FORMAL DEFORMATIONS OF CATEGORIES

FORML DEFORMTIONS OF CTEGORIES NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT bstract. Let 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 -linear - category. Our main result states that if C is a -linear -category which has a compact generator 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 formal group actions. Contents 1. Introduction 1 1.1. Main results 4 1.2. Outline of this paper 5 1.3. Conventions 5 2. Formal moduli and approximation after Lurie 6 2.1. Formal moduli problems 6 2.2. xiomatic deformation theory 9 2.3. Classification of E n -formal moduli problems 13 2.4. Proximate formal moduli problems and loop spaces 16 3. Deformations of linear -categories 22 3.1. Linear -categories 22 3.2. Deformations of objects 24 3.3. Deformations of categories 27 3.4. Deformations as Ind-coherent loop actions 33 3.5. Deformations of associative algebras 38 3.6. Formal deformations 40 References 45 1. Introduction Let be a field of characteristic 0 and let B be an associative algebra over. It is nown since the wor of Gerstenhaber that the deformation theory of B up to isomorphism has a close relation with the Hochschild cohomology of B. First order infinitesimal deformations (=over [t]/t 2 ) of B are classified by the group HH 2 (B). If µ 0 denotes the multiplication of B, a 2-cocycle φ defines a first order deformation of µ 0 given by µ = µ 0 + ɛφ. Obstructions to extend infinitesimal deformations live in the group HH 3 (B). Moreover if L = HH (B)[1] denote the dg-lie algebra given by the shifted Hochschild complex together with the Gerstenhaber bracet, formal deformations (=over [[t]]) of B are classified by the set MC(L t[[t]]) of solutions of the Maurer Cartan equation in (L t[[t]]) 1 up to gauge equivalence. Date: May 4, 2016. 1

2 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT In [LVdB05] and [LVdB06], W. Lowen and M. Van den Bergh study deformations of an abelian category up to equivalence of categories and its relation to a suitably defined Hochschild cohomology, as well as obstructions for infinitesimal deformations. However, as in the case of associative algebras, there is no mathematical statement which exhibits the precise relation between the corresponding deformation functor and the Hochschild cohomology complex. Deformations of dg-categories. In geometry we study derived categories attached to manifolds and varieties (e.g. local systems, coherent sheaves...). Moreover many invariants which play a central role in mirror symmetry lie cyclic homology are invariant under Morita equivalence. This forces us, in order to have a well behaved Morita theory (see [Toë07]), to wor in the setting of differential graded categories (or linear -categories). Given a dg-category C, it is nown that the group HH 2 (C) parametrizes more than the dg-category deformations of C, namely every 2-cocycle corresponds to a curved -deformation of C (see [Low08] and [KL09]). From the purpose of homotopy theory, curved -categories are considered as pathological objects since no they do not possess a good notion of equivalence nor derived categories. It is therefore useful from this point of view to have another description of the space of deformations controlled by the Hochschild cohomology complex. We will see that such a new description as well as its relation with the space of deformations of a dg-category was given by J. Lurie using his theory of formal moduli problems and Koszul duality. Derived deformation theory and formal moduli problems. dopting the foncteur des points point of view, deformation theory was first formalized through the notion of functors from artinian local algebras to sets satisfying the Schlessinger conditions. fter the wor of Drinfeld, Kontsevitch, and more recently of Hinich, Manetti, Kapranov and Ciocan-Fontanine, and finally Pridham and Lurie, it is now understood that deformation theory is in essence derived, and that it is naturally formalized using certain derived functors from artinian dg-algebras to simplicial sets satisfying derived analogs of the Schlessinger conditions. In [Lurb], J. Lurie maes a systematic study of these functors under the name of formal moduli problems, and through a spectacular use of the theory of -categories and their monoidal structures, achieves the classification of formal moduli problems by dg-lie algebras, maing a mathematical statement out of what was then a principle. This result was also 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/} {formal moduli problems/} This 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 Cartan solutions. The inverse functor is given by the sifted tangent complex F T F [ 1] which has therefore the structure of a dg-lie algebra. Moreover, Lurie develops an axiomatic approach for proving this equivalence, which therefore applies in other contexts. In particular, if n 0 is an integer, he defines formal moduli problems over E n -algebras and obtains an equivalence {augmented E n -algebras/} {E n -formal moduli problems/} where E n is the -operad of little n-cubes. Here the equivalence is valid for a field of arbitrary characteristic. The functor above is defined using the self duality functor of the E n -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 deformation problem is defined, precisely because it gives a finer structure on the tangent complex, as for example in the case of deformations of a category.

FORML DEFORMTIONS OF CTEGORIES 3 Formal moduli problem of a linear -category. Let be a field. In this paper we have choosen to wor in the setting of -linear -categories instead of -linear dg-categories mainly to fit with [Lurb] whose constructions and results are used in this paper. The relation between these two classes of categories will be mentionned in 3.1. -linear -category is roughtly a presentable -category which is enriched over the -category Mod of -module spectra. ny dg-category gives a linear -category by applying the dg-nerve construction. If is an E 1 -algebra, the -category LMod of left -module spectra has in general no natural monoidal structure. However if is an E 2 -algebra, LMod ihnerits a natural monoidal structure and we can spea of -linear -categories. For this reason the deformation functor of a -linear -category is defined over artinian E 2 -algebras. If C is a (compactly generated a ) -linear -category, we can define an -functor CatDef c C from artinian E 2 -algebras to spaces such that for each artinian E 2 -algebra, the space CatDef c C() is a classifying space of (compactly generated) deformations of C over. In general, the -functor CatDef c C does not satisfy the derived Schlessinger conditions and is therefore not a formal moduli problem. The moral reason for this is that compact objects of C do not always deform. The typical example is the -category Mod B where B = [u ±1 ] is the free graded commutative algebra with u of degree 2 and zero differential where any B-module has 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 Hochschild cohomology complex HH (C) of C admits the structure of an E 2 -algebra by [Lurc, 5.3]. Therefore the augmented E 2 -algebra HH (C) corresponds to a formal moduli problem Ψ HH (C) defined over artinian E 2 -algebras through the equivalence of the previous paragraph. Using Koszul duality, Lurie defines a natural transformation θ : CatDef c C Ψ HH (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 π 0 CatDef c C([t]/t 2 ) HH 2 (C) studied in [KL09]. Moreover Lurie shows that the formal moduli problem Ψ HH (C) is the best approximation of CatDef c C by a formal moduli problem (see also [Pre12, Lem 5.3.3.6]), and it should therefore be regarded as the correct space of deformations of C, as well as a substitute of the space of curved -deformations. To emphasize this, we set the notation CatDef C = Ψ HH (C). By the universal property of Hochschild cohomology, for each artinian E 2 -algebra, the space CatDef C () has the following interpretation (see [Lurc, 5.3]): CatDef C () {D (2) () linear structures on C} where D (2) () is the E 2 -Koszul dual of. The interesting question becomes: does there exists a good class of categories C for which θ is an equivalence? Killing curvatures. The example B = [u ±1 ] above shows that there might exist Hochschild cocycles which do not correspond to any uncurved deformations of C through the map θ. However because the map θ is 0-truncated, it is not far from being an equivalence, and therefore CatDef c C 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 latter condition on C is rather reasonable and is satisfied by the derived category of quasi-coherent sheaves on a finite type scheme over, but also excludes the example above. Moreover he shows ([Lurb, Thm 5.3.33]) that if in addition C is generated by a set of unobstructible compact generators, the map θ is an equivalence, or in other words CatDef c C is a formal a This 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 lie the derived category of quasicoherent sheaves on a finite type scheme.

4 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT moduli problem. However this latter condition is not satisfied by most derived categories of varieties of interest. 1.1. Main results. In this paper we study the space of formal deformation of a -linear -category C defined as CatDef c C([[t]]) = lim i CatDef c C([t]/t i ). In this case the (formal) E 2 -Koszul dual of [[t]] is equivalent to the underlying E 2 -algebra of the commutative graded alegbra [β] with β of cohomological degree 2. Therefore CatDef C ([[t]]) is equivalent to the space of [β]-linear structures on C (see [Toë14, Thm 5.1]). We obtain the following. Theorem 1.1. (See Theorem 3.63). Let be a field and C a -linear -category. Suppose that C admits a single compact generator E such that Ext m C (E, E) = 0 for m 0. Then the natural transformation θ induces an equivalence CatDef c C([[t]]) CatDef C ([[t]]) {[β] linear structures on C}. Moreover, for any formal deformation {C i } i 0 of C, each C i has a compact generator which satisfies the same condition as E. The assumptions on C in this theorem are rather reasonable. For example if C is the derived category of quasi-coherent sheaves on a finite type scheme over, we now by a theorem of Bondal Van-den-Bergh that C admits a compact generator E which satisfies the assumption of Theorem 1.1. To prove this theorem we use a new description of the space CatDef C ([[t]]) in terms of actions of the loop group ΩÂ1 on C where Â1 = Spf([[t]]) is the formal completion of the affine line at 0. similar description appears in [Pre12, Lem 5.3.3.6]. Theorem 1.2. (See Corollary 3.52). Let be field and C a compactly generated -linear -category whose spaces of morphisms are cohomologically bounded above. Then there exists a natural equivalence CatDef C ([[t]]) {ΩÂ1 ctions on C}. Moreover we obtain a similar formula not only for [[t]] but for deformations over any pro-artinian E 2 -algebra. We derive this result from a fundamental fact concerning the - category of formal moduli problems over any reasonable deformation context in the sense of Lurie: we show that the loop functor Ω : {formal moduli problems over } {Group objects in formal moduli problems over } is an equivalence of -categories (see Proposition 2.44). This fact implies a general formula for the best approximation to a deformation functor (see Theorem 2.48). Consider for example the deformation functor lgdef B of an E 1 -algebra B over. It can be shown that if B is connective, lgdef B is a formal moduli problem (see Proposition 3.56) whose tangent complex is given by the shifted derived derivations Der(B, B)[1]. Moreover this fact about the loop functor above implies that there exists an equivalence lgdef B ([[t]]) {ΩÂ1 ctions on B}. Setch of proof of Theorem 1.1. Under these assumptions on C, we now by [Lurb, Prop 5.3.21] that the map θ induces an isomorphism on π i for i > 0 and an injection on π 0. Therefore for the first part, it only remains to prove the surjectivity on π 0. If E C is an object, we can define an pre-formal moduli ObjDef E encoding deformations of E in C. We can define as well a pre-formal moduli Def c (C,E) encoding simulnateous deformations C and

FORML DEFORMTIONS OF CTEGORIES 5 E. point in Def (C,E) () corresponds to a compactly generated deformation C of C and of a deformation E of E in C. There exists a map Def c (C,E) CatDef c C which forget the deformation of the object. This latter map induces a map on the loops ut (C,E) ut C which taes an infinitesimal autoequivalence of C fixing E to the underlying autoequivalence of C (forgetting that it fixes E). Let α π 0 CatDef C ([[t]]) which corresponds to an action ρ : ΩÂ1 ut C via Theorem 1.2. If E is a compact generator of C, it might not be fixed under ρ. However because ρ corresponds to a D (2) ([[t]]) [β]-linear structure on C via Theorem 1.1, it is possible to find another compact generator of C fixed under ρ. For this we can show that it suffices to tae the cofiber E of the map β : E[ 2] E given by the multiplication by β. Therefore there exists a lift ρ : ΩÂ1 ut (C,E ) to autoequivalences of C fixing E. But such autoequivalences restrict to algebra autoequivalences of the E 1 - algebra B = End C (E ). Hence we have an action ρ : ΩÂ1 ut alg B which corresponds to a formal deformation B t of B through the equivalence above. By construction, left modules over B t are sent to α through the map θ, hence we have surjectivity on π 0. It also proves that any formal deformation of C is equivalent to left modules over a formal deformation of an E 1 -algebra, and has therefore a compact generator. This argument is deeply connected to the fact that D (2) ([[t]]) [β] is a graded polynomial algebra and cannot be adapted to the case of infinitesimal deformations where D (2) ([t]/t 2 ) [ 2]. 1.2. Outline of this paper. Section 2.1 contains some motivations and a first definition of 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 mae recalls from Lurie s fundamental wor in [Lurb], in a form that is suitable for our use. The corresponding references are written. We apologize for the formal aspect of the exposition. Section 2.4 contains some reminders about proximate formal moduli problems but also contains new materials concerning the loop functor on formal moduli as well as our first main results concerning the explicit formula for the associated formal moduli in terms of group 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 follore facts about deformations of associative algebras which we use 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 formal deformations as well as its proof. 1.3. Conventions. Let U be a Grothendiec universe, with U satisfying the axiom of infinity. The U-small mathematical objects will be called only small. We assume the axiom of Universes. Some arguments in this article will require to enlarge the universe U, which is always possible by assuming the axiom of Universes. If V is such an enlargement in which U is small, the V-small mathematical objects will be called not necessarily small. We wor within the theory of -categories in the sense of Lurie [Lur09], a..a quasicategories. We follow the terminology and the conventions of loc.cit. regarding the theory of -categories.

6 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT 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, we denote 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 objects in C is by definition the full subcategory of Fun( 1, C) consisting of morphisms C 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 small Kan complexes. We denote by S big the -category of not necessarily small spaces. The word space means an object of the -category S. The expression not necessarily small space means an object of S big 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 be an integer. space X is n-connected if π i (X, x) = 0 for every i < n and every vertex x X. Every space is declared to be ( 1)-connective. We follow the terminology of [Lurc] concerning n-truncated spaces. Let n 1 be an integer. space X is n-truncated if π i (X, x) = 0 for every i > n and every vertex x X. By convention a space is ( 2)-truncated if it is contractible. map of 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 review in 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. spectrum X is n-connective (resp. n-truncated) if π i X = 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 formal moduli problems, and we define them and their tangent complex in the context of commutative dg-algebras. The introduction of [Lurb] is highly recommended. Let be a field of characteristic 0 and B an associative -algebra. Let art denote the ordinary category of local artinian commutative -algebras and Gpd the category of groupoids. The ordinary deformation theory of B is encoded in a functor def B : art Gpd such that for each art, the set def B () is the set of isomorphism classes of deformations of B over and functoriality is given by base change. In the introduction we saw that the tangent space of def B is given by π 0 def B ([t]/t 2 ) HH 2 (B). Moreover it can be shown that the group of automorphisms of any first order deformation B 1 of B is given by π 1 (def B ([t]/t 2 ), B 1 ) HH 1 (B). The natural question in deformation theory is to as whether a first order deformation B 1 of B can be extended to a second order deformation. It is possible to prove by direct computation that for every first order deformation B 1 of B, there exists a class o(b 1 ) HH 3 (B) such that there exists a lift of B 1 to a second order deformation of B if and only if o( 1 ) = 0. This latter fact is much worse than the facts above giving an identification of Hochschild cohomology with a space of deformations. Indeed it does not give a natural way to obtain the obstruction and it does not give a deformation theoretic interpretation of the whole group HH 3 (B).

FORML DEFORMTIONS OF CTEGORIES 7 t the cost of passing from deformations over artinian algebras to deformations over artinian 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 deformation theory gives an answer to another related important question (which was perhaps its original motivation): the problem of classifying deformation problems by differential graded Lie algebras. Recall that the category of commutative dg-algebras cdga admits a simplicial combinatorial model structure with equivalences being quasi-isomorphisms and fibrations being levelwise surjective maps. Recall as well that for any model category M with subclass of wea equivalences W, we have an associated -category N(M)[W 1 ] to M which is by definition 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 Clg dg the -category associated to the model category cdga. If n 0 is an integer, we denote by [n] the shifted trivial square zero extension of by an element in degree n. If n = 0, [n] = [t]/t 2 and if n > 0, the underlying complex of [n] consists of in degree 0 and in degree n, with zero differential. It is well nown that there exists a diagram [t]/t 3 [t]/t 2 [1] in the -category Clg dg, which is a cartesian diagram. This important fact provides a way to analyze more carefully the behavior of the projection map def B ([t]/t 3 ) def B ([t]/t 2 ). Deformation theory was at first formalized through the notion of functors defined on local artinian algebras satisfying some exactness properties called the Schlessinger conditions. Nevertheless to mae the above idea concrete, we need to allow our functors to be defined on artinian dg-algebras. In practise, most if not all the deformation functors naturally extend to dg-algebras. commutative -dg-algebra is artinian if H 0 () is a local artinian ring with residue field, H i () = 0 for i > 0 and i 0 and if for every i, the vector space H i () is finite dimensional over. The trivial square zero extensions [n] are examples of such. We denote by dgart the full subcategory of Clg dg consisting of artinian dg-algebras. The deformation functor Def B extends naturally on artinian dg-algebras. If dgart, a deformation of B over is given by an associative dg-algebra B together with an equivalence B B in Clg dg. It is possible to define an -functor Def B : dgart S such that for each dgart, the space Def B () is the classifying space of deformation of B over up to equivalence. It can be shown (see section 3.5) that the induced diagram Def B ([t]/t 3 ) Def B ([t]/t 2 ) Def B () Def B ( [1])

8 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT is cartesian in S and that there exists a bijection π 0 Def B ( [1]) HH 3 (B). Moreover Def B () so that we obtain a fiber sequence of pointed sets π 0 Def B ([t]/t 3 ) π 0 Def B ([t]/t 2 ) HH 3 (B). The map on the right associates to each (equivalence class of) first order deformation the obstruction of lifting it to a second order deformation. This motivates the following central definition introduced in [Lurb, Def 0.0.8]. Definition 2.1. n -functor F : dgart S is called a formal moduli problem if it satisfies the following two conditions: (1) The space F () is contractible. (2) For every cartesian diagram of the form in dgart, the induced diagram F () [n] F ( ) is cartesian in S. F () F ( [n]) The first condition means that we are looing at the deformation of one object (up to equivalence). Indeed we should thin about a formal moduli problem as a formal neibourghood of a point in a derived moduli space. The second condition, which is a derived analog of the Schlessinger conditions, provides a way to relate the spaces of deformations over various rings with the spaces of obstructions, and provides as well the tangent complex associated to the formal moduli problem. Indeed for each m 1 the diagram [m 1] [m] is cartesian in dgart. Therefore if F is a formal moduli problem, the induced map F ( [m 1]) F ( [m]) ΩF ( [m]) is a homotopy equivalence, and provides the bonding maps of a spectrum T F with (T F ) m = F ( [m]). Here the loop space is taen with respect to the point F () F ( [m]). The zero level (T F ) 0 = F ([t]/t 2 ) is the usual Zarisi tangent space. The level 1 is the obstruction space for lifting infinitesimal deformations and the higher positive levels are higher obstruction spaces. s expected, it can be shown that a map of formal moduli problems f : F G is an equivalence if and only if the induced map of spectra T F T G is an equivalence (see [Lurb, Prop 1.2.10]). It can be shown that the tangent complex functor commutes with finite limits so for any formal moduli problem we obtain an equivalence T ΩF ΩT F = T F [ 1]. It is therefore not suprising that the shifted tangent complex T F [ 1] has the structure of a dg-lie algebra being the tangent complex of a formal group. Moreover the central theorem in the theory of formal moduli problems is the following classification result, and which

FORML DEFORMTIONS OF CTEGORIES 9 shows the power of this derived point of view. If FMP denotes the -category of formal moduli problems over, Lurie defines an -functor dglie FMP which is proven to be an equivalence of -categories, and whose inverse equivalence is given on objects by F T F [ 1] (see [Lurb, Thm 0.0.13]). We will review in 2.3 this classification and as well as in less commutative deformation contexts. 2.2. xiomatic deformation theory. s explained in the introduction, some deformation functors are often defined on E n -algebras for some n <, and it is therefore useful to have a theory of such formal moduli problems at our disposal. In [Lurb], an axiomatic approach to formal moduli problems is given, and allows the author to obtain results in all possible deformation context at the same time, in particular the context of E n -algebras for every 0 n. In this section we recall definitions and results from [Lurb, 1] related to the axiomatic approach to deformation theory. The main result is the classification of formal moduli problems relative to a deformation context which admits a deformation theory. We start 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 subcategory M Ω of the -category of square diagrams Fun( 1 1, C) consisting of pullbac diagrams Y 0 0 X where 0 and 0 are zero objects of C. pplying [Lur09, Prop 4.3.2.15] we have that the -functor e : M Ω C given by evaluation at the final vertex is a trivial fibration. Let s be a section of e and let e : M Ω C be the evaluation at the initial vertex. The loop functor of 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 finite limits. Let S fin denote the full subcategory of S which contains and is stable under finite colimits, or in other words the -category of finite spaces. Let S fin be the -category of pointed finite spaces and let be an -category admitting finite limits. Recall (see [Lurc, Def 1.4.2.1]) that an -functor X : S fin is pointed excisive if it carries pushout diagrams to pullbac diagrams in and if X( ) is a final object of. The -category of spectra objects in is defined to be the full subcategory Sp() of Fun(S fin, ) consisting of pointed excisive -functors. If X : S fin is a spectrum in and n 0, we denote by Ω n X the evaluation of X on the pointed n-sphere S n (for which we choose a model in S fin ). Remar 2.4. In the situation of Notation 2.3, denote by Ω : the loop functor of. Unwinding the definition, a spectrum object X : S fin is essentially given by a sequence of finite pointed spaces Ω n X, n 0, together with equivalences ΩΩ n X Ω n 1 X. Indeed by [Lurc, Prop 1.4.2.24], there exists an equivalence of -categories Sp() lim(... Ω Ω Ω ). Definition 2.5. deformation context is a pair (, (E α ) α T ) where is a presentable -category and (E α ) α T is a sequence of objects of the -category of spectra objects Sp() in.

10 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT map f : in is called elementary if it can be written as a pullbac f E α [n] for an index α T and an integer n. map f : in is called small if it can be written as a finite composition of elementary maps 0 = 1... m = in. n object is called artinian if the map is small. We denote by art the full subcategory of spanned by artinian objects. Example 2.6. Commutative dg-algebras. Let be a commutative ring. Consider the simplicial model category of unbounded dg-modules over (we consider differential Z-graded modules) dgmod with class of equivalences W 1 being the class of quasi-isomorphisms and fibrations are levelwise surjective morphisms. Let cdga = lg Comm (dgmod ) denote the category of commutative algebra objects in dgmod, its objects are commutative dg-algebras over. We endow cdga with its usual simplicial model structure with equivalences W 2 being quasi-isomorphisms and fibrations are levelwise surjectives morphisms. Let Clg dg := N(cdgac 1 )[W2 ] be the associated -category. Let Clg dg,aug := (Clg dg ) / be the -category of augmented commutative dg-algebras over. The -category of spectra in Clg dg and in Clgdg,aug is equivalent to the -category Mod dg := N(dgmodc 1 )[W1 ] of dg-modules over. The presentable -category Clg dg,aug together with T = a one point set and E = Mod dg concentrated in degree zero form a deformation context (Clgdg,aug, {}). The deloopings E[n] of E are the square zero extensions [n] and the elementary maps are given by the inclusion maps [n] for n 0. The artinian objects are given by local artinian commutative dg-algebras (similar to [Lurb, Prop 1.1.11]), which are by definition commutative dg-algebras satisfying the following conditions: H i () = 0 for i < 0 and for i 0 For every i, the -vector space H i () is finite dimensional The commutative ring H 0 () is local with residue field. We denote by Clg dg,art the subcategory of artinian commutative dg-algebras over. map B in Clg dg,art is small if and only if the induced ring map H 0 () H 0 (B) is surjective (similar to [Lurb, Lem 1.1.20]). Definition 2.7. (Lurie [Lurb, Def 1.1.14]). Let (, (E α ) α T ) be a deformation context and let denote a final object of. n -functor F : art S is called a pre-formal moduli problem if the space F ( ) is contractible. n -functor F : art S is called a formal moduli problem if it is a pre-formal moduli problem and if in addition for every pullbac square 0 1 01

FORML DEFORMTIONS OF CTEGORIES 11 in art such that the map 0 01 is small, the induced diagram F () F ( 0 ) is a pullbac in S. F ( 1 ) F ( 01 ) The following result allows us to test the second condition in the previous definition on a smaller class of pullbac diagrams of artinian objects. Proposition 2.8. [Lurb, Prop 1.1.15]. Let (, (E α ) α T ) be a deformation context. n -functor F : art S is a formal moduli problem if and only if the two conditions are satisfied: 1) The space F ( ) is contractible. 2) For every pullbac diagram of the form Ω m E α in art, for some α T, the induced diagram F () is a pullbac in S. F ( ) F (Ω m E α ) Example 2.9. Let (Clg dg,aug, {}) be the deformation context of Example 2.6 formed by augmented commutative dg-algebras. By Proposition 2.8, a pre-formal moduli problem F : Clg dg,art 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 (, (E α ) α T ) be a deformation context. We denote by PFMP the -category of pre-formal moduli problem and by FMP the -category of formal moduli problems. Let i : FMP PFMP denote the natural inclusion -functor. Both FMP and PFMP are presentable -categories, and i preserves small limits. Hence by the adjoint -functor theorem [Lur09, Cor 5.5.2.9] i has a left adjoint denoted by FMP L i PFMP. For every pre-formal moduli problem F we therefore have a natural map F i (L (F )) in PFMP given by the unit map of this adjunction, which is universal among maps from F to formal moduli problems. The formal moduli problem L (F ) is therefore the best approximation we were refering to in the introduction. For convenience, when the deformation context is clear, we denote L simply by L. Notation 2.11. Let (, (E α ) α T ) be a deformation context. For each α, the object E α is a spectrum in the -category, in particular we have E α ( ). By [Lurb, Prop 1.2.3], for every α T and every pointed finite set S, the object E α (S) is artinian. Hence for each α we have a spectrum object E α : S fin art.

12 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT Proposition 2.12. [Lurb, Prop 1.2.4]. Let (, (E α ) α T ) be a deformation context and let F : art S be a formal moduli problem. For every α T the composite -functor S fin E α art F S is pointed excisive in the sense of Notation 2.3 and therefore defines a spectrum in S denoted by F (E α ). Remar 2.13. The idea behind Proposition 2.12 is to consider for each α and each integer m 0 the pullbac diagram Ω m E α Ω m 1 E α in art. If F is a formal moduli problem the induced diagram of spaces F (Ω m E α ) F (Ω m 1 E α ) is a pullbac, and therefore gives the bonding maps of the tangent spectrum of F at α: ΩF (Ω m E α ) F (Ω m 1 E α ). Definition 2.14. [Lurb, Def 1.2.5]. Let (, (E α ) α T ) be a deformation context and let F : art S be a formal moduli problem. The tangent spectrum of F at α is by definition the spectrum F (E α ). Proposition 2.15. [Lurb, Prop 1.2.10]. Let (, (E α ) α T ) be a deformation context and let f : F G be a map in FMP. Then f is an equivalence if and only if for each α T the induced map on tangent spectra F (E α ) G(E α ) is an equivalence. Theorem 2.16. (Lurie [Lurb, Thm 1.3.12]). Let (, (E α ) α T ) be a deformation context which admits a deformation theory D : op B in the sense of [Lurb, Def 1.3.9]. Then there exists an -functor Ψ : B FMP which to an object B B assigns the formal moduli problem Ψ B defined by Ψ B () = Map B (D(), 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 (, (E α ) α T ) be a deformation context. Let art be an artinian object. We denote by Spf() : art S the -functor j () where j the co-yoneda embedding. For every R art we have therefore an equivalence Spf()(R) Map (, R). By the -categorical co-yoneda lemma, the pre-formal moduli problem Spf() is a formal moduli problem. By the -categorical Yoneda lemma ([Lur09, Prop 5.1.3.1]) this construction defines a fully faithful -functor Spf : ( art ) op FMP. Remar 2.18. In the deformation context (Clg dg,aug, {}) of example 2.6, suppose is a local artinian discrete commutative -algebra with residue field viewed naturally as an artinian commutative dg-algebra. Then the formal moduli problem Spf() parametrizes deformations of the point corresponding to the augmentation.

FORML DEFORMTIONS OF CTEGORIES 13 Notation 2.19. Let C be a small -category. We denote by Ind(C) the -category of Ind-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 : C op S which are left exact. By [Lur09, Prop 5.3.5.10] the -category Ind(C) is equivalent to the -category freely generated by C under filtered colimits. Namely if D is a small -category which admits filtered colimits, then the Yoneda embedding C Ind(C) induces an equivalence of -categories Fun ω (Ind(C), D) Fun(C, D) where the left handside is the -category of -functors which commute with filtered colimits. The dual notion to Ind-objects is that of Pro-objects. The -category of pro-objects associated to C is the -category P ro(c) := Ind(C op ) op. If D is a small -category which admits cofiltered limits, then the Yoneda embedding C Ind(C) induces an equivalence of -categories Fun ω (P ro(c), D) Fun(C, D) where the left handside is the -category of -functors which commute with cofiltered limits. Notation 2.20. Let (, (E α ) α T ) be a deformation context. We consider the -category P ro( art ) of pro-artinian objects. The -functor Spf : ( art ) op FMP gives an essentially unique -functor which commutes with filtered colimits Ind(Spf) : Ind(( art ) op ) P ro( art ) op FMP By abusing notations, we denote again the functor Ind(Spf) by Spf. If = lim i i is a pro-object in art, then there is an equivalence in FMP. Spf() colim i Spf( i ) Notation 2.21. Let (, (E α ) α T ) be a deformation context and F : art S a preformal moduli problem. Then we denote again by F the natural extension of F to proartinian objects F : P ro( art ) S whose existence is ensured by the universal property of the Pro completion (see Notation 2.19). Its value on a pro-artinian object = lim i i is given informally by F () = lim i F ( i ). For example if we wor in the deformation context (Clg dg,aug, {}) of augmented commutative dg-algebras over, the discrete algebra of formal power series over gives a pro-object [[t]] := lim i [t]/t i. If F is a formal moduli problem, the space F ([[t]]) := lim i F ([t]/t i ) is often called the space of formal arcs in F. 2.3. Classification of E n -formal moduli problems. We recall from [Lurb, 2 and 4] the classification of formal moduli problems related to the deformation contexts formed by E -algebras and by E n -algebras (see examples 2.22 and 2.27). Example 2.22. E -algebras. Let be a field. We denote by Mod the -category of -modules spectra whose objects are decribed by spectra together with an action of the ring spectrum H associated to. Let Clg := lg E (Mod ) be the -category of E -algebras over and Clg aug := (Clg ) / the -category of augmented E -algebras over. By [Lurc, Thm 4.5.4.7], if is of characteristic zero, there exists an equivalence of -categories Clg Clg dg and therefore an equivalence Clgaug Clg dg,aug. The stabilization Stab(Clg aug ) is equivalent to the -category Mod of -module spectra.

14 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT Together with T = a one point set and E = as discrete module, the presentable - category Clg aug form a deformation context (Clg aug, {}). The deloopings E[n] of E are given by the square zero extensions [n]. The artinian objects (see [Lurb, Prop 1.1.11]) are given by E -algebras which satisfy the following conditions: π i = 0 for i < 0 and for i 0. For every i, the -vector space π i is finite dimensional. The commutative ring π 0 is local with residue field. We denote by Clg art Clg aug the subcategory of artinian E -algebras. map B in Clg art is small if and only if the induced ring map π 0 π 0 B is surjective (see [Lurb, Lem 1.1.20]). Definition 2.23. n E -formal moduli problem over is a formal moduli problem relative to the deformation context (Clg aug, {}) formed by augmented E -algebras (example 2.22). We call them only formal moduli problems when the context is clear. We denote by FMP E () the -category of E -formal moduli problems over. Notation 2.24. Let be a field of characteristic zero and let dglie denote the category of differential Z-graded Lie algebras over. There exists a combinatorial model structure on dglie with equivalences W being quasi-isomorphisms and fibrations are given by levelwise surjective morphisms (see [Lurb, Prop 2.1.10]). We denote by lg Lie := N(dglie c )[W 1 ] the -category associated to dglie. Notation 2.25. Koszul duality between E and Lie. Let be a field of characteristic zero and let D : (Clg aug ) op lg Lie denote the -functor left adjoint to the -functor C : lg Lie (Clg aug ) op We call D the Koszul duality functor given by Lie algebra cochains (see [Lurb, 2.3]). between E and Lie-algebras. For a differential graded Lie algebra g over and an artinian E -algebra, the space Map lg Lie(D(), g) is a model for the space MC(g m ) of solutions of the Maurer Cartan equation in the Lie algebra g m where m is the augmentation ideal of. Theorem 2.26. (Lurie [Lurb, Thm 2.0.2]). Let be a field of characteristic zero. The -functor D : (Clg aug ) op lg Lie defined above is a deformation theory in the sense of [Lurb, Def 1.3.9]. s a corollary to Theorem 2.16 we obtain an equivalence of -categories Ψ : lg Lie Moreover the diagram of -categories lg Lie where U is the forgetful functor is commutative. FMP E (). Ψ U FMP E () Sp T [ 1] We now recall the classification of formal moduli problems defined over E n -algebras. In the sequel, is a field of any characteristic and n 0 is an integer. Example 2.27. E n -algebras. Let be a field. Let n 0 be an integer and let lg (n) := lg En (Mod ) denote the -category of E n -algebras over. Let moreover lg (n),aug be the -category of augmented E n -algebras over. The stabilization Stab(lg (n),aug ) is

FORML DEFORMTIONS OF CTEGORIES 15 equivalent to the -category Mod of -module spectra. Together with T = and E = as a discrete module, the presentable -category lg (n),aug form a deformation context (lg (n),aug, {}). The deloopings E[n] of E are given by the square zero extensions [n]. The artinian objects (see [Lurb, Prop 4.5.1] are given by E n -algebras which satisfy the following conditions: π i = 0 for i < 0 and for i 0. For every i, the -vector space π i is finite dimensional. Let r denote the radical of the algebra π 0, then the natural map (π 0 )/r is an isomorphism. We denote by lg (n),art lg (n),aug the full subcategory of artinian E n -algebras. map B in lg (n),art is small if and only if the induced ring map π 0 π 0 B is surjective (see [Lurb, Prop 4.5.3]). In this paper we will study the deformation functor of a fixed -linear -category which is naturally defined in the deformation context formed by augmented E 2 -algebras. Definition 2.28. Let n 1. n E n -formal moduli problem over is a formal moduli problem relative to the deformation context (lg (n),aug, {}) formed by augmented E n - algebras over (exemple 2.27). We call them only formal moduli problems when the context is clear. We denote by FMP En () the -category of E n -formal moduli problems over and by PFMP En () the -category of pre-e n -formal moduli problems over. Notation 2.29. Let lg (n) be an E n -algebra over. We denote by ug() = (, ) the space of augmentations of. Let now, B be two augmented E n - Map lg (n) algebras over with augmentations ɛ : and η : B respectively. Note that the natural maps of E n -algebras B B induce a map of spaces ug( B) ug() ug(b). We recall that the tensor product B differs a priori from the coproduct in lg (n). We denote by P air(, B) the space of pairings between and B; it is by definition the homotopy fiber of ug( B) ug() ug(b) over the point (ɛ, η). The space P air(, B) have points corresponding to augmentations B which extends the given augmentations ɛ : and η : B. Proposition 2.30. [Lurb, Prop 4.4.1]. Let lg (n),aug be an augmented E n -algebra over. The construction B P air(, B) defines an -functor (lg (n),aug ) op S. Moreover this -functor is representable. In other words, there exists an augmented E n -algebra D (n) () and a universal pairing ν : D (n) () such that for every augmented E n -algebra B the map ν induces an equivalence of spaces Map (n),aug lg (B, D (n) ()) P air(, B). Notation 2.31. In the situation of Proposition 2.30 we refer to D (n) () as the E n -Koszul dual to. Proposition 2.30 moreover implies that the construction D (n) () determines an -functor D (n) : (lg (n),aug ) op lg (n),aug called the Koszul duality functor. Because the construction (, B) P air(, B) is symmetric in and B, the -functor D (n) is self adjoint, meaning that there exists an equivalence of spaces Map lg (n),aug (B, D (n) ()) Map (n),aug lg (, 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) : (lg (n),aug ) op lg (n),aug defined above is a deformation theory in the

16 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT sense of [Lurb, Def 1.3.9]. -categories s a corollary to Theorem 2.16 we obtain an equivalence of Ψ : lg (n),aug FMP En (). Moreover there exists a commutative diagram of -categories lg (n),aug m Mod Ψ FMP En () Sp T [ n] where m B is the augmentation ideal of an augmented E n -algebra B. The following corollary will be useful in the sequel and follows from the fact that the Koszul duality functor D (n) is a deformation theory. Corollary 2.33. Let n 1 be an integer and denote by t : FMP En () lg (n),aug the equivalence given by Theorem 2.32. Let B be any pro-artinian E n -algebra over. Then there exists a natural equivalence t Spf(B) D (n) f (B) in lg(n),aug or equivalently an equivalence of formal moduli problems Spf(B) Ψ (n) D. In particular if B is artinian we have an f (B) equivalence of augmented E n -algebras t Spf(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 E n -algebra, we have a natural map Spf(B)() = Map lg (n),aug (B, ) Map (n),aug lg (D (n) (), D (n) (B)) = Ψ D (B)(). (n) By [Lurb, Prop 4.4.21], this map is an equivalence when is n-coconnective (the local finiteness assumption is satisfied whenever is artinian). The E n -algebra [m] is n- coconnective as long as m > n. This implies that the map above is an equivalence when = [m] for every m > n, which in turn implies that the natural transformation Spf(B) Ψ D (n) (B) is an equivalence on tangent spaces and therefore an equivalence of formal moduli problems. 2.4. Proximate formal moduli problems and loop spaces. We recall from the notion of n-proximate formal moduli problem from [Lurb, 5.1], which permits to study deformation functors which do not satisfy the derived Schlessinger conditions, but are very close to. n example is provided by the deformation functor CatDef C of a -linear -category C. We then describe a general formula for the approximation of some formal moduli problems with respect to a general deformation context. Definition 2.34. [Lurb, Def 5.1.5]. Let (, (E α ) α T ) be a deformation context. pre-formal moduli problem F : art S is an n-proximate formal moduli problem if for every pullbac square 0 1 01 in art such that the map 0 01 is small, the induced map of spaces is (n 2)-truncated. F () F ( 0 ) F (01) F ( 1 )

FORML DEFORMTIONS OF CTEGORIES 17 Remar 2.35. By [Lurb, Prop 5.1.4] the condition of Definition 2.34 for being n-proximate can be tested on the smaller class of pullbac diagram of artinian objects the form for α T and m 0. B Ω m E α Notation 2.36. We denote by FMP (n) PFMP the full subcategory of n-proximate formal moduli problems. There is a tower of -categories FMP = FMP (0) FMP(1)... FMP(n) FMP(n+1)... PFMP For example, we will see below that the deformation functor of an object in a stable -linear -category is in general 1-proximate and that the deformation functor of a fixed stable -linear -category itself is in general 2-proximate. Theorem 2.37. [Lurb, Thm 5.1.9]. Let (, (E α ) α T ) be a deformation context which admits a deformation theory and let F : art S be a pre-formal moduli problem. Then the 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 L (F ) is (n 2)-truncated. For all the sequel of this subsection we wor in a fixed deformation context (, (E α ) α T ) and we suppose that the -category is pointed. This will be the case in all our applications where will be an -category of augmented E n -algebras for some n. Remar 2.38. Let be an -category having an initial object. Let Fun (, S) denote the subcategory of Fun(, S) consisting of -functors F : S such that F ( ) is a contractible space. It can be shown that the -category Fun (, S) is pointed, or in other words has an object which is both initial and final. This object is described informally by the -functor sending every object of to a fixed final object of S. Notation 2.39. Let (, (E α ) α T ) be a deformation context such that is a pointed - category. Fix a final object of which is therefore an initial object. Then the -category PFMP of pre-formal moduli problem is by definition the -category Fun ( art, S) of -functors sending to a contractible space. In this situation, by Remar 2.38 the - category PFMP is pointed and we denote by Ω : PFMP PFMP its loop functor in the sense of Notation 2.2. Because the inclusion FMP PFMP and the loop functor above commute with limits, for every formal moduli problem F then ΩF is again a formal moduli problem. For an integer n 0, we denote by Ω = Ω... Ω ( times) the iterated loop functor. Remar 2.40. In the situtation of Notation 2.39, if F : art S is a pre-formal moduli problem, the loop object ΩF is the pre-formal moduli problem described on objects by (ΩF )() = Ω F () where the loop space Ω is the loop space taen at the base point F () corresponding to the map F ( ) F () induced by the essentially unique map. If we denote by E the element in the one point set π 0 F ( ) = {E}, which we imagine being an object of an -category C. Informally, the point F () corresponds to the trivial deformation of E over. Moreover the -functor ΩF parametrizes the infinitesimal automorphisms of E in C, or in other words the deformations of id E in C.

18 NTHONY BLNC, LUDMIL KTZRKOV, PRNV PNDIT Proposition 2.41. Let (, (E α ) α T ) be a deformation context such that is pointed. Let F : art S be a pre-formal moduli problem and let n 0. Then F is an n-proximate formal moduli problem if and only if Ω n F is a formal moduli problem. Proof. It is a direct consequence of the fact that the loop space functor commutes with small 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 cartesian symmetric monoidal structure C on C (see [Lurc, 2.4.1]). The -category Mon E (C) of E -monoids objects in C is the -category lg E (C ) of E -algebra objects with respect to the cartesian symmetric monoidal structure on C. n E -monoid with multiplication map m : in C is called grouplie if the maps (m, p 1 ) :, (m, p 2 ) : are equivalences in C. We denote by Gp E (C) Mon E (C) the full subcategory of grouplie E -monoids in C, whose objects are also called E -groups in the sequel. Notation 2.43. Let (, (E α ) α T ) be a deformation context. The -category PFMP is an -topos. We consider the cartesian symmetric monoidal structure PFMP on PFMP. Because the inclusion FMP PFMP commutes with small products, the -category FMP inherits this cartesian symmetric monoidal structure and we have a symmetric monoidal -category FMP with a symmetric monoidal inclusion FMP PFMP. Let 0, we denote by Ĝp E () the -category Gp E (FMP ) and we called its objects formal E -groups relative to or just formal E -groups when the context is clear. consequence of the construction of Bar/Cobar functors (see [Lurc, Not 5.2.6.11]) is that the loop functor PFMP PFMP given by Ω factorizes through the -category of E -group objects Gp E (PFMP ) giving an -functor still denoted by Ω : PFMP Gp E (PFMP ). Remar that if F : art S is a -proximate formal moduli problem then Ω F is a formal moduli problem which is moreover a formal E -group. We obtain a commutative diagram of -categories PFMP Ω Gp E (PFMP ) Ω FMP () Ω FMP FMP We adopt the notation Ω = Ω FMP. The following result gives conditions on under which Ω is an equivalence and is the first ey result of this paper. Proposition 2.44. Let (, (E α ) α T ) be a deformation context which admits a deformation theory D : op B and such that is pointed. We suppose moreover that there exists an -functor U : B such that is a stable -category. U is conservative, commutes with small limits and with sifted colimits.