φE′ is the image of the class idE ⊗ [β] by themap Ext0

C(E,E) ⊗k π2(k[β]/β) → Ext2C(E

′, E′), showing that φE′ = 0. This proves thatE′ is fixed by ρ. Hence there exists a lift ρ : G → Aut(C,E′) of ρ through the natural mapAut(C,E′) → AutC .

Let B = E(C, E′) ' EndC(E′) be the E1-algebra of endomorphisms of E′ in C. Because

C is tamely compactly generated, we have ExtmC (E′, E′) = 0 for m 0, hence there existsan n ≥ 0 such that B is n-connective. Because E′ is a compact generator of C, the k-linear∞-functor LModB → C sending B to E′ is an equivalence of k-linear ∞-categories. Let

ρ′ : G → AutalgB be the composition of ρ with the natural map Aut(C,E′) → Autalg

B fromRemark 3.74. By Corollary 3.58, there exists an equivalence of spaces

AlgDefB(k[[t]]) ' MapGpE2

(G,AutalgB )

therefore the formal action ρ′ : G → AutalgB corresponds to an n-connective deformation

Bt = Bii≥0 of B over k[[t]] well defined up to equivalence in AlgDefB(k[[t]]). Considerthe commutative diagram from Notation 3.59:

AlgDefB(k[[t]])

mB

θB // AlgDef∧B(k[[t]])

m∧B

σ∧B // MapGpE2

(G,AutalgB )

ν

CatDefcC(k[[t]])θC // CatDef∧C (k[[t]])

ϕ∧C // MapGpE2

(G,AutC)


where we identify CatDefC with CatDefLModB and where θB , σ∧B and ϕ∧C are equivalences.By construction of Bt, we have ν(σ∧B(θB(Bt))) = ν(ρ′) = ρ and therefore θC(mB(Bt)) = αin π0CatDef∧C (k[[t]]).

Proof. of Theorem 3.63. For the first part, Proposition 3.66 shows that the map

CatDefcC(k[[t]])→ CatDef∧C (k[[t]])

is (−1)-truncated and Proposition 3.75 shows that it induces a surjection on π0, thereforeit is an equivalence.

For the second part, let Ct = Cii≥0 be a compactly generated formal deformationof C. Let Bt = Bii≥0 be the n-connective E1-algebra formal deformation providedby Proposition 3.75, for some integer n. We have θC(mB(Bt)) = θC(Ct). Because themap π0θC : π0CatDefcC(k[[t]]) → π0CatDef∧C (k[[t]]) is bijective, we have mB(Bt) = Ct inπ0CatDefcC(k[[t]]). This signifies that there exists for each i ≥ 0 a k-linear equivalenceCi ' LModBi , proving that each Ci has a compact generator Ei ∈ Ci. Because each Bi isn-connective, we have ExtmCi(Ei, Ei) = 0 for every m > n and therefore each Ci is tamelycompactly generated.

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Documents

Contents DEFORMATIONS OF CATEGORIES ANTHONY BLANC, LUDMIL KATZARKOV, PRANAV PANDIT Abstract. Let k be a eld. In this paper we use the theory of formal moduli problems developped by