Derived deformation theory of algebraic structures
aa r X i v : . [ m a t h . A T ] O c t DERIVED DEFORMATION THEORY OF ALGEBRAICSTRUCTURES
GRÉGORY GINOT, SINAN YALIN
Abstract.
The main purpose of this article is to develop an explicit deriveddeformation theory of algebraic structures at a high level of generality, en-compassing in a common framework various kinds of algebras (associative,commutative, Poisson...) or bialgebras (associative and coassociative, Lie,Frobenius...), that is algebraic structures parametrized by props.A central aspect is that we define and study moduli spaces of deformationsof algebraic structures up to quasi-isomorphisms (and not only up to isomor-phims or ∞ -isotopies). To do so, we implement methods coming from derivedalgebraic geometry, by encapsulating these deformation theories as classifying(pre)stacks with good infinitesimal properties and derived formal groups. Inparticular, we prove that the Lie algebra describing the deformation theory ofan object in a given ∞ -category of dg algebras can be obtained equivalentlyas the tangent complex of loops on a derived quotient of this moduli space bythe homotopy automorphims of this object.Moreover, we provide explicit formulae for such derived deformation prob-lems of algebraic structures up to quasi-isomorphisms and relate them in aprecise way to other standard deformation problems of algebraic structures.This relation is given by a fiber sequence of the associated dg-Lie algebrasof their deformation complexes. Our results provide simultaneously a vastgeneralization of standard deformation theory of algebraic structures which issuitable (and needed) to set up algebraic deformation theory both at the ∞ -categorical level and at a higher level of generality than algebras over operads.In addition, we study a general criterion to compare formal moduli problemsof different algebraic structures and apply our formalism to E n -algebras andbialgebras. Contents
Introduction 30.1. Motivations 40.2. Main results 60.3. Further applications and perspectives 10Notations and conventions 101. Recollections 111.1. Symmetric monoidal categories over a base category 121.2. Props, properads and their algebras 131.3. Algebras and coalgebras over operads 161.4. Homotopy algebras 172. Derived deformation theory of algebraic structures 182.1. A brief preliminary about cdgas 182.2. Relative categories versus ∞ -categories 192.3. Formal moduli problems and (homotopy) Lie algebras 222.4. Moduli spaces of algebraic structures and their formal moduli problems 263. Derived formal groups of algebraic structures and associated formal moduli problems 293.1. Generalities on derived formal groups 293.2. Derived prestack group and their tangent L ∞ -algebras 323.3. Prestacks of algebras and derived groups of homotopy automorphisms 343.4. The fiber sequence of deformation theories 403.5. Equivalent deformation theories for equivalent (pre)stacks of algebras 414. The tangent Lie algebra of homotopy automorphims 434.1. Homotopy representations of L ∞ -algebras and a relevant application 444.2. ∞ -actions in infinitesimally cohesive presheaves 454.3. The Lie algebra of homotopy automorphisms as a semi-direct product 474.4. The operad of differentials 524.5. Computing the tangent Lie algebra of homotopy automorphims 545. Examples 615.1. Deformations of E n -algebras 615.2. Deformation complexes of P ois n -algebras 625.3. Bialgebras 646. Concluding remarks and perspectives 656.1. Algebras over operads in vector spaces 656.2. Differential graded algebras over operads 666.3. Algebras over properads 67References 67 ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 3
Introduction
Deformations of algebraic structures of various kind, both classical and homo-topical, have played a central role in mathematical physics and algebraic topologysince the pioneering work of Drinfeld [16, 17] in the 80s as well as the work ofKontsevich [56, 57] or Chas-Sullivan [12] in the late 90s. For instance, in classicaldeformation quantization, a star-product is a deformation of the commutative alge-bra of functions to an associative algebra while a quantum group is a deformationof the cocommutative bialgebra structure of a universal envelopping algebra.In most applications, one consider deformations of algebraic structures up tosome equivalence relations, usually called gauge equivalences. In particular, differ-ent gauge equivalences on the same algebraic structure lead to different deformationtheories. This data is organized into a moduli space of deformations whose con-nected components are the gauge equivalence classes of the deformed structure.Their higher homotopy groups encode (higher) symmetries which are becoming in-creasingly important in modern applications. By the Deligne philosophy, now adeep theorem by Lurie [65] and Pridham [78] using derived geometry ideas, such amoduli space is equivalent to the data of a homotopy Lie algebra.The emergence of derived/higher structures techniques allows not only to con-sider general moduli spaces of deformations (derived formal moduli problems), butalso to consider deformations of algebraic structures more general than those givenby Quillen model categories of algebras over operads. In particular, it allows toconsider bialgebraic structures, that is algebras over props, in high generality .The main goal of this paper is to exploit these techniques to prove several newresults about deformation theory of algebraic structures. In particular, we seek toprovide appropriate extension of classical algebraic deformation theory simultane-ously in two directions:(1) By considering very general kinds of algebraic structures parametrized byprops, which are of crucial importance in various problems of topology,geometry and mathematical physics where such structures appear;(2) By considering derived formal moduli problems controlling the deforma-tion theory of algebras in the ∞ -category of algebras , that is up to quasi-isomorphism , contrary to the setting of standard operadic deformation the-ory which considers deformations up to ∞ -isotopies (see § 0.2 below andsection 5,6.1, 6.2 as well for detailed comparison and examples) .Both directions require to work out new methods:(1) by getting rid of the stantard use of Quillen model structures to describemodel categories of algebras, which does not make sense anymore for alge-bras over props: one has to work directly at an ∞ -categorical level.(2) by replacing the classical gauge group action and classical deformation func-tors by appropriate derived moduli spaces of algebraic structures and de-rived formal groups of homotopy automorphisms, which requires to importmethods coming from derived algebraic geometry.We now explain in more details the motivations and historical setting for ourwork in § 0.1 and then our contributions and main results in § 0.2. for instance, algebraic structures up to quasi-isomorphisms form precisely Kontsevich settingencompassing deformation of functions into star-products in the analytic or algebraic geometrycontext as well as for smooth manifolds, where it boils down to ”up to isomorphism“ ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 4
Motivations.
As already mentioned, many algebraic structures of varioustypes play a key role in algebra, topology, geometry and mathematical physics.This is the case of associative algebras, commutative algebras, Lie algebras, andPoisson algebras to name a few. All these kinds of algebra share a common fea-ture, being defined by operations with several inputs and one single output (theassociative product, the Lie bracket, the Poisson bracket). The notion of operadis a unifying approach to encompass all these structures in a single formalism,and has proven to be a very powerful tool to study these structures, both from acombinatorial perspective and in a topological or dg-context . The first historicalexamples, of topological nature, are the operads of little n -disks discovered in thestudy of iterated loop spaces in the sixties. They are the first known examplesof E n -operads. Algebras governed by E n -operads and their deformation theoryplay a prominent role in a variety of topics such as the study of iterated loopspaces, Goodwillie-Weiss calculus for embedding spaces, deformation quantizationof Poisson manifolds and Lie bialgebras, factorization homology and derived sym-plectic/Poisson geometry [56, 57, 64, 67, 11, 28, 27, 35, 43, 47, 53, 58, 69, 77, 83, 90].In a highly non trivial way, one can prove in the K -linear setting that for n ≥ and sev-eral outputs also appear naturally in a variety of topics related to the same fields ofmathematics. A standard example of such structure is the associative and coasso-ciative bialgebra, which consists of two structures, an associative algebra structureand a coassociative coalgebra structure, related by a compatibility condition suchthat bialgebras can be defined as algebras in coalgebras or vice-versa. Bialgebrasare central in various topics of algebraic topology, representation theory and math-ematical physics [16, 17, 5, 25, 26, 37, 72, 73]. In particular, the work of Drinfeldin quantum group theory ([16, 17]) puts emphasis on their strong relationship withLie bialgebras (which are determined by a Lie bracket, a Lie co-bracket and a com-patibility relation as well). Here the formalism of props, which actually goes backto [68], is the convenient unifying framework to handle such structures. Props playsa crucial role in the deformation quantization process for Lie bialgebras, as shownby Etingof-Kazdhan ([25], [26]), and more generally in the theory of quantizationfunctors [22, 74]. Props also appear naturally in topology, for example the Frobe-nius bialgebra structure on the cohomology of compact oriented manifolds comingfrom Poincaré duality, and the involutive Lie bialgebra structure on the equivarianthomology of loop spaces on manifolds, which lies at the heart of string topology([13],[14]) and are also central in symplectic field theory and Lagrangian Floer the-ory by the work of Cielebak-Fukaya-Latsheev [15]. Props also provide a concise wayto encode various field theories such as topological quantum field theories and con-formal field theories, and have recently proven to be the kind of algebraic structureunderlying the topological recursion phenomenom, as unraveled by Kontsevich andSoibelman in [59] (see also [9] for a survey of the connections with mathematicalphysics and algebraic geometry).A meaningful idea to understand the behavior of these various structures and,accordingly, to get more information about the mathematical objects on which they where the strict algebraic structure are no longer invariant under the natural equivalence ofthe underlying object and need to be replaced by their homotopy enhancement ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 5 act, is to organize all the possible deformations of a given structure into a single geo-metric object which encapsulates not only the deformations but also an equivalencerelation between these deformations. That is, to define a formal moduli problem .Such ideas goes back to the pioneering work of Kodaira-Spencer in geometry andthe work of Gerstenhaber on associative algebras and the Gerstenhaber algebrastructure on Hochschild cohomology. A convergence of ideas coming (among oth-ers) from Grothendieck, Artin, Quillen, Schlessinger, Deligne and Drinfeld led tothe modern formulation of deformation theory in terms of formal moduli problems(also called deformation functors), and in the eighties, to the groundbreaking ideasupported by Deligne and Drinfeld that any formal moduli problem corresponds toa certain differential graded Lie algebra which parametrizes algebraically the cor-responding deformation theory. The deformations correspond to special elementsof this Lie algebra called the Maurer-Cartan elements, and equivalences of defor-mations are determined by a quotient under the action of a gauge group. Thisprinciple had major applications among which one can pick deformation theory ofcomplex manifolds, representation spaces of fundamental groups of projective va-rieties in Goldman-Millson’s papers, and Kontsevich deformation quantization ofPoisson manifolds.However, this theory of “classical” or “underived” formal moduli problems hadits limitations and was not satisfactory for several deep reasons: • It is impossible to get an equivalence between dg Lie algebras and underivedformal moduli problems, in particular because two Lie algebras which arenot quasi-isomorphic can nevertheless describe the same moduli problem(a famous example is the deformation theory of a closed subscheme, seeneither as a point of a Hilbert scheme or as a point of a Quot scheme). Evenworse, their is no systematic recipe to build a Lie algebra out of a moduliproblem; • Deformation problems for which the equivalence relation is given by weakequivalences (say, quasi-isomorphisms between two deformations of a dgalgebra) do not fit in the framework of classical algebraic geometry (thatis, deformations which manifests a non trivial amount of homotopy theory); • There is no natural interpretation of the obstruction theory in terms of thecorresponding moduli problem.These difficulties pointed towards the necessity to introduce both features fromhomotopy theory and more flexibility in the geometric conditions defining formalmoduli problems, that is, the use of ∞ -category theory and derived algebraic ge-ometry . The appropriate formalism is then the theory of derived formal moduliproblems , which are simplicial presheaves over augmented artinian cdgas satisfyingsome extra properties with respect to homotopy pullbacks (a derived version ofthe Schlessinger condition). According to Lurie and Pridham equivalence theorems[64, 78], (derived) formal moduli problems and dg Lie algebras are equivalent as ∞ -categories, and this theorem as well as its numerous consequences solves theaforementioned difficulties.In this paper, we use rather systematically these ideas of derived formal moduliproblems and derived techniques to study deformation theory of algebraic struc-tures. In particular, we give a conceptual explanation of the differences betweenvarious deformation complexes appearing in the literature by explaining which kindof derived moduli problem each of these complexes controls. A key part of our study ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 6 is that we study algebras over very general props and that we consider moduli spacesof deformations of algebraic structures up to quasi-isomorphisms .0.2.
Main results.
Deformation theory is classicaly encoded by moduli problems(see [46, 57, 64, 78])) which appear often as formal neighbourhoods controlingthe infinitesimal deformation theory of points on a given moduli space (variety,scheme, derived stack). Since (derived) formal moduli problems and dg Lie algebrasare equivalent as ∞ -categories, to a given formal moduli problem controlling theinfinitesimal neighbourhood of a point on a moduli space corresponds a dg Liealgebra called the deformation complex of this point.Here we are interested in moduli spaces of algebraic structures and formal mod-uli problems controlling their deformations. A convenient formalism to deal withalgebraic structures at a high level of generality, in order to encompass not onlyalgebras but also bialgebras, is the notion of properad [92], a suitable generalizationof operads. Briefly, given a properad P and a complex X , a P -algebra structure on X is given by a properad morphism P → End X where End X ( m, n ) = Hom ( X ⊗ m , X ⊗ n ) is the endomorphism properad of X . In thedifferential graded setting, algebraic structures are deformed as algebraic structures up to homotopy , for instance dg associative algebras are deformed as A ∞ -algebras.A standard (pr)operadic approach to define a deformation complex of those struc-ture is as follows. Dg properads form a model category in the sense of Quillen, sothat a notion of homotopy P -algebra (or P -algebra up to homotopy) can be de-fined properly by considering cofibrant resolutions of properads. Given a cofibrantdg properad P ∞ and a P ∞ -algebra structure ψ : P ∞ → End X on a complex X ,there is a formal moduli problem P ∞ { X } ψ controlling the deformation theory of ψ . The associated deformation complex is an explicit dg Lie algebra noted g ψP,X .However, we can also construct a derived formal moduli problem controling thedeformation theory of a P ∞ -algebra A in the ∞ -category P ∞ − Alg , which is not thesame as deforming the morphism ψ (in a way precised below, the Maurer-Cartanelements are the same in both cases but the gauge equivalence relation differs ). Toset up the appropriate framework for such a deformation theory, we introduce thenotion of derived prestack group, which can be thought as a family of homotopyformal groups parametrized by a base space. In Section 3, we apply this formalismto the deformation theory of algebras over properads. Briefly, one associates to A an ∞ -functor G P ( A ) : CDGA K → E − Alg gp (Spaces) R haut P ∞ − Alg ( Mod A ) ( A ⊗ R )where haut P ∞ − Alg ( Mod A ) ( A ⊗ R ) is the ∞ -group of self equivalences of A ⊗ R inthe ∞ -category of P ∞ -algebras in A -modules. This is the derived prestack group ofhomotopy automorphisms of A . Taking homotopy fibers over augmented Artiniancdgas, we obtain a derived formal group \ G P ( A ) id ( R ) = hof ib ( G P ( A )( R ) → G P ( A )( K ))which associates to any augmented Artinian cdga R the space of R -deformationsof A . Precisely, we prove ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 7
Theorem 0.1 (See Theorem 3.20) . The simplicial presheaf G P ( A ) defines a grou-plike E -monoid object in the ∞ -category of infinitesimally cohesive simplicial ∞ -presheaves. In particular \ G P ( A ) id is a derived formal group. By the equivalence between derived formal groups and derived formal moduliproblems , these deformations are parametrized by a dg Lie algebra
Lie ( \ G P ( A ) id ).Two natural questions arise from these constructions. • First, can we relate the classical deformation theory of the morphism ψ : P ∞ → End X , controled by g ψP,X , to the deformation theory of ( X, ψ ) in P ∞ − Alg , controled by
Lie ( \ G P ( X, ψ )) ? • Second, is there an explicit formula computing
Lie ( \ G P ( X, ψ )) for general P and ( X, ψ ) ?The answer to the first question is the following natural homotopy fiber sequencerelating these two deformation complexes (see Theorem 3.24):
Theorem 0.2.
There is a homotopy fiber sequence of L ∞ -algebras g ψP,X −→ Lie ( \ G P ( X, ψ )) −→ Lie ( haut ( X )) where Lie ( haut ( X )) is the Lie algebra of homotopy automorphisms of X as a com-plex. To explain concretely how this fiber sequence explains the difference between g ψP,X and Lie ( \ G P ( X, ψ )), let us start with the following observation. One shouldnote that the deformation complex g ψP,X does not give exactly the usual cohomologytheories of algebras. As a motivating example, let us consider the case of theHochschild cochain complex of a dg associative algebra A which can be written asHom( A ⊗∗ , A ). This Hochschild complex is bigraded, with a cohomological gradinginduced by the grading of A and a weight grading given by the tensor powers A ⊗• .It turns out that the classical deformation complex g ψAss,A is Hom( A ⊗ > , A ) and inparticular misses the summand Hom( A, A ) of weight 1; which is precisely the oneallowing to consider algebras up to (quasi-)isomorphisms.To correct this, we use the “plus” construction g ψ + P + ,X . This is a functorialconstruction which allows to modify any dg-Prop to get the right cohomology theoryand can be obtained by a slight modification of the properad P , see § 4. Moreover,this “plus” construction gives us an explicit model of the deformation complexof ( X, ψ ) in the ∞ -category of P ∞ -algebras up to quasi-isomorphisms and thusanswers the second question: Theorem 0.3 (See Theorem 4.26) . There is an equivalence of L ∞ -algebras Lie ( \ G P ( X, ψ )) ≃ g ψP,X ⋊ h End ( X ) ≃ g ψ + P + ,X . The middle term of this equivalence is a homotopical semi-direct product of g ψP,X with the Lie algebra End ( X ) of endomorphisms of X (equipped with the commu-tator of the composition product as Lie bracket). It reinterprets the deformationcomplex Lie ( haut P ∞ − Alg ( X, ψ )) as the tangent Lie algebra of a homotopy quotientof P ∞ { X } by the ∞ -action of haut ( X ).To summarize, the conceptual explanation behind this phenomenon is as follows.On the one hand, the L ∞ -algebra g ψP,X controls the deformations of the P ∞ -algebra ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 8 structure over a fixed complex X , that is, the deformation theory of the properadmorphism ψ . On the other hand, we built a derived formal group \ haut P ∞ ( X, ψ ) id whose corresponding L ∞ -algebra Lie ( \ haut P ∞ ( X, ψ ) id ) describes another deriveddeformation problem : an R -deformation of a P -algebra A in the ∞ -category of P ∞ -algebras up to quasi-isomorphisms is a an R -linear P ∞ -algebra ˜ A ≃ A ⊗ R with a K -linear P ∞ -algebra quasi-isomorphism ˜ A ⊗ R K ∼ → A . The later L ∞ -algebraadmits two equivalent descriptions Lie ( \ haut P ∞ ( A ) id ) ≃ g ϕP,X ⋉ hol End ( X ) ≃ g ϕ + P + ,X where the middle one exhibits this moduli problem as originating from the homo-topy quotient of the space of P ∞ -algebra structures on X by the homotopy action ofself-quasi-isomorphisms haut ( X ), that is, deformations of the P ∞ -algebra structure up to self quasi-isomorphisms of X , and the right one encodes this as simultaneouscompatible deformations of the P ∞ -algebra structure and of the differential of X .We will go back to this in full details in Sections 4 and 6.Returning to the Hochschild complex example, we now see the role of the weight1 part Hom( A, A ). Indeed, in the case of a an associative dg algebra A , the com-plex g ψ + Ass + ,A ∼ = Hom ( A ⊗ > , A )[1] computes the reduced Hochschild cohomology of A , where the right hand side is a sub-complex of the standard Hochschild cochaincomplex shifted down by 1 equipped with its standard Lie algebra structure.Thecomplex g ψAss,A ∼ = Hom( A ⊗ > , A )[1] is the one controlling the formal moduli prob-lem of deformations of A with fixed differential , where the right hand side is thesubcomplex of the previous shifted Hochschild cochain complex where we haveremoved the Hom ( A, A ) component .In addition, we prove a general criterion to compare formal moduli problemsbetween two kinds of algebras (see Theorem 3.25): Theorem 0.4.
Let F be an equivalence of presheaves of ∞ -categories F : P ∞ − Alg ∼ −→ Q ∞ − Alg.
Then F induces an equivalence of fiber sequences of derived formal moduli problems P ∞ { X } ψ ∼ (cid:15) (cid:15) / / B fmp \ haut P ∞ − Alg ( X, ψ ) Id ( X,ψ ) ∼ (cid:15) (cid:15) / / B fmp \ haut ( X ) Id X = (cid:15) (cid:15) Q ∞ { F ( X ) } F ( ψ ) / / B fmp \ haut Q ∞ − Alg ( F ( X, ψ )) Id ( X,ψ ) / / B fmp \ haut ( X ) Id X where F ( ψ ) is the Q ∞ -algebra structure on the image of ( X, ψ ) under F . Here B fmp is the inverse ∞ -functor of the equivalence between formal moduliproblems and formal group. These constructions are the content of Section 3. Thus, when A is an ordinary, non dg, vector space, the complex g Ass,A parametrizes themoduli space of associative algebra structures on A , while g + Ass + ,A parametrizes the moduli spaceof asociative algebra structures up to isomorphism of algebras there is also a third complex, the full shifted Hochschild complex Hom( A ⊗≥ , A )[1), whichcontrols not the deformations of A itself but the linear deformations of its dg category of modulesMod A [55, 77] ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 9
In Section 5, we apply our machinery to get new results on the derived defor-mation theory of n -shifted Poisson algebras or n -Poisson (Poisson algebras with aPoisson bracket of degree 1 − n ): Theorem 0.5 (See Corollary 5.7) . (1) The Tamarkin deformation complex [85]controls deformations of A in P ois n, ∞ − Alg [ W − qiso ], that is, in homotopy dg- P ois n -algebras up to quasi-isomorphisms . It is thus equivalent to the tangent Lie algebra g ψ + P ois + n ,A of G P ois n ( A ) .(2) For n ≥ the Tamarkin deformation complex of A is equivalent, as an L ∞ -algebra, to the E n -tangent complex of A seen as an E n -algebra via the formality of E n -operads. To the best of the authors knowledge, the proof that this complex is indeed adeformation complex in the precise meaning of formal derived moduli problemsis new, as well as the concordance with the L ∞ -structure induced by the higherDeligne conjecture (which provides an E n +1 -algebra structure on the E n -tangentcomplex of an E n -algebra). We also prove that the deformation complex g ψP ois n ,A of the formal moduli problem P ois n ∞ { A } ψ of homotopy n -Poisson algebra struc-tures deforming ψ is given by the L ∞ -algebra CH ( • > P ois n ( A )[ n ], which is a furthertruncation of CH P ois n ( A )[ n ].Concerning the full shifted Poisson complex, we conjecture the following: Conjecture.
Let n ≥ A be an n -Poisson algebra. The L ∞ -algebra struc-ture of the full shifted Poisson complex CH ∗ P ois n ( A )[ n ] controls the deformationsof M od A into E n − -monoidal dg categories.This conjecture is deeply related to the deformation theory of shifted Poissonstructures in derived algebraic geometry, in the sense of [11]. Precisely, if X is aderived Artin stack locally of finite presentation and equipped with an n -shiftedPoisson structure, then its sheaf of principal parts (which controls the local defor-mation theory on X and whose modules describe the quasi-coherent complexes over X ) forms a sheaf of mixed graded P ois n +1 -algebras. The deformation theory ofthe category of quasi-coherent complexes should then be controled by a full shiftedPoisson complex.Concerning bialgebras, we obtain the first theorem describing precisely why (asuitable version of) the Gerstenhaber-Schack complex C ∗ GS ( B, B ) ∼ = Y m,n ≥ Hom dg ( B ⊗ m , B ⊗ n )[ − m − n ]is the appropriate deformation complex of a dg bialgebra up to quasi-isomorphisms in terms of derived moduli problems: Theorem 0.6 (See Teorem 5.9) . The Gerstenhaber-Schack complex is quasi-isomorphicto the L ∞ -algebra controlling the deformations of dg bialgebras up to quasi-isomorphisms: C ∗ GS ( B, B ) ∼ = g ϕ + Bialg + ∞ ,B ≃ Lie ( haut Bialg ∞ ( B )) . which we denote CH ( • > Pois n ( A )[ n ] since it is the part of positive weight in the full Poissoncomplex [10] there are several closely related versions of the Gerstenhaber-Schack, depending on how wetruncate them ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 10
Note that our theorem 0.3 implies that the L ∞ -algebra structure induced on C ∗ GS ( B, B ) contains as a sub L ∞ -algebra the Merkulov-Vallette deformation com-plex [71].In Section 6, we give an overview and comparison of various (derived or not)deformation problems of algebraic structures arising in our work and the litterature.0.3. Further applications and perspectives.
A first major application appearedearlier in our preprint [45], where some of the results of the present article wereannounced. Our article provides complete proofs of these results and add somenew ones as well. In this related work [45], the authors use them crucially to provelongstanding conjectures in deformation theory of bialgebras and E n -algebras aswell as in deformation quantization. We prove a conjecture enunciated by Ger-stenhaber and Schack (in a wrong way) in 1990 [37], whose correct version is thatthe Gerstenhaber-Schack complex forms an E -algebra, hence unraveling the fullalgebraic structure of this complex which remained mysterious for a while. It isa “differential graded bialgebra version” of the famous Deligne conjecture for as-sociative differential graded algebras (see for instance [83] and [56]). The secondone, enunciated by Kontsevich in his celebrated work on deformation quantizationof Poisson manifolds [57] in 2000, is the formality, as an E -algebra, of the de-formation complex of the symmetric bialgebra which should imply as a corollaryDrinfeld’s and Etingof-Kazdhan’s deformation quantization of Lie bialgebras (see[16], [25] and [26]). We solve both conjectures actually at a greater level of gener-ality than the original statements. Moreover, we deduce from it a generalizationof Etingof-Kadhan’s celebrated deformation quantization in the homotopical anddifferential graded setting.The new methods developed here to approach deformation theory and quan-tization problems have several possible continuations. In particular, we aim toinvestigate in future works how our derived algebraic deformation theory could beadapted to provide new deformation theoretic approach, formality statements anddeformation quantization of shifted Poisson structures in derived algebraic geom-etry. This problem is of crucial importance to understand quantum invariants ofvarious moduli spaces of G -bundles over algebraic varieties and topological mani-folds, which are naturally shifted Poisson stacks.Moreover, there is no doubt that our framework for derived algebraic deforma-tion theory will also be useful to study deformation problems related to the variouskinds of (bi)algebras structures mentionned in this introduction, occuring in math-ematical physics, algebraic topology, string topology, symplectic topology and soon. Acknowledgement.
The authors wish to thank V. Hinich, S. Merkulov, P. Safra-nov and T. Willwacher for their useful comments. They were also partially sup-ported by ANR grants CHroK and CatAG and the first author benefited from thesupport of Capes-Cofecub project 29443NE and Max Planck Institut fur Mathe-matik in Bonn as well.
Notations and conventions
The reader will find below a list of the main notations used at several places inthis article. • We work over a field of characteristic zero denoted K . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 11 • We work with cochain complexes and a cohomological grading. • Ch K is the category of Z -graded cochain complexes over K . • Let ( C , W C ) be a relative category, also called a category with weak equiv-alences. Here C is a category and W C its subcategory of weak equivalences.The hammock localization (see [19]) of such a category with respect toits weak equivalences is noted L H ( C , W C ), and the mapping spaces of thissimplicial localization are noted L H ( C , W C )( X, Y ). • We will note L ( M ) the ∞ -category associated to a model category M ,which is the coherent nerve of its simplicial localization. • Given a relative category (
M, W ), we denote by M [ W − ] its ∞ -categoricallocalization. • Several categories of algebras and coalgebras will have a dedicated notation: cdga for the category of commutative differential graded algebras, dgArt forthe category of Artinian cdgas, dgCog for the category of dg coassociativecoalgebras and dgLie for the category of dg Lie algebras. • Given a cdga A , the category of A -modules is noted M od A . More generally,if C is a symmetric monoidal category tensored over Ch K , the category of A -modules in C is noted M od A ( C ). • Given a dg Lie algebra g , its Chevalley-Eilenberg algebra is noted C ∗ CE ( g )and its Chevalley-Eilenberg coalgebra is noted C CE ∗ ( g ). • More general categories of algebras and coalgebras over operads or proper-ads will have the following generic notations: given a properad P , we willnote P − Alg the category of dg P -algebras and given an operad P we willnote P − Cog the category of dg P -coalgebras. • Given a properad P , a cofibrant resolution of P is noted P ∞ . • When the base category is a symmetric monoidal category C other than Ch K , we note P − Alg ( C ) the category of P -algebras in C and P − Cog ( C )the category of P -coalgebras in C . • Algebras over properads form a relative category for the weak equivalencesdefined by chain quasi-isomorphisms. The subcategory of weak equivalencesof P − Alg is noted wP − Alg . • Given a properad P and a complex X , we will consider an associated con-volution Lie algebra noted g P,X which will give rise to two deformationcomplexes: the deformation complex g ϕP,X controling the formal moduliproblem of deformations of a P -algebra structure ϕ on X , and a variant g ϕ + P + ,X whose role will be explained in Section 3. • We will consider various moduli functors in this paper, defined as simpli-cial presheaves over Artinian augmented cdgas: the simplicial presheaf of P ∞ -algebra structures on X noted P ∞ { X } , the formal moduli problem ofdeformations of a given P ∞ -algebra structure ϕ on X noted P ∞ { X } ϕ , andthe derived prestack group of homotopy automorphisms of ( X, ϕ ) noted haut P ∞ ( X, ϕ ). The derived prestack group of automorphisms of X as achain complex will be denoted haut ( X ).1. Recollections
The goal of this section is to briefly review several key notions and results frommodel categories and props that will be used in the present paper.
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 12
Symmetric monoidal categories over a base category.
Symmetric monoidalcategories over a base category formalize how a given symmetric monoidal categorycan be tensored and enriched over another category, in a way compatible with themonoidal structure:
Definition 1.1.
Let C be a symmetric monoidal category. A symmetric monoidalcategory over C is a symmetric monoidal category ( E , ⊗ E , E ) endowed with a sym-metric monoidal functor η : C → E , that is, an object under C in the 2-category ofsymmetric monoidal categories.This defines on E an external tensor product ⊗ : C×E → E by C ⊗ X = η ( C ) ⊗ E X for every C ∈ C and X ∈ E . This external tensor product is equipped with thefollowing natural unit, associativity and symmetry isomorphisms:(1) ∀ X ∈ E , C ⊗ X ∼ = X ,(2) ∀ X ∈ E , ∀ C, D ∈ C , ( C ⊗ D ) ⊗ X ∼ = C ⊗ ( D ⊗ X ),(3) ∀ C ∈ C , ∀ X, Y ∈ E , C ⊗ ( X ⊗ Y ) ∼ = ( C ⊗ X ) ⊗ Y ∼ = X ⊗ ( C ⊗ Y ).The coherence constraints of these natural isomorphisms (associativity pen-tagons, symmetry hexagons and unit triangles which mix both internal and externaltensor products) come from the symmetric monoidal structure of the functor η .We will implicitly assume throughout the paper that all small limits and smallcolimits exist in C and E , and that each of these categories admit an internalhom bifunctor. We suppose moreover the existence of an external hom bifunctor Hom E ( − , − ) : E op × E → C satisfying an adjunction relation ∀ C ∈ C , ∀ X, Y ∈ E , M or E ( C ⊗ X, Y ) ∼ = M or C ( C, Hom E ( X, Y ))(so E is naturally an enriched category over C ).Throughout this paper we will deal with symmetric monoidal categories equippedwith a model structure. We assume that the reader is familiar with the basics ofmodel categories. We refer to to Hirschhorn [50] and Hovey [49] for a compre-hensive treatment of homotopical algebra. We just recall the axioms of symmetricmonoidal model categories formalizing the interplay between the tensor and themodel structures (in a word, these conditions ensure that the tensor product formsa Quillen bifunctor). From the point of view of ∞ -categories, if a model category isequipped with a compatible symmetric monoidal structure (that is, satisfying theconditions below), then its associated ∞ -category is symmetric monoidal as well(as an ∞ -category). Definition 1.2. (1) A symmetric monoidal model category is a symmetric monoidalcategory C equipped with a model category structure such that the following axiomsholds: MM0.
For any cofibrant object X of C , the map Q C ⊗ X → C ⊗ X ∼ = X induced by a cofibrant resolution Q C → C of the unit 1 C is a weak equivalence. MM1.
The pushout-product ( i ∗ , j ∗ ) : A ⊗ D ⊕ A ⊗ C B ⊗ C → B ⊗ D of cofibrations i : A B and j : C D is a cofibration which is also acyclic as soon as i or j isso.(2) Suppose that C is a symmetric monoidal model category. A symmetricmonoidal category E over C is a symmetric monoidal model category over C ifthe axiom MM1 holds for both the internal and external tensor products of E . Example 1.3.
The usual projective model category Ch K of unbounded chain com-plexes over a field K forms a symmetric monoidal model category. ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 13
A property of the pushout-product axiom MM1 which will be useful later is thatit is equivalent to the following standard dual version:
Lemma 1.4. (cf. [49, Lemma 4.2.2] ) In a symmetric monoidal model category C ,the axiom MM1 is equivalent to the following one: MM1’.
The morphism ( i ∗ , p ∗ ) : Hom C ( B, X ) → Hom C ( A, X ) × Hom C ( A,Y ) Hom C ( B, Y ) induced by a cofibration i : A B and a fibration p : X ։ Y is a fibration in C which is also acyclic as soon as i or p is so. Props, properads and their algebras.
Props generalize operads, so thatalgebras over props can be defined by operations with multiple outputs, contraryto operads which parametrize only operations with one single output. For thisreason, the formalism of props is particularly adapted to the study of bialgebra-like structures. Properads are an intermediate object between operads and props,which are close enough to operads in the sense that they are defined, like operads, asmonoids in a category of symmetric sequences (contrary to props), but are sufficientto encode several interesting bialgebra-like structures. Contrary to props, they fitwell in a theory of bar-cobar constructions and Koszul duality which is useful to getexplicit resolutions in deformation theory of algebraic structures. We detail someof these ideas below.1.2.1.
Props and their algebras.
Let C be a symmetric monoidal category. A Σ-biobject is a double sequence { M ( m, n ) ∈ C} ( m,n ) ∈ N where each M ( m, n ) isequipped with a right action of Σ m and a left action of Σ n commuting with eachother. Definition 1.5.
A prop is a Σ-biobject endowed with associative horizontal com-position products ◦ h : P ( m , n ) ⊗ P ( m , n ) → P ( m + m , n + n ) , associative vertical composition products ◦ v : P ( k, n ) ⊗ P ( m, k ) → P ( m, n )and units 1 → P ( n, n ) which are neutral for ◦ v . These products satisfy the exchangelaw ( f ◦ h f ) ◦ v ( g ◦ h g ) = ( f ◦ v g ) ◦ h ( f ◦ v g )and are compatible with the actions of symmetric groups. The elements of P ( m, n )are said to be of arity ( m, n ).Morphisms of props are equivariant morphisms of collections compatible withthe composition products.There is a functorial free prop construction F leading to an adjunction F : C S ⇄ P rop : U with the forgetful functor U . Also, like in the case of operads, there is a notion ofideal in a prop, so that one can define a prop by generators and relations. Thisapproach is particularly useful considering the definition of algebras over a prop: ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 14
Definition 1.6. (1) To any object X of C we can associate an endomorphism prop End X defined by End X ( m, n ) = Hom C ( X ⊗ m , X ⊗ n ) . (2) A P -algebra is an object X ∈ C equipped with a prop morphism P → End X .Operations of P are sent to operations on tensor powers of X , and the com-patibility of a prop morphism with composition products on both sides impose therelations that such operations satisfy. This means that given a presentation of aprop P by generators and relations, the P -algebra structure on X is determinedby the images of these generators and their relations. Let us give some motivatingexamples related to our article: Example 1.7.
A differential graded associative and coassociative bialgebra is atriple (
B, µ, ∆) such that:(i) (
B, µ ) is a dg associative algebra;(ii) ( B, ∆) is a dg coassociative coalgebra;(iii) the map ∆ : B → B ⊗ B is a morphism of algebras and the map µ : B ⊗ B → B is a morphism of coalgebras. That is, bialgebras can be defined equivalently ascoalgebras in algebras or algebras in coalgebras.We describe by generators and relations the prop encoding such bialgebras. Theprop Bialg of associative-coassocative bialgebras is generated by the two degreezero operations ❏❏ tt tt ❏❏ where the graphs have to be read from top to bottom, so we have one generator ofarity (2 ,
1) and one generator of arity (1 , ✿✿✿ ✿✿✿ ☎☎☎ ✿✿✿ ☎☎☎ ☎☎☎✿✿✿ ☎☎☎ − ✿✿✿ ☎☎☎ ☎☎☎ ✿✿✿ − ☎☎☎ ✿✿✿③③ ❉❉ ❉❉ ❉❉③③ ❉❉ Compatibility relation ✿✿✿ ☎☎☎ − ☎☎☎ ❄❄❄❄❄⑧⑧⑧⑧⑧ ✿✿✿❉❉ ③③③③ ❉❉ In the unitary and counitary case, one adds a generator for the unit, a generatorfor the counit and the necessary compatibility relations with the product and thecoproduct. We note the corresponding properad
Bialg . Example 1.8.
Lie bialgebras originate from mathematical physics, in the studyof integrable systems whose gauge groups are not only Lie groups but Poisson-Liegroups. A Poisson-Lie group is a Poisson manifold with a Lie group structure suchthat the group operations are morphisms of Poisson manifolds, that is, compatiblewith the Poisson bracket on the ring of smooth functions. The tangent space at theneutral element then inherits a Lie cobracket from the Poisson bracket on smoothfunctions which satisfies some compatibility relation with the Lie bracket. Liebialgebras are used to build quantum groups and appeared for the first time in theseminal work of Drinfeld [16].
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 15
The prop
BiLie encoding Lie bialgebras is generated by1 ❈❈ ④④ ④④ ❈❈ , that is, they areantisymmetric. It is quotiented by the ideal generated by the following relationsJacobi 1 ❈❈ ❈❈ ④④❑❑❑ sss + 3 ❈❈ ❈❈ ④④❑❑❑ sss + 2 ❈❈ ❈❈ ④④❑❑❑ sss co-Jacobi sss ❑❑❑④④ ❈❈ ❈❈ sss ❑❑❑④④ ❈❈ ❈❈ sss ❑❑❑④④ ❈❈ ❈❈ ❈❈ ④④④④ ❈❈ − ❖❖❖ ♦♦♦ ❖❖❖ ❖❖❖ ♦♦♦ ❖❖❖ − ♦♦♦ ❖❖❖ ♦♦♦ ♦♦♦ ❖❖❖ ♦♦♦ g is a cocyclein the Chevalley-Eilenberg complex C ∗ CE ( g, Λ g ), where Λ g is equipped with thestructure of g -module induced by the adjoint action.We can also define a P -algebra in a symmetric monoidal category over C : Definition 1.9.
Let E be a symmetric monoidal category over C .(1) The endomorphism prop of X ∈ E is given by End X ( m, n ) = Hom E ( X ⊗ m , X ⊗ n )where Hom E ( − , − ) is the external hom bifunctor of E .(2) Let P be a prop in C . A P -algebra in E is an object X ∈ E equipped with aprop morphism P → End X .This definition will be useful, for instance, in the case where P is a dg prop (aprop in Ch K ) but algebras over P lie in a symmetric monoidal category over Ch K .To conclude, props enjoy nice homotopical properties. Indeed, the category ofΣ-biobjects C S is a diagram category over C , so it inherits the usual projective modelstructure of diagrams, which can be transferred along the free-forgetful adjunction: Theorem 1.10. (cf. [32, Theorem 5.5] ) The category of dg props
P rop equippedwith the classes of componentwise weak equivalences and componentwise fibrationsforms a cofibrantly generated model category.
Properads.
Composing operations of two Σ-biobjects M and N amounts toconsider 2-levelled directed graphs (with no loops) with the first level indexed byoperations of M and the second level by operations of N . Vertical composition bygrafting and horizontal composition by concatenation allows one to define props asbefore. The idea of properads is to mimick the construction of operads as monoidsin Σ-objects, by restricting the vertical composition product to connected graphs.The unit for this connected composition product ⊠ c is the Σ-biobject I given by I (1 ,
1) = K and I ( m, n ) = 0 otherwise. The category of Σ-biobjects then forms asymmetric monoidal category ( Ch SK , ⊠ c , I ). ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 16
Definition 1.11.
A dg properad (
P, µ, η ) is a monoid in ( Ch SK , ⊠ c , I ), where µ denotes the product and η the unit. It is augmented if there exists a morphism ofproperads ǫ : P → I . In this case, there is a canonical isomorphism P ∼ = I ⊕ P where P = ker ( ǫ ) is called the augmentation ideal of P .Morphisms of properads are morphisms of monoids in ( Ch SK , ⊠ c , I ).Properads have also their dual notion, namely coproperads: Definition 1.12.
A dg coproperad ( C, ∆ , ǫ ) is a comonoid in ( Ch SK , ⊠ c , I ).As in the prop case, there exists a free properad functor F forming an adjunction F : Ch SK ⇄ P roperad : U with the forgetful functor U [92]. Dually, there exists a cofree coproperad functordenoted F c ( − ) having the same underlying Σ-biobject. Moreover, according to [71],this adjunction equips dg properads with a cofibrantly generated model categorystructure with componentwise fibrations and weak equivalences. There is also anotion of algebra over a properad similar to an algebra over a prop, since theendomorphism prop restricts to an endomorphism properad. Moreover, properadsalso form a model category for the same reasons as props: Theorem 1.13. (cf. [71, Appendix A] ) The category of dg props
P rop equippedwith the classes of componentwise weak equivalences and componentwise fibrationsforms a cofibrantly generated model category.
Properads are general enough to encode a wide range of bialgebra structuressuch as associative and coassociative bialgebras, Lie bialgebras, Poisson bialgebras,Frobenius bialgebras for instance. A main advantage of properads compared toprops lies in the existence of bar, cobar constructions and Koszul duality for suchobjects, which have as direct application the construction of explicit cofibrant reso-lutions in the model category of properads. Such resolutions are in particular usefulto describe the up to homotopy versions of the aforementioned algebraic structures.1.3.
Algebras and coalgebras over operads.
Operads are used to parametrizevarious kind of algebraic structures consisting of operations with one single output.Fundamental examples of operads include the operad As encoding associative alge-bras, the operad Com of commutative algebras, the operad
Lie of Lie algebras andthe operad
P ois of Poisson algebras. Dg operads form a model category with bar-cobar resolutions and Koszul duality [61]. An algebra X over a dg operad P can bedefined in any symmetric monoidal category E over Ch K , alternatively as an algebraover the corresponding monad P ( − ) : Ch K → Ch K , which forms the free P -algebrafunctor, or as an operad morphism P → End X where End X ( n ) = Hom E ( X ⊗ n , X )and Hom E is the external hom bifunctor. Remark . There is a free functor from operads to props, so that algebras overan operad are exactly the algebras over the corresponding prop. Hence algebrasover props include algebras over operads as particular cases.Dual to operads is the notion of cooperad, defined as a comonoid in the cate-gory of Σ-objects. A coalgebra over a cooperad is a coalgebra over the associatedcomonad. We can go from operads to cooperads and vice-versa by dualization.Indeed, if C is a cooperad, then the Σ-module P defined by P ( n ) = C ( n ) ∗ = Hom K ( C ( n ) , K ) form an operad. Conversely, suppose that K is of characteristic ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 17 zero and P is an operad such that each P ( n ) is finite dimensional. Then the P ( n ) ∗ form a cooperad in the sense of [61]. The additional hypotheses are needed becausewe have to use, for finite dimensional vector spaces V and W , the isomorphism( V ⊗ W ) ∗ ∼ = V ∗ ⊗ W ∗ to define properly the cooperad coproduct. We also give thedefinition of coalgebras over an operad: Definition 1.15. (1) Let P be an operad. A P -coalgebra is a complex C equipedwith linear applications ρ n : P ( n ) ⊗ C → C ⊗ n for every n ≥
0. These maps areΣ n -equivariant and associative with respect to the operadic compositions.(2) Each p ∈ P ( n ) gives rise to a cooperation p ∗ : C → C ⊗ n . The coalgebra C is usually said to be conilpotent if for each c ∈ C , there exists N ∈ N so that p ∗ ( c ) = 0 when we have p ∈ P ( n ) with n > N .If K is a field of characteristic zero and the P ( n ) are finite dimensional, thenit is equivalent to define a P -coalgebra via a family of applications ρ n : C → P ( n ) ∗ ⊗ Σ n C ⊗ n .1.4. Homotopy algebras.
Given a prop, properad or operad P , a homotopy P -algebra, or P -algebra up to homotopy, is an algebra for which the relations arerelaxed up to a coherent system of higher homotopies: Definition 1.16.
A homotopy P -algebra is an algebra over a cofibrant resolution P ∞ of P .Properads have a well defined theory of bar-cobar constructions and Koszulduality [92], which allows to produce cofibrant resolutions of properads. The bar-cobar resolution is a functorial cofibrant resolution but of a rather big size, whereasthe resolution obtained from the Koszul dual (when P is Koszul) is not functorialbut smaller and better suited for computations. These resolutions are of the form P ∞ = ( F ( V ) , ∂ ) where ∂ is a differential obtained by summing the differentialinduced by the Σ-biobject V with a certain derivation. When P is concentratedin degree zero, all the higher homology groups of P ∞ vanish, so that H ∗ P ∞ ∼ = P .To sum up, a homotopy P -algebra is an algebra over a quasi-free resolution of P ,in which the generators give the system of higher homotopies and the relationsdefining a strict P -algebra become coboundaries.To make this definition meaningful, one has to prove that the notion of homotopy P -algebra does not depend (up to homotopy) on a choice of resolution: Theorem 1.17. (cf. [96] ) A weak equivalence of cofibrant dg props P ∞ ∼ → Q ∞ induces an equivalence of the corresponding ∞ -categories of algebras P ∞ − Alg [ W − qiso ] ∼ → Q ∞ − Alg [ W − qiso ] , where P ∞ − Alg [ W − qiso ] denotes the ∞ -categorical localization of P ∞ − Alg withrespect to its subcategory of quasi-isomorphisms.
For algebras over operads, this was already a well-known result formulated asa Quillen equivalence of model categories, but in the case of algebras over propsthe problem is far more subtle because of the absence of model category structureon such algebras, and requires different methods based on simplicial homotopy and ∞ -category theory [95, 96].Homotopy algebras are central to deformation theory, since the deformationsof algebraic structures in a differential graded setting appear to be naturally up ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 18 to homotopy. For example, the Hochschild complex of a dg associative algebra A controls the deformations of A as an A ∞ -algebra . So the coherence theorem abovegives a meaning to the deformation theory of dg algebras over props, that will becentral in our paper.2. Derived deformation theory of algebraic structures
A brief preliminary about cdgas.
Before getting to the heart of the sub-ject, let us precise that, as usual in deformation theory and (derived) algebraicgeometry, the commutative differential algebras (cdga for short) that we considerhere are unital. That is, we consider the category of unital commutative monoids inthe symmetric monoidal model category Ch K and note it CDGA K . Such monoidsenjoy many useful homotopical properties, as they form a homotopical algebra con-text in the sense of [87, Definition 1.0.1.11]. We will not list all the propertiessatisfied by cdgas, but here is a non-exhaustive one that will be useful in thisarticle:(1) The category CDGA K forms a cofibrantly generated model category withfibrations and weak equivalences being the degreewise surjections and quasi-isomorphisms.(2) Given a cdga A , its category of dg A -modules M od A forms a cofibrantlygenerated symmetric monoidal model category. The model structure is,again, right induced by the forgetful functor, and the tensor product isgiven by − ⊗ A − . In particular, we have a Quillen adjunction( − ) ⊗ A : Ch K ⇆ M od A : U with a strong monoidal left adjoint (hence the right adjoint is lax monoidal).The unit η of this adjunction is defined, for any complex X , by the chainmorphism η ( X ) : X → X ⊗ Ax x ⊗ A where 1 A is the unit element of A (the image of 1 K by the unit map of A ).(3) Base changes are compatible with the homotopy theory of modules. Pre-cisely, a morphism of cdgas f : A → B induces a Quillen adjunction f ! : M od A ⇆ M od B : f ∗ where f ∗ equip a B -module with the A -module structure induced by themorphism f and f ! = ( − ) ⊗ A B . Moreover, if f is a quasi-isomorphism ofcdgas then this adjunction becomes a Quillen equivalence.(4) The category of augmented cdgas CDGA aug K is the category under K asso-ciated to CDGA K , so it forms also a cofibrantly generated model category.Moreover, this model category is pointed with K as initial and terminal ob-ject, so that one can alternately call them pointed cdgas. Let us note alsothat augmented unital cdgas are equivalent to non-unital cdgas CDGA nu K via the Quillen equivalence( − ) + : CDGA nu K ⇆ CDGA aug K : ( − ) − where A + = A ⊕ K for A ∈ CDGA nu K and A − is the kernel of the augmen-tation map of A for A ∈ CDGA aug K . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 19
There is a simplicial cdga called the
Sullivan cdga of polynomial forms on thestandard simplices . It is given by(2.1) Ω n := Sym n M i =0 (cid:0) K t i ⊕ K dt i (cid:1)! / (cid:18) t + · · · + t n = 1 dt + · · · + dt n = 0 (cid:19) which is precisely the algebra of piecewise linear forms on the standard simplex ∆ n ,the differential being in particular defined, on the generators, by d ( t i ) = dt i . Thesimplicial structure is induced by the cosimplicial structure of n ∆ n , see [82] fordetails.For any cdga A , the category M od A of left dg A -modules is a (cofibrantly gener-ated) symmetric monoidal model category tensored over chain complexes. There-fore one can define the category P ∞ − Alg ( M od A ) of P ∞ -algebras in M od A , forany cofibrant prop P ∞ as in section 1.4 and Theorem 1.17 extends to this context.An important subcategory of augmented cdgas is the one of artinian algebras,which are the coaffine formal moduli problems. Definition 2.1.
An augmented cdga A is Artinian if • its cohomology groups H n ( A ) vanish for n > n <<
0, and eachof them is finite dimensional over k ; • the (commutative) ring H ( A ) is artinian in the standard meaning of com-mutative algebra.We denote dgArt aug K the full subcategory of CDGA aug K of Artinian cdgas.2.2. Relative categories versus ∞ -categories. There are many equivalent waysto model ∞ -categories. Precisely, there are several Quillen equivalent models for ∞ -categories we can choose to work with [7], for instance quasi-categories [66],complete Segal spaces [80], simplicial categories [6], or relative categories [3, 4]. Inthis paper, it will often be convenient to consider ∞ -functors which are associatedto “naive” functors, provided-of course-that they preserve weak equivalences. Thisis not necessarily posible to do that in a straightforward naive way depending on themodel chosen for ∞ -categories. Therefore, here, we choose to work in the homotopytheory of relative categories as developed recently by Barwick-Kan [3, 4]. Thiswill allow us to define more easily ∞ -functors starting from classical constructions,instead of going through, for instance, the cartesian fibration/opfibration formalismof [66]. For the sake of clarity, we start by recalling the main features of this theoryand refer to [3, 4] for more details. Then we state some technical lemmas that willhelp us to go from equivalences of relative categories to equivalences of ∞ -categories.2.2.1. ∞ -categories associated to relative categories or model categories. We nowrecall and compare various standard ways to construct ∞ -categories. Definition 2.2.
A relative category is a pair of categories ( C , W C ) such that W C is a subcategory of C containing all the objects of C . We call W C the category ofweak equivalences of C . A relative functor between two relative categories ( C , W C )and ( D , W D ) is a functor F : C → D such that F ( W C ) ⊂ W D . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 20
We note
RelCat the category of relative categories and relative functors. ByTheorem 6.1 of [3], there is an adjunction between the category of bisimplicial setsand the category of relative categories K ξ : sSets ∆ op ⇆ RelCat : N ξ (where K ξ is the left adjoint and N ξ the right adjoint) which lifts any Bousfieldlocalization of the Reedy model structure of bisimplicial sets into a model structureon RelCat . In the particular case of the Bousfield localization defining the completeSegal spaces [80], one obtains a Quillen equivalent homotopy theory of the homotopytheories in
RelCat [3]. In particular, a morphism of relative categories is a weakequivalence if and only if its image under N ξ is a weak equivalence of complete Segalspaces. We refer the reader to Section 5.3 of [3] for the definition of the functor N ξ .Let us just mention that it is weekly equivalent to the classifying diagram functor N defined in [80], which is a key tool to construct complete Segal spaces.A simplicial category is a category enriched over simplicial sets. We denote by SCat the category of simplicial categories. There exists functorial cosimplicial res-olutions and simplicial resolutions in any model category ([20],[50]), so model cat-egories provide examples of (weakly) simplicially enriched categories. One recoversthe morphisms of the homotopy category from a cofibrant object to a fibrant objectby taking the set of connected components of the corresponding simplicial mappingspace. Another more general example is the simplicial localization developed byDwyer and Kan [18]. To any relative category Dwyer and Kan associates a simplicialcategory L ( C , W C ) called its simplicial localization. They developed also anothersimplicial localization, the hammock localization L H ( C , W C ) [19]. By taking thesets of connected components of the mapping spaces, we get π L ( C , W C ) ∼ = C [ W − C ]where C [ W − C ] is the localization of C with respect to W C (i.e. the homotopy cat-egory of ( C , W C )). The simplicial and hammock localizations are equivalent in thefollowing sense: Proposition 2.3. (Dwyer-Kan [19] , Proposition 2.2) Let ( C , W C ) be a relativecategory. There is a zigzag of Dwyer-Kan equivalences L H ( C , W C ) ← diagL H ( F ∗ C , F ∗ W C ) → L ( C , W C ) where F ∗ C is a simplicial category called the standard resolution of C (see [18] Section 2.5).
Let us precise the definition of Dwyer-Kan equivalences:
Definition 2.4.
Let C and D be two simplicial categories. A functor F : C →D is a Dwyer-Kan equivalence if it induces weak equivalences of simplicial sets
M ap C ( X, Y ) ∼ → M ap D ( F X, F Y ) for every
X, Y ∈ C , as well as inducing an equiv-alence of categories π C ∼ → π D .Let us compile some useful results: first, every Quillen equivalence of model cate-gories gives rise to a Dwyer-Kan equivalence of their simplicial localizations, as wellas a Dwyer-Kan equivalence of their hammock localizations (see [20] Proposition5.4 in the case of simplicial model categories and [48] in the general case). By The-orem 1.1 of [6], there exists a model category structure on the category of (small)simplicial categories with the Dwyer-Kan equivalences as weak equivalences. Everysimplicial category is Dwyer-Kan equivalent to the simplicial localization of a cer-tain relative category (see for instance [4], Theorem 1.7) and the associated model ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 21 structure is also a homotopy theory of homotopy theories. The Reedy weak equiva-lences between two complete Segal spaces are precisely the Dwyer-Kan equivalencesbetween their associated homotopy theories (Theorem 7.2 of [80]).Therefore the ∞ -category associated to a relative category is thus, equivalently,the ∞ -category associated to its simplicial localization or the ∞ -category associ-ated to its corresponding complete Segal space. The same construction applies toturn relative functors into ∞ -functors. Moreover, it can be made functorial. Forinstance, given a relative category ( C , W C ), the associated quasi-category is givenby the composite N coh L H ( C , W C ) f , where L H ( − ) is the Dwyer-Kan localizationfunctor, ( − ) f is a functorial fibrant resolution in the Bergner model structure [6],and N coh is the coherent nerve. In the following, given a relative category ( M, W ),where W is the subcategory of weak equivalences, we will denote by M [ W − ] its ∞ -categorical localization.2.2.2. From relative categories to homotopy automorphisms.
We now collect twolemmas that will be useful in the next section to obtain equivalences between ∞ -categories of algebras, and see under which conditions they induce equivalencesbetween the formal moduli problems controlling deformations of such algebras: Lemma 2.5.
Let F : ( C , W C ) ⇄ ( D , W D ) : G be an adjunction of relative categories(that is, the functors F and G preserves weak equivalences) such that the unitand counit of this adjunction are pointwise weak equivalences. Then F induces anequivalence of ∞ -categories with inverse G .Proof. Let us denote by
RelCat the category of relative categories. The objectsare the relative categories and the morphisms are the relative functors, that is, thefunctors restricting to functors between the categories of weak equivalences. By [3,Theorem 6.1], there is an adjunction between the category of bisimplicial sets andthe category of relative categories K ξ : sSets ∆ op ⇆ RelCat : N ξ (where K ξ is the left adjoint and N ξ the right adjoint) which lifts any Bousfieldlocalization of the Reedy model structure of bisimplicial sets into a model structureon RelCat . In the particular case of the Bousfield localization defining the modelcategory
CSS of complete Segal spaces [80, Theorem 7.2], one obtains a Quillenequivalent homotopy theory of ∞ -categories in RelCat [3].As recalled in 2.2.1, a way to build the ∞ -category associated to a relative cate-gory ( C , W C ) is to take a functorial fibrant resolution N ξ ( C , W C ) f of the bisimplicialset N ξ ( C , W C ) in CSS to get a complete Segal space. So we want to prove that N ξ F f is a weak equivalence of CSS . For this, let us note first that the assumptionon the adjunction between F and G implies that F is a strict homotopy equiva-lence in RelCat in the sense of [3]. By [3, Proposition 7.5 (iii)], the functor N ξ preserves homotopy equivalences, so N ξ F is a homotopy equivalence of bisimpli-cial sets, hence a Reedy weak equivalence. Since CSS is a Bousfield localizationof the Reedy model structure on bisimplicial sets, Reedy weak equivalences areweak equivalences in
CSS , then by applying the fibrant resolution functor ( − ) f weconclude that N ξ F f is a weak equivalence of complete Segal spaces. (cid:3) In the formalism of Dwyer-Kan’s hammock localization, an equivalence of sim-plicial categories F : C → D satisfies in particular the following property: for every
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 22 two objects X and Y of C , it induces a weak equivalence of simplicial mappingspaces L H ( C , W C )( X, Y ) ∼ → L H ( D , W D )( F ( X ) , F ( Y )) . (in particular, the associated functor Ho ( F ) at the level of homotopy categoriesis an equivalence). We would like this weak equivalence to restrict at the level ofhomotopy automorphisms: Lemma 2.6.
Let F : ( C , W C ) ⇄ ( D , W D ) : G be an adjunction of relative cate-gories satisfying the assumptions of Lemma 2.5. Then the restriction of F to thesubcategories of weak equivalences wF : W C → W D is an equivalence of simplicial localizations (actually an equivalence of ∞ -groupoids)inducing a weak equivalence of homotopy automorphisms L H W C ( X, X ) ∼ → L H W D ( F ( X ) , F ( X )) , where L H W C is Dwyer-Kan’s hammock localization of W C with respect to itself.Proof. This adjunction of relative categories induces, by Lemma 2.5, an equivalenceof simplicial localizations between L H ( C , W C ) and L H ( D , W D ). By construction,this implies that the simplicial categories L H W C and L H W D are equivalent aswell. Alternately, one could say that an equivalence of ∞ -categories induce anequivalence of the associated ∞ -groupoids of weak equivalences. By definition ofan equivalence of simplicial categories, we get the desired equivalence between thesimplicial mapping spaces of L H W C and their images under F in L H W D (that is,an equivalence of homotopy automorphisms). (cid:3) Formal moduli problems and (homotopy) Lie algebras.
Formal moduliproblems arise when one wants to study the deformation theory of an object in acategory, of a structure on a given object, of a point in a given moduli space (variety,scheme, stack, derived stack). The general principle of moduli problems is that thedeformation theory of a given point in its formal neighbourhood (that is, the formalcompletion of the moduli space at this point) is controlled by a certain tangent dgLie algebra. This idea of a correspondence between formal moduli problems anddg Lie algebras arose from unpublished work of Deligne, Drinfed and Feigin, andwas developed further by Goldman-Millson, Hinich, Kontsevich, Manetti amongothers. However, there was no systematic recipe to build a dg Lie algebra fora given moduli problem, and even worse, different dg Lie algebras can representthe same (underived) moduli problem (a famous example is the construction oftwo non quasi-isomorphic dg Lie algebras for the deformation problem of a closedsubscheme, seen either as a point of a Hilbert scheme or as a point of a Quotscheme). To overcome these difficulties, one has to consider moduli problems in aderived setting. The rigorous statement of an equivalence between (derived) formalmoduli problems and dg Lie algebras was proved independently by Lurie in [64] andby Pridham in [78]. In this paper, what we will call moduli problems are actuallyderived moduli problems.
Definition 2.7.
Formal moduli problems are functors F : dgArt aug K → sSet fromaugmented Artinian commutative differential graded algebras to simplicial sets sat-isfying the following conditions: ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 23 (1) The functor F preserves weak equivalences (that is, quasi-isomorphisms ofcdgas are sent to weak equivalences of simplicial sets);(2) There is a weak equivalence F ( K ) ≃ pt ;(3) The functor F is infinitesimally cohesive : Given any (homotopy) pull-back diagram A ′ / / (cid:15) (cid:15) A (cid:15) (cid:15) B ′ / / B in dgArt K such that the induced maps π ( A ) → π ( B ) and π ( B ′ ) → π ( B ) are surjective, the induced diagram F ( A ′ ) / / (cid:15) (cid:15) F ( A ) (cid:15) (cid:15) F ( B ′ ) / / F ( B )is a (homotopy) pullback in sSet . Remark . Condition (3) is a derived version of the classical
Schlessinger condition introduced in [64] and developped in [65] The notion of (infinitesimally) cohesivegeneralizes to any functor from connective dg-commutative algebras to sSet . In thatgeneral setting functors satisfying condition (3) in Definition 2.7 are called cohesive,while infinitesimally cohesive stands for those functors satisfying this condition onlywhen the the maps π ( A ) → π ( B ) and π ( B ′ ) → π ( B ) are further required tohave nilpotent kernels. For Artinian cdgas, the latter condition is automatic andtherefore infinitesimally cohesive and cohesive are the same. We stick to the longername to recall that special property of the Artinian context.The value F ( K ) corresponds to the point of which we study the formal neigh-bourhood, the evaluation F ( K [ t ] / ( t )) on the algebra of dual numbers encodesinfinitesimal deformations of this point, and the F ( K [ t ] / ( t n )) are polynomial de-formations of a higher order, for instance. Formal moduli problems form a fullsub- ∞ -category noted F M P K of the ∞ -category of simplicial presheaves over aug-mented Artinian cdgas. By [64, Theorem 2.0.2], this ∞ -category is equivalent tothe ∞ -category dgLie K of dg Lie algebras. Moreover, one side of the equivalence ismade explicit, and is equivalent to the nerve construction of dg Lie algebras studiedthoroughly by Hinich in [46]. The homotopy invariance of this nerve relies on nilpo-tence conditions on the dg Lie algebra. In the case of formal moduli problems, thisnilpotence condition is always satisfied because one tensors the Lie algebra withthe maximal ideal of an augmented Artinian cdga.It turns out that this nerve construction can be extended to homotopy Lie alge-bras, that is, L ∞ -algebras: Definition 2.9. (1) An L ∞ -algebra is a graded vector space g = { g n } n ∈ Z equippedwith maps l k : g ⊗ k → g of degree 2 − k , for k ≥
1, satisfying the following properties: • l k ( ..., x i , x i +1 , ... ) = − ( − | x i || x i +1 | l k ( ..., x i +1 , x i , ... ) • for every k ≥
1, the generalized Jacobi identities k X i =1 X σ ∈ Sh ( i,k − i ) ( − ǫ ( i ) l k ( l i ( x σ (1) , ..., x σ ( i ) ) , x σ ( i +1) , ..., x σ ( k ) ) = 0 ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 24 where σ ranges over the ( i, k − i )-shuffles and ǫ ( i ) = i + X j
3. Note that in this case, we have that (
Sym •≥ ( g [1]) , Q ) isexactly the reduced Chevalley-Eilenberg complex computing the homology of theLie algebra g .This last remark justifies to call the dg-coalgebra of (2) the reduced Chevalley-Eilenberg chain complex of the L ∞ -algebra g , denoted C CE ∗ ( g ). The dg-algebra C ∗ CE ( g ) obtained by dualizing the dg coalgebra of (2) is called the (reduced) Chevalley-Eilenberg cochain algebra of g . Definition 2.11. A L ∞ algebra g is filtered if it admits a decreasing filtration g = F g ⊇ F g ⊇ ... ⊇ F r g ⊇ ... compatible with the brackets: for every k ≥ l k ( F r g, g, ..., g ) ∈ F r g. We require moreover that for every r , there exists an integer N ( r ) such that l k ( g, ..., g ) ⊆ F r g for every k > N ( r ).A filtered L ∞ algebra g is complete if the canonical map g → lim r g/F r g is anisomorphism.In particular a nilpotent L ∞ -algebra is complete and, if m is the augmentationideal of an Artinian CDGA, then g ⊗ m is also complete for any L ∞ -algebra g .The completeness of a L ∞ algebra allows to define properly the notion of Maurer-Cartan element: Definition 2.12. (1) Let g be a complete L ∞ -algebra and τ ∈ g , we say that τ is a Maurer-Cartan element of g if X k ≥ k ! l k ( τ, ..., τ ) = 0 . The set of Maurer-Cartan elements of g is noted M C ( g ).(2) The simplicial Maurer-Cartan set is then defined by M C • ( g ) = M C ( g ˆ ⊗ Ω • ) , , where Ω • is the Sullivan cdga of de Rham polynomial forms on the standardsimplex ∆ • (see 2.1 and [82]) and ˆ ⊗ is the completed tensor product with respectto the filtration induced by g . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 25
The simplicial Maurer-Cartan set is a Kan complex, functorial in g and preservesquasi-isomorphisms of complete L ∞ -algebras. The Maurer-Cartan moduli set of g is MC ( g ) = π M C • ( g ): it is the quotient of the set of Maurer-Cartan elements of g by the homotopy relation defined by the 1-simplices. When g is a complete dgLie algebra , it turns out that this homotopy relation is equivalent to the action ofthe gauge group exp ( g ) (a prounipotent algebraic group acting on Maurer-Cartanelements), so in this case this moduli set coincides with the one usually known forLie algebras. We refer the reader to [96] for more details about all these results.We also recall the notion of Definition 2.13 (Twisting by a Maurer-Cartan element) . . The twisting of acomplete L ∞ algebra g by a Maurer-Cartan element τ is the complete L ∞ algebra g τ with the same underlying graded vector space and new brackets l τk defined by l τk ( x , ..., x k ) = X i ≥ i ! l k + i ( τ, ..., τ | {z } i , x , ..., x k )where the l k are the brackets of g .Let us explain briefly why Lurie’s equivalence [64, Theorem 2.0.2] lifts from the ∞ -category of dg Lie algebras dgLie to the ∞ -category of L ∞ -algebras L ∞ − Alg .Let p : L ∞ ∼ → Lie be the cofibrant resolution of the operad
Lie encoding L ∞ -algebras. This morphism induces a functor p ∗ : dgLie → L ∞ − Alg which associatesto any dg Lie algebra the L ∞ -algebra with the same differential, the same bracketof arity 2 and trivial higher brackets in arities greater than 2 as in remark 2.10.It turns out that this functor is a right Quillen functor belonging to a Quillenequivalence p ! : L ∞ − Alg ⇆ dgLie : p ∗ , since p is a quasi-isomorphism of Σ-cofibrant operads (see [30, Theorem 16.A] forthe general result). Quillen equivalences induce equivalences of the ∞ -categoriesassociated to these model categories. Therefore, we have a commutative triangle of ∞ -categories L ∞ − Alg ˜ ψ & & ▲▲▲▲▲▲▲▲▲▲ dgLie p ∗ O O ψ / / F M P K where ψ maps a Lie algebra to its nerve functor, and ˜ ψ maps an L ∞ -algebra to itsMaurer-Cartan space defined as dgArt aug K ∋ R M C • ( g ⊗ m R ) (where m R is themaximal ideal of R ).The maps p ∗ and ψ are weak equivalences of ∞ -categories (the model of quasi-categories is used in [66], but actually any model works). By the two-out-of-threeproperty of weak equivalences, this implies that ˜ ψ : L ∞ − Alg → F M P K is a weakequivalence of ∞ -categories. Definition 2.14.
Let F be a formal moduli problem. We denote F
7→ L F ∈ L ∞ − Alg an inverse of the equivalence ˜ ψ : L ∞ − Alg → F M P K .To conclude, let us say a word about formal deformations. Although the ringof formal power series in one variable K [[ t ]] is not Artinian, given a formal moduli ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 26 problem F , one can properly define the notion of formal deformation, or deforma-tion over K [[ t ]], by setting F ( K [[ t ]]) := lim i F ( K [ t ] /t i ) . (where we consider a homotopy limit in the ∞ -category of simplicial sets). Let usnote g F the dg Lie (or L ∞ ) algebra of F via the Lurie-Pridham correspondence.By [91, Corollaire 2.11] (or [64]), there is an natural weak-equivalence F ( K [[ ~ ]]) ≃ M ap ( K [ − , g F )where K [ −
1] is the one dimensional Lie algebra concentrated in degree 1 with trivialLie bracket. Here
M ap denotes the derived mapping space in the ∞ -category of dgLie algebras, which can be explicited in the corresponding model category by takinga cofibrant resolution of K [ − F ( K [[ t ]]) ≃ M C • ( tg F [[ t ]]) . This means that formal deformations of a point can be explicitely described interms of the corresponding Lie algebra.2.4.
Moduli spaces of algebraic structures and their formal moduli prob-lems.
Moduli spaces of algebraic structures were originally defined by Rezk as sim-plicial sets, in the setting of simplicial operads [79]. This notion can be extendedto algebras over differential graded props as follows (see [96]):
Definition 2.15.
Let P ∞ be a cofibrant prop and X be a complex. The modulispace of P ∞ -algebra structures on X is the simplicial set P ∞ { X } defined by P ∞ { X } = M or
P rop ( P ∞ , End X ⊗ Ω • ) , where the prop End X ⊗ Ω • is defined by End X ⊗ Ω • )( m, n ) = End X ( m, n ) ⊗ Ω • and Ω • is the Sullivan cdga of the standard simplex ∆ • (see 2.1).Given a cofibrant properad P ∞ and any properad Q , we will denote M ap
P rop ( P, Q ) :=
M or − P rop ( P, Q ⊗ Ω • ), the mapping space of properads morphisms.This simplicial set enjoys the following key properties, see [96]: Proposition 2.16. (1) The simplicial set P ∞ { X } is a Kan complex and π P ∞ { X } = [ P ∞ , End X ] Ho ( P rop ) is the set of homotopy classes of P ∞ -algebra structures on X .(2) Any weak equivalence of cofibrant props P ∞ ∼ → Q ∞ induces a weak equiva-lence of Kan complexes Q ∞ { X } ∼ → P ∞ { X } . We can extend the moduli space of P ∞ -structure to a simplicial presheaf by basechange from K to any Artinian cdga. ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 27
Definition 2.17.
Let P ∞ be a cofibrant properad and X be a complex. We definea simplicial presheaf P ∞ { X } : dgArt aug K → sSet by the formula P ∞ { X } : A ∈ dgArt aug K P ∞ ⊗ A { X ⊗ A } Mod A where P ∞ ⊗ A { X ⊗ A } Mod A is the mapping space of dg props in A -modules M ap ( P ∞ ⊗ A, End
Mod A X ⊗ A ) and End
Mod A X ⊗ A is the endormorphism prop of X ⊗ A takenin the category of A -modules.In other words, P ∞ { X } ( A ) is the simplicial moduli space of P ∞ -algebra struc-tures on X ⊗ A in the category of A -modules. Indeed, since M od A is tensoredover Ch K , on can make P ∞ act on A -modules either by morphisms of dg props in A -modules from P ∞ ⊗ A to the endomorphism prop defined by the internal hom of M od A , or by morphisms of dg props from P ∞ to the endomorphism prop definedby the external hom of M od A . See for instance [96, Lemma 3.4].By Proposition 2.16, the simplicial set P ∞ { X } ( A ) classifies P ∞ ⊗ A -algebrastructures on X ⊗ A . However, the simplicial presheaf P ∞ { X } is not a formalmoduli problem, since P ∞ { X } ( K ) is in general not contractible.The formal moduli problem P ∞ { X } ψ controlling (a certain type of) formal de-formations of a P ∞ -algebra structure ψ : P ∞ → End X on X is defined, on anyaugmented Artinian cdga A , as the homotopy fiber (2.2) P ∞ { X } ψ ( A ) = hof ib ( P ∞ { X } ( A ) → P ∞ { X } ( K ))taken over the base point ψ , the map being induced by the augmentation A → K .The moduli spaces of algebraic structures gives and its associated formal moduliproblem are encoded by L ∞ -algebras according to Lurie - Pridham Theorem. Wewill now explain how those L ∞ -structures can be described explicitly using dg-properads following [70] and [96].Cofibrant resolutions of a properad P can always be obtained as a cobar con-struction Ω( C ) on some coproperad C (which is usually the bar construction or theKoszul dual if P is Koszul). Given a cofibrant resolution P ∞ := Ω( C ) ∼ → P of P andanother properad Q , one constructs the convolution dg Lie algebra Hom Σ ( C, Q ): Definition 2.18.
Let C be an augmented coproperad and Q be a properad. Theirassociated convolution dg Lie algebra is the dg K -module Hom Σ ( C, Q )of morphisms of Σ-biobjects from the augmentation ideal of C to Q endowed withthe differential induced by the internal ones of C and Q . It is equipped with theLie bracket given by the antisymmetrization of the convolution product.This convolution product is defined similarly to the convolution product of mor-phisms from a coalgebra to an algebra, using the infinitesimal coproduct of C andthe infinitesimal product of Q .The total complex Hom Σ ( C, Q ) is a complete dg Lie algebra. More generally,if P is a properad with minimal model ( F ( s − C ) , ∂ ) ∼ → P for a certain homotopycoproperad C (see [70, Section 4] for the definition of homotopy coproperads), and Q is any properad, then the complex Hom Σ ( C, Q ) is a complete dg L ∞ algebra (which is not a dg-Lie algebra in general).The simplicial mapping space of morphisms P ∞ → Q is computed by the con-volution L ∞ -algebra Hom Σ ( C, Q ) thanks to the foillowing theorem:
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 28
Theorem 2.19. (cf. [96, Theorem 2.10,Corollary 4.21] ) Let P be a dg properadequipped with a minimal model P ∞ := ( F ( s − C ) , ∂ ) ∼ → P and Q be a dg properad.The simplicial presheaf M ap ( P ∞ , Q ) : A ∈ dgArt aug K M ap
P rop ( P ∞ , Q ⊗ A ) is equivalent to the simplicial presheaf M C • ( Hom Σ ( C, Q )) : A ∈ dgArt aug K M C • ( Hom Σ ( C, Q ) ⊗ A ) associated to the complete L ∞ -algebra Hom Σ ( C, Q ) . Note that by [95, Corollary 2.4], the tensor product
M C • ( Hom Σ ( C, Q ) ⊗ A )does not need to be completed because A is Artinian. In order to get a fromalmoduli problem, we also consider the simplicial presheaf M C fmp • ( Hom Σ ( C, Q )) : A ∈ dgArt aug K M C • ( Hom Σ ( C, Q ) ⊗ m A ) , where m A is the maximal ideal of A . This presheaf is a formal moduli problemassociated to Hom Σ ( C, Q ). In the case where Q = End X , Theorem 2.19 impliesthat the the simplicial presheaf M C • ( Hom Σ ( C, End X )) is equivalent to P ∞ { X } (definition 2.17).This theorem applies in particular to the case of a Koszul properad, which in-cludes for instance Frobenius algebras, Lie bialgebras and their variants such asinvolutive Lie bialgebras in stSS:DefALgStructClassicring topology. It applies alsoto more general situations such as the properad Bialg encoding associative andcoassociative bialgebras, which is homotopy Koszul [70, Proposition 41].We now describe the L ∞ -algebra structure encoding this formal moduli problem.It is given by twisting the convolution Lie algebra by Ψ as follows. The twistingof Hom Σ ( C, End X ) by a properad morphism ψ : P ∞ → End X is often called thedeformation complex of ψ , and we have an isomorphism g ψP,X = Hom Σ ( C, End X ) ψ ∼ = Der ψ (Ω( C ) , End X )where the right-hand term is the complex of derivations with respect to ψ [71,Theorem 12]. Proposition 2.20.
The tangent L ∞ -algebra of the formal moduli problem P ∞ { X } ψ is given by g ψP,X = Hom Σ ( C, End X ) ψ . Proof.
Let A be an augmented Artinian cdga. By Theorem 2.19, we have thehomotopy equivalences P ∞ { X } ψ ( A ) ≃ hof ib ( M C • ( g P,X )( A ) → M C • ( g P,X )( K ))= hof ib ( M C • ( g P,X ⊗ A ) → M C • ( g P,X )) ≃ M C • ( hof ib L ∞ ( g P,X ⊗ A → g P,X ))where hof ib L ∞ ( g P,X ⊗ A → g P,X ) is the homotopy fiber, over the Maurer-Cartanelement ψ , of the L ∞ -algebra morphism g P,X ⊗ A → g P,X given by the tensorproduct of the augmentation A → K with g P,X . This homotopy fiber is nothing Proposition 2.20 belows justifying the name, though of course one has to be careful aboutwhich kind of deformation it encodes
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 29 but g ϕP,X ⊗ m A , where m A is the maximal ideal of A , so there is an equivalence offormal moduli problems P ∞ { X } ψ ≃ M C fmp • ( g ψP,X ) . By Lurie’s equivalence theorem, this means that g ψP,X is the Lie algebra of theformal moduli problem P ∞ { X } ψ . (cid:3) Derived formal groups of algebraic structures and associatedformal moduli problems
In this subsection, we explain how the theory of formal moduli problems isrelated to derived formal groups, and how this allows to state the correspondencebetween formal groups and Lie algebras at a higher (and derived) level of generality.This correspondence is suitable for us to define a natural deformation problem ofhomotopy P -algebras structures on a complex X up to quasi-isomorphisms and tounderstand how it relates to those associated moduli space of algebraic structuresfrom section 2.4. Definition 3.1.
Let C be a stable ∞ -category. We denote by M on gpE ( C ) the ∞ -category of grouplike E -monoids in C , that is the subcategory of grouplikeobjects in the ∞ -category of E -algebras in C equipped with the cartesian monoidalstructure. Here an E -monoid G is said to be grouplike if the two canonical maps( µ, π i ) : G × G → G × G (induced by the multiplication µ : G × G → G and the thetwo canonical projections π , π : G × G → G ) are equivalences.A group object of C is an object of M on gpE ( C ). Example 3.2.
Loop spaces provide the main source of examples of group objects intopology (which are also called H -groups in this particular setting). A topologicalmonoid M is said to be grouplike if π M is a group, and since any grouplike topo-logical monoid is equivalent to a loop space, grouplike topological monoids modelgroup objects in the ∞ -category of topological spaces in the sense of Definition 3.1.The same holds true for grouplike simplicial monoids, which model group objectsin the ∞ -category of simplicial sets and will be especially useful for us to studyhomotopy automorphisms of algebras.3.1. Generalities on derived formal groups.
First, let us remark that the cat-egory of formal moduli problems is pointed. In fact, the whole sub ∞ -category of SP sh (( dgArt aug K ) op ) consisting of those ∞ -functors F from augmented dg Artinianalgebras to simplicial sets such that F ( K ) is contractible is pointed. We will de-note it SP sh pt (( dgArt aug K ) op ). To see that it is pointed, let us first note pt the ∞ -functor sending any augmented Artinian cdga to the simplicial set generated bya single vertex. Now let R and R ′ be two augmented Artinian cdgas, let us write η R , η R ′ for their respective unit morphisms and ǫ R , ǫ R ′ their respective augmenta-tions. Let f : R → R ′ be a morphism of augmented Artinian cdgas. A morphismof augmented Artinian cdgas commutes with units, so the diagram pt ( R ) ∼ / / = (cid:15) (cid:15) F ( K ) = (cid:15) (cid:15) F ( η R ) / / F ( R ) f (cid:15) (cid:15) pt ( R ′ ) ∼ / / F ( K ) F ( η R ′ ) / / F ( R ′ ) ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 30 commutes as well, hence we get a unique morphism of ∞ -functors pt → F . A mor-phism of augmented Artinian cdgas commute with augmentations, so the diagram F ( R ) f (cid:15) (cid:15) F ( ǫ R ) / / F ( K ) = (cid:15) (cid:15) ∼ / / pt ( R ) = (cid:15) (cid:15) F ( R ′ ) F ( ǫ R ′ ) / / F ( K ) ∼ / / pt ( R ′ )commutes as well, hence a unique morphism F → pt . Consequently, one can formthe pointed loop space functor as the homotopy pullback(3.1) Ω ∗ F := pt × hF pt in SP sh pt (( dgArt aug K ) op ). Let us note that since dgArt aug K and sSet are pre-sentable, the ∞ -category of pointed ∞ -functors is presentable as well. There-fore SP sh pt (( dgArt aug K ) op ) is a presentable pointed ∞ -category, and F M P K is apresentable pointed sub- ∞ -category of it. Therefore the universal property of ho-motopy pushouts makes Ω ∗ F into a group-like E -monoid in simplicial presheaves(see 3.4.(1) below).Moreover, the inclusion i : F M P K ֒ → SP sh pt (( dgArt aug K ) op )commutes with small homotopy limits, and moreover small homotopy limits in SP sh pt (( dgArt aug K ) op ) are determined pointwise, so that we have proved Lemma 3.3.
For any derived formal moduli problem F and any augmented Ar-tinian cdga R , we have (Ω ∗ F )( R ) ∼ = Ω η R F ( R ) where the base point of F ( R ) isgiven by the morphism F ( η R ) : pt ≃ F ( K ) → F ( R ) induced by the unit η R of R . The base point of F ( R ) corresponds to the “trivial R -deformation” of the uniquepoint of F ( K ). It is important to mention that F M P K is presentable [64, Remark1.1.17] and that the inclusion of F M P K in pointed ∞ -functors admits a left adjoint(applying the ∞ -categorical adjoint functor theorem) L : SP sh pt (( dgArt aug K ) op ) → F M P K making a simplicial presheaf canonically into a formal moduli problem. When F is a formal moduli problem, then LF ∼ = F , otherwise LF is the (best) formalmoduli problem approximating the pointed ∞ -functor F . The functor L is hardto understand explicitely in general, but is related to the (standard) pointwiseclassifying space functor, in the sense that we have a natural equivalence(3.2) L ( B Ω ∗ F ) ∼ = LF where B is given by applying objectwise the classifying space functor from E -monoids in spaces to spaces.The loop space functor enjoys the following properties (as a consequence ofLurie’s work [64], see for example [8, Proposition 2.15] for a proof): Proposition 3.4. (1)
Let C be a pointed presentable ∞ -category. The pointedloop space ∞ -functor lifts to a ( ∞ -categorical) limit preserving functor Ω ∗ : C →
M on gpE ( C ) where M on gpE ( C ) (see 2.11) is the ∞ -category of grouplike E -monoids in C . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 31 (2)
In the case C = F M P K , the loop space functor is an equivalence. Definition 3.5. A derived formal group is an object of M on gpE ( F M P K ), that is agroup object in the stable pointed ∞ -category of formal moduli problems.By proposition 3.4.(2), the functor Ω ∗ has a left adjoint(3.3) B fmp : M on gpE ( F M P K ) −→ F M P K . The functor B fmp is obtained as a generalized bar construction given by the real-ization of a simplicial object in derived formal moduli problems, hence a homotopycolimit corresponding to a classifying space ∞ -functor for derived formal groups(see [8, Lemma 2.16] and [67, Remark 5.2.2.8]). Composing equivalence (2) withLurie’s equivalence theorem [64] result into the equivalence M on gpE ( F M P K ) ∼ = F M P K ∼ = L ∞ − Alg between the ∞ -category of dg-Lie algebras and derived formal groups. This equiv-alence is an analogue to the classical correspondence between formal/algebraic/Liegroups and Lie algebras. This equivalence holds true not only in the commutativecase but also for iterated loop spaces and noncommutative moduli problems, see[8, Proposition 2.15]. Remark . Note that B fmp is not defined pointwise by the standard classifyingspace. If it was so, then, given a formal moduli problem F , for any Artinian cdga R ,there would be an equivalence B Ω η R F ( R ) ≃ F ( R ), which would imply that F ( R )is connected. This is not the case, since F ( R ) is equivalent to the nerve of thedg Lie algebra L F ⊗ m R (where L F is the dg Lie algebra of F via Lurie-Pridhamcorrespondence), and the connected components of the later are the equivalenceclasses of Maurer-Cartan elements of L F .The tangent complex of a formal moduli problem F has a canonical Ω-spectrumstructure. Indeed, for any integer n , one has a homotopy pullback of augmentedArtinian cdgas K ⊕ K [ n ] (cid:15) (cid:15) / / K (cid:15) (cid:15) K / / K ⊕ K [ n + 1]where K ⊕ K [ n ] is the square zero extension of K by K [ n ]. Such a square satisfiesthe conditions required to apply the infinitesimal cohesiveness property of F , andmoreover F ( K ) is contractible, hence inducing a weak equivalence of simplicial sets F ( K ⊕ K [ n ]) ∼ → pt × h K ⊕ K [ n +1] pt ≃ Ω ∗ F ( K ⊕ K [ n + 1]) . We recognize here the structure of an Ω-spectrum T F , whose associated complexis T F , the tangent complex of F (at its unique points). Now, recall that thepointed loop space functor for formal moduli problems is determined pointwise bythe standard pointed loop space, so T Ω ∗ F ≃ Ω T F , which means that(3.4) T Ω ∗ F ∼ = T F [ − ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 32
Remark . Since (Ω ∗ F )( R ) ≃ Ω η R F ( R ) (lemma 3.3), the derived formal group ofa formal moduli problem F seems to retain, for any R , only the informations aboutthe connected component of the trivial R -deformation. However, all the informationof the deformation problem is in fact contained here, since its tangent complex givesthe dg Lie algebra controling it. To understand how this is possible, let us remindthat by infinitesimal cohesiveness of F we have, for example, equivalences F ( K ⊕ K [ n ]) ≃ Ω ∗ F ( K ⊕ K [ n + 1])which means that the space of K ⊕ K [ n ]-deformations is equivalent to the space ofself-equivalences of the trivial K ⊕ K [ n + 1]-deformation. For example, deformationsover the algebra of dual numbers K [ t ] / ( t ) are recovered as loops over the trivial K [ ǫ ] / ( ǫ )-deformation with ǫ of degree 1.More generally, if F is a pointed ∞ -functor such that Ω ∗ F is a derived formalgroup (e.g. a 1-proximate moduli problem in the sense of [64, Definition 5.1.5], seealso cite[Lemma 2.11]BKP), then T Ω ∗ F ≃ T LF [ − . This comes from [64, Lemma 5.1.12]. In other words,
The derived formal group Ω ∗ F controls the deformations parametrized by the formal moduli completion of F .Note that many functors are not representable by a derived stack via Lurie’s rep-resentability theorem [65], but produce nethertheless derived formal moduli prob-lems when restricted to Artinian cdgas, so one can associate a Lie algebra to themwithout any representability condition. Example 3.8.
A case of interest for us is when F is an infinitesimally cohesive (inthe sense of [65, Definition 2.1.1]) simplicial ∞ -presheaf over ( dgArt aug K ) op . Thatis a simplicial presheaf preserving weak equivalences and satisfying the derivedSchlessinger condition ( 2.7), but such that F ( K ) is not (necessarily) contractible.Then one can nethertheless attach to any K -point x ∈ F ( K ) a derived formal moduliproblem c F x by setting c F x ( R ) = hof ib x ( F ( R ) → F ( K )) , where the map is induced by the augmentation R → K of the Artinian cdga R (see the proof of 3.10). Thus, one attaches to any x ∈ F ( K ) a derived formalgroup by taking the pointed loop space of the construction above. Hence, such a F parametrizes a family of derived formal moduli problems over F ( K ).3.2. Derived prestack group and their tangent L ∞ -algebras. We will nowstudy families of derived formal groups, which we call derived prestack groups.These are analogues of Lie groups but in the context of infinitesimally cohesiveprestacks instead of manifolds. In particular, they have an associated L ∞ -algebragiven by their tangent space at the neutral element.Let us denote by SP sh infcoh ∞ (( dgArt aug K ) op ) the ∞ -category of infinitesimallycohesive ∞ -functors on dgArt aug K with values in simplicial sets. Here we note SP sh for simplicial presheaves and inf coh for the infinitesimal cohesiveness of the cor-responding ∞ -functors. We can consider its ∞ -category of group objects (Defini-tion 3.1); that is we introduce the following definition: Definition 3.9. A derived prestack group is a group object in the ∞ -category SP sh infcoh ∞ (( dgArt aug K ) op ). More precisely, the ∞ -category of derived prestackgroups is M on gpE ( SP sh infcoh ∞ (( dgArt aug K ) op )). ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 33
Lemma 3.10.
For any x ∈ G ( K ) , the completion c G x := (cid:16) R hof ib x (cid:0) G ( R ) → G ( K ) (cid:1)(cid:17) is a formal derived group.Proof. The map G ( R ) → G ( K ) is induced by the augmentation R → K of R . Sincethe homotopy fiber is an ∞ -limit, it preserves the infinitesimally cohesive conditionand weak equivalences. By definition, the homotopy fiber computed for R = K is apoint and therefore c G x is a formal moduli problem according to definition 2.7. (cid:3) In other words, a derived prestack group G is a family of derived formal groupsparametrized by G ( K ). In what follows, we will by especially interested in theformal completion at the neutral element. The pointed loop space constructioncommutes with homotopy fibers, so for any F ∈ SP sh infcoh ∞ (( dgArt aug K ) op ) andany x ∈ F ( K ), we have \ (Ω x F ) e = Ω ∗ c F x . So the derived formal group associated to the derived prestack group Ω x F by com-pletion at the constant loop is the derived formal group corresponding to the formalmoduli problem c F x . Remark . One cannot expect the formal completion of any derived stack at apoint to produce a derived formal group and a corresponding tangent Lie algebra,because of the lack of cohesiveness. However, any derived Artin stack (that is,geometric for smooth morphisms) is in particular cohesive, see for instance [63,Corollary 6.5] and [65, Lemma 2.1.7].To legitimate constructions we are going to use in the next section, it is worthmentionning the following properties of infinitesimally cohesive simplicial presheaves:
Lemma 3.12. (1) The ∞ -category SP sh infcoh ∞ (( dgArt aug K ) op ) is stable under smalllimits.(2) If C and D are two equivalent ∞ -categories, the ∞ -categories SP sh infcoh ∞ ( C op ) and SP sh infcoh ∞ ( D op ) are equivalent as well.Proof. (1) Follows from the definition of infinitesimally cohesive ∞ -functors [65,Remark 2.1.11].(2) This is just a particular case of an equivalence of ∞ -categories of sheavesinduced by an equivalence of their ∞ -sites, here with the discrete Grothendiecktopology. (cid:3) Definition 3.13. (Tangent Lie algebra of derived groups) • Let b G be a derived formal group (3.5). Its tangent homotopy Lie algebra is Lie (cid:0) b G (cid:1) := L B fmp (cid:0)b G (cid:1) ∈ Lie ∞ − Alg where B fmp is the equivalence (3.3) and L ( − ) the one of 2.14. • Let G be a derived prestack group (3.9). Its tangent homotopy Lie algebra is Lie ( G ) := Lie ( b G )where b G is the formal completion at the unit of G (3.10). ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 34
The following result shows that the tangent at the identity of a derived prestackgroup inherits a canonical structure of homotopy Lie algebra (which completelydetermines it if it is actually a derived formal group).
Proposition 3.14.
Let G be a derived prestack group. (1) There is an equivalence of underlying complexes
Lie ( G ) ∼ = ( T G ) betweenits Lie algebra and its tangent space at . (2) If F is a formal moduli problem and G ∼ = Ω F , then Lie ( G ) = L F . (3) For any point x in G ( K ) , ( T G ) x ∼ = ( T G ) .Proof. By Proposition 3.4 and (3.4) we have equivalences of complexes T b G ∼ = T Ω B fmp b G ∼ = T B fmp b G [ − ∼ = L B fmp b G where the first equivalence follows from the fact that Ω B fmp is equivalent to theidentity, the second equivalence from 3.4 and the third equivalence from Lurie’sresult [65] asserting that the underlying complex of the Lie algebra L F of a for-mal moduli problem F is equivalent to T F [ − B fmp is the identity and Lemma 3.3,using the sequence of equivalences Lie (Ω F ) Q = L B fmp Ω F ∼ = L using that B fmp Ω is equivalent to the identity.To conclude, since G is a grouplike monoid object, the map G → G induces bymultiplication by x is an equivalence which proves the last statement. (cid:3) In this paper, we are interested in derived groups analogue to the “derived alge-braic groups” in the sense of [27] and deformation complexes associated to homotopyautomorphisms of algebras over properads.
Example 3.15.
Easy examples of derived prestack groups G are given by infinites-imally cohesive ∞ -functors G : dgArt aug K → Ω- Spaces where Ω-
Spaces is the ∞ -category M on gpE ( T op ) of grouplike E -monoids in spaces,i.e., group objects in topological spaces. By May’s recognition principle, the latterare (weakly) equivalent to loop spaces, hence the terminology. Our examples ofinterests will take place in the ∞ -category of grouplike simplicial monoids sM on gl asa model for M on gpE ( T op ) (i.e. we use the equivalence between the model categoriesof topological spaces and simplicial sets and strictification to model Ω-
Spaces ). Aswe explained, a derived prestack group G gives rises to a family of derived formalgroups parametrized by G ( K ).In the next section we will focus the formal neighourhood of the identity inhomotopy automorphism groups, and see how this formalism applies to homotopyautomorphisms of algebras over properads.3.3. Prestacks of algebras and derived groups of homotopy automor-phisms.
The self equivalences of an object in an ∞ -category are canonically agroup object in space (as in example 3.15). When the ∞ -category comes froma model category, strict models for those self equivalences are given by simplicialmonoids of homotopy automorphisms. We refer the reader to [35, Section 2.2] for a ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 35 detailed account on simplicial monoids of homotopy automorphisms in model cate-gories and to [18, 19, 20] for the generalization to homotopy automorphisms in thesimplicial localization of any relative category.
Definition 3.16.
Let X be a chain complex. Let P be a properad, P ∞ a cofibrantresolution of P , and ( X, ψ : P ∞ → End X ) be a P ∞ -algebra structure on X . • We denote haut ( X ) the derived prestack group of homotopy automorphismsof the underlying complex X taken in the model category of chain com-plexes . It is defined by dgArt aug K ∋ A haut Mod A ( X ⊗ A ) , where haut Mod A is the simplicial monoid of homotopy automorphisms inthe category of A -modules. • We define haut P ∞ ( X, ψ ) to be the derived prestack group associated to theautomorphisms of (
X, ψ ) in the ∞ -category P ∞ − Alg [ W − qiso ]: dgArt aug K ∋ A Iso P ∞ − Alg ( Mod A )[ W − qiso ] (cid:0) X ⊗ A, X ⊗ A (cid:1) where, for any ∞ -category C , we write Iso C for the space of maps in theunderlying maximum ∞ -groupoid of C .Note that, since X is cofibrant (like any chain complex over a field) and ( − ) ⊗ A is a left Quillen functor, the homotopy automorphisms haut ( X ) above are exactlythe self quasi-isomorphisms of X ⊗ A . We prove in Theorem 3.20 below that haut P ∞ ( X, ψ ) is indeed a derived prestack group.Let us describe more precisely this derived group: consider the presheaf of ∞ -categories over CDGA op K defined by P ∞ − Alg : CDGA K → Cat ∞ R P ∞ − Alg ( M od cofR )[ W − qiso ]where Cat ∞ is the ∞ -category of ∞ -categories. Here M od cofR is the subcate-gory of cofibrant R -modules in the projective model structure. Let us take thenthe maximal sub- ∞ -groupoid of P ∞ − Alg ( M od cofR )[ W − qiso ] for each R , gettingan ∞ -groupoid valued presheaf. Then, the based loop space at a point ( X, ψ )is exactly haut P ∞ ( X, ψ ). An explicit construction for this is given by, for anycdga R , the Dwyer-Kan simplicial loop groupoid [21] of the quasi-category P ∞ − Alg ( M od R )[ W − qiso ]. Then the Kan complex of paths from ( X ⊗ R, φ ⊗ R ) to itselfin this simplicial loop groupoid is a model for haut P ∞ ( X, ψ )( R ) (this is similar toexample 3.15).We now describe a “point-set” model for the construction of those ( ∞ -categorical)derived groups of P ∞ -algebras automorphisms. First, we introduce a related anduseful construction. The presheaf of Dwyer-Kan classification spaces.
The assignment A wP ∞ − Alg ( M od cofA ) , where the w ( − ) stands for the subcategory of weak equivalences and cof for cofi-brant A -modules, defines a weak presheaf of categories in the sense of [1, Definition Precisely we consider the projective model structure that is, the automorphisms or weak self-equivalences of ( X, ψ ) in this ∞ -category ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 36
I.56]. It sends a morphism A → B to the functor − ⊗ A B . which is symmetricmonoidal, hence lifts at the level of P ∞ -algebras. This is not a strict presheaf, sincethe composition of morphisms A → B → C is sent to the functor ( − ) ⊗ A B ⊗ B C ,which is naturally isomorphic (but not equal) to ( − ) ⊗ A C . This weak presheaf canbe strictified into a presheaf of categories. Applying the nerve functor to this thendefines an ∞ -groupoid:(3.5) the simplicial presheaf of Dwyer-Kan classification spaces is N wP ∞ − Alg.
We also denote(3.6) N wCh K : A M od cofA the simplicial presheaf of quasi-coherent modules of [87, Definition 1.3.7.1]. Theloop space on N wP ∞ − Alg based at a P ∞ -algebra ( X, ψ ) is then the strictificationof the weak simplicial presheafΩ ( X,ψ ) N wP ∞ − Alg : A Ω ( X ⊗ A,ψ ⊗ A ) N wP ∞ − Alg ( M od cofA ) . Lemma 3.17.
The pointwise loop space defined above is pointwise equivalent to theloop space functor in the projective model category of simplicial presheaves, wherewe consider simplicial presheaves with values in pointed simplicial sets.Proof.
The pointed loop space functor on the projective model category of simplicialpresheaves
SP sh ( C ) on a model category C is defined on any simplicial presheaf F as the homotopy pullback pt × hF pt . In the model category setting, a homotopypullback is computed as the limit of a fibrant resolution of the pullback diagramin SP sh ( C ) Iinj , where I is the small category {• → • ← •} and inj means thatwe consider this diagram category equipped with the injective model structure.Moreover, we have a Quillen equivalence SP sh ( C ) Iproj ⇆ SP sh ( C ) Iinj where
SP sh ( C ) Iproj is the projective model category of I -diagrams and the adjunc-tion is given by the identity functors. In particular, this implies that every fibrantresolution in SP sh ( C ) Iinj is a fibrant resolution in
SP sh ( C ) Iproj . In the projectivemodel structure
SP sh ( C ) Iproj , fibrations are the same as in the projective modelcategory of functors
F un ( C ×
I, sSet ) proj . So a fibrant resolution in SP sh ( C ) Iinj ispointwise a fibrant resolution in sSet
Iinj . Moreover, limits in
SP sh ( C ) are deter-mined pointwise. This implies that the pullback of a fibrant resolution of a pullbackdiagram in simplicial presheaves is given, pointwise, by the pullback of a fibrantresolution of a pullback diagram in simplicial sets. That is, the homotopy pullbackdefining the loop space functor for simplicial presheaves, when valued at a givenobject of C , gives the homotopy pullback defining the loop space functor for pointedsimplicial sets. (cid:3) Homotopy automorphism presheaves as loops over the presheaf of Dwyer-Kan clas-sification spaces.
In the case of an operad O , there is an easy model for haut O ∞ .Indeed, in that case, O ∞ -algebras inherits a canonical model category structureand haut O ∞ is the ∞ -functor associated to a simplicial presheaf given by the sim-plicial monoid of homotopy automorphisms of ( X, ψ ) in the model category of O ∞ -algebras. That is the simplicial sub-monoid of self weak equivalences in the ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 37 usual homotopy mapping space
M ap O ∞ − Alg ( X, X ) (see for instance [50, Chapter17]). Thus this weak simplicial presheaf is A haut O ∞ ( X ⊗ A, ψ ⊗ A ) Mod A where haut O ∞ ( X ⊗ A, ψ ⊗ A ) Mod A is the simplicial monoid of homotopy auto-morphisms of ( X ⊗ A, ψ ⊗ A ) ∈ O ∞ − Alg ( M od cofA ). Note that by definition,this homotopy automorphism are computed by taking a cofibrant resolution of( X ⊗ A, ψ ⊗ A ) to get a cofibrant-fibrant object (all algebras are fibrant), andthen considering weak self-equivalences of it. Our simplicial presheaf is then itsstrictification (see [1, Section I.2.3.1]).In the case of a general properad P , there is no model category structure anymoreon the category of P ∞ -algebras. However, we can still define the simplicial monoid L H wP ∞ − Alg ( X, ψ ) of homotopy automorphisms in the simplicial or hammocklocalization (with respect to quasi-isomorphisms) of P ∞ -algebras, following Dwyer-Kan [19, 20]. Note that by [20], in the case when P ∞ − Alg is a model category(that is, P is an operad), we have a homotopy equivalence haut P ∞ ( X, ψ ) ≃ L H wP ∞ − Alg ( X, ψ )(taking the model category construction for the left side of this equivalence), so thetwo constructions agree. In both cases, these are models of the pointed loop spaceΩ ( X,ψ ) N wP ∞ − Alg on the simplicial presheaf of Dwyer-Kan classification spaces:
Lemma 3.18.
Let P ∞ be a cofibrant prop. Then haut P ∞ ( X, ψ ) is equivalent to A ∈ dgArt aug K Ω ( X ⊗ A,ψ ⊗ A ) (cid:16) N wP ∞ − Alg ( M od cofA ) (cid:17) . Further, haut ( X ) is equivalent to A ∈ dgArt aug K Ω X ⊗ A (cid:16) N wM od cofA (cid:17) . Proof.
This comes from the fact that, for any relative category (
C, W ) and anyobject X of C , the connected component of X in N W is equivalent to the classifyingspace BLW ( X, X ). Therefore there is an equivalence LW ( X, X ) ≃ Ω X N W ofsimplicial monoids. Hence we can define the presheaf of homotopy automorphisms,or self-weak equivalences, haut P ∞ ( X, ψ ) is equivalent to the following simplicialpresheaf haut P ∞ ( X, ψ ) : A ∈ dgArt aug K Ω ( X ⊗ A,ψ ⊗ A ) N wP ∞ − Alg ( M od cofA ) . The proof for haut ( X ) is similar. (cid:3) Prestacks of algebras.
We will now prove that what we called the derived group ofautomorphisms of an algebra is indeed a derived prestack group. As a first step,we need the following version of Rezk’s homotopy pullback theorem [79]:
Proposition 3.19.
Let P ∞ be a cofibrant prop and X be a chain complex. Theforgetful functor P ∞ − Alg → Ch K induces a homotopy fiber sequence P ∞ { X } → N wP ∞ − Alg → N wCh K of simplicial presheaves over augmented Artinian cdgas. ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 38
Proof.
We explain briefly how [97, Theorem 0.1] can be transposed in the contextof simplicial presheaves of cdgas. The identification of the homotopy fiber of theforgetful map N wP ∞ − Alg → N wCh K with the simplicial presheaf P ∞ { X } follows from the two following facts. First,we can identify it pointwise with M ap ( P ∞ ⊗ A, End
Mod A X ⊗ A ), where End
Mod A X ⊗ A is theendomorphism prop of X ⊗ A in the category of A -modules. This comes from theextension of [97, Theorem 0.1] to A -linear P ∞ -algebras, which holds true triviallyby replacing chain complexes by A -modules as target category in the universalfunctorial constructions of [97, Section 2.2] ( A -modules are equipped with exactlythe same operations than chain complexes which are needed in this construction:directs sums, suspensions, twisting cochains). Second, for any morphism of cdgas f : A → B , the tensor product ( − ) ⊗ A B induces an isomorphism of simplicial sets M ap ( P ∞ ⊗ A, End
Mod A X ⊗ A ) ∼ = M ap ( P ∞ ⊗ B, End
Mod B X ⊗ B )fitting in a commutative square M ap ( P ∞ ⊗ A, End
Mod A X ⊗ A ) ∼ = / / ( − ) ⊗ A B (cid:15) (cid:15) P ∞ { X } ( A ) f (cid:15) (cid:15) M ap ( P ∞ ⊗ A, End
Mod A X ⊗ A ) ∼ = / / P ∞ { X } ( B )(see for instance [96, Section 3]) so that we get a morphism of homotopy fibersequences P ∞ { X } ( A ) / / (cid:15) (cid:15) N wP ∞ − Alg ( M od A ) / / ( − ) ⊗ A B (cid:15) (cid:15) N wM od A ( − ) ⊗ A B (cid:15) (cid:15) P ∞ { X } ( B ) / / N wP ∞ − Alg ( M od B ) / / N wM od B . (cid:3) Theorem 3.20.
The simplicial presheaf haut P ∞ ( X, ψ ) defines indeed a derivedprestack group in the sense of Definition 3.9 .In particular \ haut P ∞ ( X, ψ ) id is a derived formal group.Proof. First, recall that haut P ∞ ( X, ψ ) is equivalent to Ω ( X,ψ ) N wP ∞ − Alg , andthat we already know it is a presheaf with values in grouplike simplicial monoids,hence a group object in simplicial presheaves. Second, we use the simplicial presheafversion of Rezk’s pullback theorem [79] for algebras over properads, that is, thehomotopy fiber sequence P ∞ { X } → N wP ∞ − Alg → N wCh K of simplicial presheaves over augmented Artinian cdgas, taken over the base point X given by Proposition 3.19. This homotopy fiber sequence induces a homotopyfiber sequence Ω ( X,ψ ) N wP ∞ − Alg → Ω X N wCh K → P ∞ { X } that is an object of Mon gpE ( SP sh infcoh ∞ (( dgArt aug K ) op )) ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 39 hence the fiber sequence haut P ∞ ( X, ψ ) → haut ( X ) → P ∞ { X } . (by Lemma 3.18). Now we combine this result with Lemma 3.12(1) to deducethat haut P ∞ ( X, ψ ) preserves weak equivalences and is infinitesimally cohesive. Forthis, we just have to check that the two right-hand terms of the fiber sequencesatisfy these properties and use that this homotopy fiber sequence is in particulara pointwise homotopy fiber sequence. Concerning haut ( X ) this is already knownfrom see [64, Section 5.2], and concerning P ∞ { X } this follows from its isomorphismwith the Maurer-Cartan simplicial presheaf in Theorem 2.19.In particular, \ haut P ∞ ( X, ψ ) id is a derived formal group. Note that we couldhave directly proved this last statement by taking the formal completion of thefiber sequence above (that is, the componentwise homotopy fiber of this diagramover each appropriate base point), and then apply Lemma 3.12(1) to \ haut ( X ) id (which is a formal derived group) and \ P ∞ { X } ϕ (which is a derived formal moduliproblem). (cid:3) Remark . The classification space N wP ∞ − Alg decomposes as a coproduct ofthe classifying spaces of homotopy automorphisms of P ∞ -algebras N wP ∞ − Alg ∼ = a [ Y,φ ] ∈ π N wP ∞ − Alg
Bhaut P ∞ − Alg ( Y, φ )where [
Y, φ ] ranges over quasi-isomorphism classes of P ∞ -algebras. Restricting thehomotopy pullback of Proposition 3.19 to the connected component Bhaut ( X ) ofthe base space, we get a homotopy pullback P ∞ { X } (cid:15) (cid:15) / / ` [ Y,φ ] ,Y ≃ X Bhaut P ∞ − Alg ( Y, φ ) (cid:15) (cid:15) pt (cid:31) (cid:127) / / Bhaut ( X )where the coproduct ` [ Y,φ ] ,Y ≃ X ranges over P ∞ -algebras so that Y ≃ X as com-plexes. So Rezk homotopy pullback theorem and its version above tells us that ` [ Y,φ ] ,Y ≃ X Bhaut P ∞ − Alg ( Y, φ ) can be seen as a homotopy quotient of P ∞ { X } bythe action of haut ( X ). From a deformation theoretic perspective, this means that ata “tangent level”, the deformation theory of ψ : P ∞ → End X corresponds to defor-mations of the P ∞ -algebra ( X, ψ ) which preserves the differential of the underlyingcomplex X , whereas the deformations associated to haut P ∞ − Alg ( X, ψ ) deform thedifferential as well (that is, it takes into account the action of haut ( X ) on P ∞ { X } ).We are going to see in Section 3 how to formalize properly this idea. Remark . Let us explain the relationship between the classifying presheaf ofalgebras and the derived formal group of homotopy automorphisms in the neigh-bourhood of the identity. Recall the construction haut P ∞ ( X, ψ ) : A ∈ dgArt aug K Ω ( X ⊗ A,ψ ⊗ A ) N wP ∞ − Alg ( M od cofA ) , from which we deduce \ haut P ∞ ( X, ψ ) id = Ω ∗ \ N wP ∞ − Alg ( X,ψ )ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 40 where Ω ∗ is the loop space for pointed functors as explained in Section 3. Usingthe decomposition of the nerve of weak equivalences into classifying spaces of ho-motopy automorphisms pointed out in Remark 3.21, we see moreover that for anyaugmented Artinian cdga R , there is a decomposition \ N wP ∞ − Alg ( X,ψ ) ( R ) ∼ = a [ Y,φ ] | ( Y,φ ) ⊗ R K ≃ ( X,ψ ) Bhaut P ∞ − Alg ( Mod R ) ( Y, φ ) . Equivalently, \ N wP ∞ − Alg ( X,ψ ) ( R ) is homotopy equivalent to the maximal ∞ -subgroupoid of the ∞ -category P ∞ − Alg ( M od R )[ W − qiso ] generated by R -linear P ∞ -algebras ( Y, φ ) such that (
Y, φ ) ⊗ R K ≃ ( X, ψ ), that is \ N wP ∞ − Alg ( X,ψ ) ( R ) ∼ = P ∞ − Alg ( M od R )[ W − qiso ] × hP ∞ − Alg ( Ch K )[ W − qiso ] { ( X, ψ ) } . The space \ N wP ∞ − Alg ( X,ψ ) encapsulates the whole deformation theory of ( X, ψ )in the ∞ -category P ∞ − Alg [ W − qiso ] as we can think of it, that is, an R -deformationof ( X, ψ ) is an R -linear P ∞ -algebra whose restriction modulo R is quasi-isomorphicto ( X, ψ ), and equivalences between R -deformations are defined by compatible R -linear quasi-isomorphisms whose restriction modulo R is homotopic to Id ( X,ψ ) .This is the natural generalization, to the differential graded setting, of classicaldeformations of degree zero algebras. Although it is not clear that such a construc-tion provides a derived formal moduli problem, one can however associates to itthe derived formal group haut P ∞ ( X, ψ ) via a loop space construction, and by thegeneral formalism explained in Section 3 we have T \ haut P ∞ ( X,ψ ) id = Lie ( L ( \ N wP ∞ − Alg ( X,ψ ) ))where L is the completion of \ N wP ∞ − Alg ( X,ψ ) in a formal moduli problem. An-other way to state this is that in general \ N wP ∞ − Alg ( X,ψ ) is 1-proximate in thesense of [64]. Remark . In the special case of operads acting on algebras concentrated indegree 0, we can say more. Let A be a P -algebra in vector spaces whose underlyingvector space is of finite dimension, then by [87, Prop.2.2.6.8], the classifying presheaf N wP − Alg is actually a derived 1-geometric stack, which implies by [63, Corollary6.5] and [65, Lemma 2.1.7] that its restriction to dgArt aug K is infinitesimally cohesive.Consequently \ N wP − Alg A is already a derived formal moduli problem in this caseand Lie (cid:0) \ haut P ( A ) id (cid:1) ∼ = Lie ( \ N wP − Alg A )by Proposition 3.14.This Lie algebra recovers in particular various known deformation complexes inthe litterature, once one has an explicit formula to compute it, as we are going todetail in Section 3.3.4. The fiber sequence of deformation theories.
We now relate preciselythe moduli problems governed by the mapping space P ∞ { X } and the homotopyautomorphisms space haut P ∞ − Alg ( X, ψ ) (Definitions 3.16 and 2.17).
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 41
Theorem 3.24.
There is a homotopy fiber sequence of derived prestack groups Ω ψ P ∞ { X } → haut P ∞ − Alg ( X, ψ ) → haut ( X ) , hence a homotopy fiber sequence of derived formal groups \ Ω ψ P ∞ { X } → \ haut P ∞ − Alg ( X, ψ ) id → \ haut ( X ) id , and equivalently of their associated L ∞ -algebras g ψP,X → Lie ( haut P ∞ − Alg ( X, ψ )) → Lie ( haut ( X )) . Proof.
Recall (see (3.1)) that the pointed loop space functor is defined on anysimplicial presheaf F as the homotopy pullback pt × hF pt . It thus commutes withhomotopy fibers, and in particular the loop space ∞ -functor commutes with fibers inthe ∞ -category of simplicial presheaves. In the fiber sequence of Proposition 3.19,we choose ψ as the base point on the left, ( X, ψ ) in the middle, and X on theright. Since fibers in the ∞ -category of presheaves valued in simplicial monoidsare determined in the underlying ∞ -category of simplicial presheaves, applying thepointed loop space ∞ -functor with respect to these base points, we deduce a fibersequence of derived prestack groupsΩ ψ P ∞ { X } → haut P ∞ − Alg ( X, ψ ) → haut ( X )(using that Ω ◦ B ≃ Id ). Hence, we get a fiber sequence of derived formal groups \ Ω ψ P ∞ { X } → \ haut P ∞ − Alg ( X, ψ ) id → \ haut ( X ) id (using that Ω c F x ≃ [ Ω x F for an infinitesimally cohesive ∞ -functor F and x ∈ F ( K ),and in particular that c G e ≃ Ω d BG for a derived prestack group G ). The corre-sponding fiber sequence of Lie algebras aasociated to this formal derived problemsidentifies with the desired one g ψP,X → Lie ( haut P ∞ − Alg ( X, ψ )) → Lie ( haut ( X ))by Lemma 3.18 and using equivalence (3.4) (cid:3) Equivalent deformation theories for equivalent (pre)stacks of alge-bras.
In derived algebraic geometry, an equivalence between two derived Artinstacks F and G induces a weak equivalence between the tangent complex over agiven point of F and the tangent complex over its image in G [87]. We now provesimilar statement about the tangent Lie algebras of our formal moduli problems ofalgebraic structures.Recall the presheaf of categories given by the ∞ -functor P ∞ − Alg : CDGA K → Cat ∞ R P ∞ − Alg ( M od R )[ W − qiso ]where Cat ∞ is the ∞ -category of ∞ -categories.The idea is that [97, Theorem 0.1] implies that the formal moduli problem P ∞ { X } ψ is “tangent” over ( X, ψ ) to the Dwyer-Kan classification space of the ∞ -category of P ∞ -algebras (see 3.3).More precisely, recall that F : P ∞ − Alg ∼ −→ Q ∞ − Alg being an an equivalenceof presheaves of ∞ -categories means that, for every augmented Artinian cdga A , F ( A ) : P ∞ − Alg ( M od A )[ W − qiso ] ∼ −→ Q ∞ − Alg ( M od A )[ W − qiso ] . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 42 is an equivalence of ∞ -categories. Relying on our previous results, we prove: Theorem 3.25.
Let F be an equivalence of presheaves of ∞ -categories F : P ∞ − Alg ∼ −→ Q ∞ − Alg.
Then F induces an equivalence of fiber sequences of derived formal moduli problems P ∞ { X } ψ ∼ (cid:15) (cid:15) / / B fmp \ haut P ∞ − Alg ( X, ψ ) Id ( X,ψ ) ∼ (cid:15) (cid:15) / / B fmp \ haut ( X ) Id X = (cid:15) (cid:15) Q ∞ { F ( X ) } F ( ψ ) / / B fmp \ haut Q ∞ − Alg ( F ( X, ψ )) Id ( X,ψ ) / / B fmp \ haut ( X ) Id X where F ( ψ ) is the Q ∞ -algebra structure on the image of ( X, ψ ) under F . Equiva-lently, F induces an equivalence of fiber sequences of the associated L ∞ -algebras g ψP,X ∼ (cid:15) (cid:15) / / Lie ( haut P ∞ − Alg ( X, ψ )) ∼ (cid:15) (cid:15) / / Lie ( haut ( X )) = (cid:15) (cid:15) g F ( ψ ) Q,F ( X ) / / Lie ( haut Q ∞ − Alg ( F ( X, ψ ))) / / Lie ( haut ( X )) . Proof.
Let F : P ∞ − Alg → Q ∞ − Alg be an equivalence of presheaves of ∞ -categories. We have a commutative triangle P ∞ − Alg F / / U ◦ F $ $ ❏❏❏❏❏❏❏❏❏ Q ∞ − Alg U z z ttttttttt Ch K . Applying the loop space functor (3.1) at the appropriate base points we get thecommutative triangleΩ ( X,ψ ) P ∞ − Alg ∼ / / ' ' ❖❖❖❖❖❖❖❖❖❖❖ Ω F ( X,ψ ) Q ∞ − Alg w w ♥♥♥♥♥♥♥♥♥♥♥♥ Ω X Ch K . But a based loop space at a point of an ∞ -category is the homotopy automorphimsgrouplike monoid of this point, so that this triangle is actually the triangle of derivedprestack groups haut P ∞ − Alg ( X, ψ ) ∼ / / ( ( PPPPPPPPPPPP haut Q ∞ − Alg ( F ( X, ψ )) v v ❧❧❧❧❧❧❧❧❧❧❧❧ haut ( X ) . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 43
By Theorem 3.24, we get the equivalence of homotopy fiber sequences of derivedprestack groupsΩ ψ P ∞ { X } (cid:15) (cid:15) ∼ / / Ω F ( ψ ) Q ∞ { F ( X ) } (cid:15) (cid:15) haut P ∞ − Alg ( X, ψ ) ∼ / / ( ( PPPPPPPPPPPP haut Q ∞ − Alg ( F ( X, ψ )) v v ❧❧❧❧❧❧❧❧❧❧❧❧ haut ( X ) . hence an equivalence of homotopy fiber sequences of the corresponding derived for-mal groups obtained by completion at the appropriate base points. This equivalenceof fiber sequences gives an equivalence of fiber sequences of the corresponding Liealgebras by the Lurie-Pridham equivalence theorem. (cid:3) Remark . There is also a “strict” version of this theorem. Let us consider amorphism of weak presheaves of relative categories, that is, given for each cdga A by a morphism of relative categories F ( A ) : ( P ∞ − Alg ( M od A ) , W qiso ) → ( Q ∞ − Alg ( M od A ) , W qiso ) . Let us suppose that F induces an equivalence of presheaves of classification spaces N wF : N wP ∞ − Alg ∼ −→ N wQ ∞ − Alg.
By [94, Section 3.3], this means that F induces an equivalence of weak presheavesof ∞ -categories as in Theorem 3.25. Then, we can mimick the proof of Theo-rem 3.25 as follows: we replace the presheaves of ∞ -categories by these presheavesof classification spaces, t ake based loop spaces which gives back the homotopyautomorphisms as well, and apply Theorem 3.24.4. The tangent Lie algebra of homotopy automorphims
The goal of this section is two-fold. First, relying on Theorem 3.24, we provethat
Lie ( haut P ∞ − Alg ( X, ψ )) is a semi-direct product, in a homotopical sense, ofthe two extremal terms of the fiber sequence, and that the later term is nothingbut
End ( X ) = Hom Ch K ( X, X ) equipped with the commutator of the compositionproduct. This is actually the transposition, at the Lie algebra level, of the homo-topy action of haut ( X ) on P ∞ { X } that we mentionned in Remark 3.21. In a fewwords, the tangent Lie algebra of homotopy automorphisms takes into account theaction of the automorphisms of the complex X on the Maurer-Cartan elements of g ψP,X , that is, on the space of P ∞ -algebra structures on X . Second, we provide anexplicit formula crucial to consider deformation complexes of algebraic structureswhich also encode compatible deformations of the differential. For this, we usea plus construction originally due to Merkulov [73]. We explain the homotopicalcounterpart of this construction, that is, how the corresponding L ∞ -algebra con-trols the derived prestack group of homotopy automorphisms of an algebra overa properad. This important result gives a conceptual explanation of how one canexpress the deformation theory inside P ∞ − Alg as deformations of a P ∞ -algebrastructure in the (pro)peradic sense plus compatible deformations of the differential,and formalizes properly Remark 3.21. ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 44
Homotopy representations of L ∞ -algebras and a relevant applica-tion. Recall (2.9) that the structure of a L ∞ -algebra g is encoded by a (cohomog-ical degree −
1) coderivation Q g of square zero on Sym • > ( g [1]). Dualizing thiscoderivation induces an augmented cdga structure on C ∗ CE ( g ) := Hom ( Sym • ( g [1]) , k )which is called the Chevalley-Eilenberg cochain algebra of g ; we denote ε the aug-mentation. For any graded module N , Hom ( Sym • ( g [1]) , N ) inherits similarly astructure of graded C ∗ CE ( g )-module. Definition 4.1.
Let ( g, Q g ) be a L ∞ -algebra and ( M, d M ) ∈ Ch K .A homotopy representation of an L ∞ -algebra g on M is a derivation D of squarezero and (cohomological) degree 1 on C ∗ CE ( g, M ) := Hom dg ( Sym • ( g [1]) , M ) suchthat M D → C CE ( g, M ) ε → M is equal to the inner differential d M of the complex M . The fact that D is a derivation means precisely that it satisfies the followingLeibniz relation: for f ∈ C nCE ( g ), Φ ∈ C ∗ CE ( g, M ), one has(4.1) D ( f · Φ) = Q g ( f ) · Φ + ( − n f · D (Φ) . Example 4.2.
A particular case of homotopy representation is the standard notionof representation, given by a (dg-) Lie algebra morphism g → End ( M ) and thestandard Chevalley-Eilenberg cochain complexes. This generalizes easily to any L ∞ -algebra g and L ∞ -morphism g → End Ch K ( M, M ).Definition 4.1 is equivalent to the data of a L ∞ -algebra structure on g ⊕ M ,that is a coderivation of square zero on Sym • > (( g ⊕ M )[1]) that vanishes on thecoideal spanned by Sym • > ( M ) and whose restriction to Sym • > ( g [1]) and M arerespectively Q g and the inner differential of M (followed by the canonical inclusionsof these complexes in Sym • > (( g ⊕ M )[1])). In other words it is a square zeroextension by M of the L ∞ -algebra structure of g . Example 4.3 (semi-direct product) . If h is a dg-Lie algebra, any dg-Lie algebrahomomorphism g → Der ( h ) induces an action of g onto C CE ∗ ( h ) as the coderivationextending the g -action on h . Similarly, if h and g are L ∞ -algebras and givenan L ∞ -algebra morphism ϕ : g → Der ( h ), we obtain a homotopy representationof g on C CE ∗ ( h ). The coalgebra structure of C CE ∗ ( h ) then yields respectively acocommutative dg-coalgebra and a cdga structure on(4.2) CE ∗ ( g, h ) := C CE ∗ (cid:0) g, C CE ∗ ( h ) (cid:1) , CE ∗ ( g, h ) := C ∗ CE (cid:0) g, C ∗ CE ( h ) (cid:1) . The augmentations yield a cofiber sequence of cdgas C ∗ CE ( g ) → CE ∗ ( g, h ) → C ∗ CE ( h ) , which is dual to a fiber sequence of dg-cocommutative coalgebras C CE ∗ ( h ) → CE ∗ ( g, h ) → C CE ∗ ( g ) , which is equivalent to a fiber sequence of L ∞ -algebras h → g ⋉ f h → g forming a split extension of g by h . The semi-direct product g ⋉ f h is the directsum g ⊕ h equipped with the L ∞ -algebra structure coming from the differential onthe coalgebra CE ∗ ( g, h ) = Sym (cid:0) g [1] ⊕ h [1] (cid:1) . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 45
Example 4.4.
In particular the adjoint action ad : g → Der ( g ) of a L ∞ -algebra g on itself yields the semi-direct product g ⋊ ad g .Semi-direct product of L ∞ -algebras will appear in the construction of the dia-gram of equivalence of fiber sequences below g ϕP,X ∼ / / (cid:15) (cid:15) g ϕP,X ∼ / / (cid:15) (cid:15) g ϕP,X (cid:15) (cid:15) g ϕ + P + ,X ∼ / / (cid:15) (cid:15) g ϕP,X ⋉ f End ( X ) ∼ / / (cid:15) (cid:15) Lie ( haut P ∞ ( X, ϕ )) (cid:15) (cid:15) End ( X ) ∼ / / End ( X ) ∼ / / End ( X )where the homotopy semi-direct product in the middle is defined in the next sec-tion 4.2. It turns out that this semi-direct product is the tangent incarnation ofthe non trivial action of haut ( X ) on the moduli space P ∞ { X } at a topologicallevel. Taking this action into account in the deformation theory of ( X, ϕ ) means,on the one hand deforming the P ∞ -algebra structure with compatible deformationsof the differential (equivalence of the middle fiber sequence with the left one), onthe other hand deforming ( X, ϕ ) in the ∞ -category of P ∞ -algebras (equivalence ofthe middle fiber sequence with the right one).The plan is as follows. First, we construct two equivalences of fiber sequences ofderived groups fitting in the diagramΩ ϕ P ∞ { X } ∼ / / (cid:15) (cid:15) Ω ϕ P ∞ { X } ∼ / / (cid:15) (cid:15) Ω ϕ P ∞ { X } (cid:15) (cid:15) Ω ϕ + P + ∞ { X } ∼ / / (cid:15) (cid:15) Ω [ ϕ ] ( P ∞ { X } //haut ( X )) ∼ / / (cid:15) (cid:15) haut P ∞ ( X, ϕ )) (cid:15) (cid:15) haut ( X ) ∼ / / haut ( X ) ∼ / / haut ( X ) , where P ∞ { X } //haut ( X ) is the appropriate homotopy quotient in the ∞ -categoryof infinitesimally cohesive ∞ -presheaves over ( dgArt aug K ) op . Secondly, the desiredfiber sequence of L ∞ -algebras is induced by this diagram (taking, as usual in this pa-per, completions at identities to get equivalences of fiber sequences of derived formalgroups). Finally, we identify the homotopy quotient Lie (Ω [ ϕ ] ( P ∞ { X } //haut ( X )))with g ϕP,X ⋉ f End ( X ).4.2. ∞ -actions in infinitesimally cohesive presheaves. In this section, theambient ∞ -category is SP sh infcoh ∞ (( dgArt aug K ) op ) and derived prestack groups areprecisely the group objects (3.1) in it. This is a particular case of infinitesimallycohesive ∞ -topos, where the theory of principal ∞ -bundles developped in [75, 76]fully applies. In this setting, the general notion of ∞ -action of a group object G in an ∞ -category on another object X provides a homotopy quotient . This which is the same as the quotient in ∞ -stack ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 46 homotopy quotient comes naturally equipped with a homotopy fiber sequence X → X//G → BG with fiber X associated to the universal G -principal ∞ -bundle • → BG . The map X//G → BG is the classifying morphism of the action of G on X . This is ananalogue of the usual quotient stack by a group stack action of [44]. Remark . When G is presented by a simplicial presheaf in grouplike simplicialmonoids (which is a model for derived prestack groups), the homotopy quotient X//G is computed by the geometric realization of the simplicial action groupoid · · · G × G × X →→→ G × X ⇒ X. (see for example [52] in the case of group actions in simplicial presheaves).Let G be a group object in SP sh infcoh ∞ (( dgArt aug K ) op ), i.e., a derived prestackgroup. Proposition 4.6.
For any X equipped with an ∞ -action of a derived prestackgroup object G , there is a fiber sequence of homotopy Lie algebras Lie (Ω ∗ X ) → Lie (Ω ∗ ( X//G )) → Lie ( G ) Proof.
Since the loop space is an homotopy pullback, the fiber sequence X → X//G → BG yields a fiber sequence of derived groupsΩ ∗ X → Ω ∗ ( X//G ) → G hence the desired fiber sequence of tangent Lie algebras by 3.13. (cid:3) We still consider an object X with an ∞ -action of a derived prestack group G . Lemma 4.7.
Assume that there exists a section of the (induced) projection map \ Ω ∗ ( X//G ) x π → b G , that is a derived formal group morphism s : b G → \ Ω ∗ ( X//G ) x such that π ◦ s is equivalent to the identity. Then there is an equivalence of L ∞ algebras Lie (cid:0) Ω ∗ ( X//G ) (cid:1) ∼ = Lie (Ω ∗ X ) ⋊ Lie ( G ) and the fiber sequence of proposition 4.6 identifies with the semi-direct product one.Proof. The Lie algebra functor (3.13) depends only on the associated formal groupat the base point. Therefore it gives a L ∞ -morphim Lie ( G ) Lie ( s ) −→ Lie (cid:0) Ω ∗ ( X//G ) (cid:1) .Composing with the adjoint action of the latter, we obtain a morphism ad ◦ Lie ( s ) : Lie ( G ) → Der (cid:16)
Lie (cid:0) Ω ∗ ( X//G ) (cid:1)(cid:17) . Since this is a map of Lie algebras, and s is a section of π , the induced action of Lie ( G ) on Lie (cid:0) Ω ∗ ( X//G ) (cid:1) restricts to ker ( π ) ∼ = Lie (Ω ∗ X ). Therefore we get an induced L ∞ -algebra map Lie ( G ) → Der (cid:0)
Lie (Ω ∗ X ) (cid:1) which defines the homotopy Lie algebra semi-direct product (4.3).It follows that the morphism Lie (Ω ∗ X ) ⋊ Lie ( G ) ∋ ( x, y ) τ x + s ( y ) ∈ Lie (cid:0) Ω ∗ ( X//G ) (cid:1) ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 47 is a L ∞ -algebra map and that we have a commutative diagram of fiber sequences Lie (Ω ∗ X ) / / Lie (Ω ∗ X ) ⋊ Lie ( G ) Lie ( π ) / / τ (cid:15) (cid:15) Lie ( G ) Lie (Ω ∗ X ) / / Lie (cid:0) Ω ∗ ( X//G ) (cid:1) Lie ( π ) / / Lie ( G )of L ∞ -algebras. The equivalence now follows from the 2 out 3 property. (cid:3) The Lie algebra of homotopy automorphisms as a semi-direct prod-uct.
The goal of this section is to prove (and makes sense of) the formula below:
Lie ( haut P ∞ − Alg ( X, ψ )) ≃ Lie (Ω [ ϕ ] ( P ∞ { X } //haut ( X ))) = g ψP,X ⋊ h End ( X ) . To do so, we will interpret
Lie ( haut P ∞ − Alg ( X, ψ )) as the tangent Lie algebra of ahomotopy quotient of P ∞ { X } by the ∞ -action of haut ( X ).An explicit model of the homotopy quotient is given by a homotopy versionof the well known Borel construction, suitably adapted for simplicial presheavesover cdgas. In [52], the Borel construction is given by the classifying space of thetranslation groupoid associated to the action of a sheaf of groups G on a sheaf X ,that is EG × G X . We adapt this construction to the case of an ∞ -action.Let P a cofibrant prop, and N w ( E cf ) ∆[ − ] ⊗ P the bisimplicial set defined by( N w ( E cf ) ∆[ − ] ⊗ P ) m,n = ( N w ( E cf ) ∆[ n ] ⊗ P ) m , where the w denotes the subcategoryof morphisms which are weak equivalences in E . We get a diagram P { X } / / (cid:15) (cid:15) diag N f w ( E cf ) ∆[ − ] ⊗ P (cid:15) (cid:15) ∼ / / diag N w ( E cf ) ∆[ − ] ⊗ P N w ( E cf ) P ∼ o o (cid:15) (cid:15) pt / / N ( f w E cf ) ∼ / / N ( w E cf ) , where the f w denotes the subcategory of morphisms which are acyclic fibrations in E . The crucial point here is that the left-hand commutative square of this diagramis a homotopy pullback, implying that we have a homotopy pullback of simplicialsets (see [96, Theorem 0.1]) P { X } (cid:15) (cid:15) / / N ( wCh P K ) (cid:15) (cid:15) { X } / / N wCh K . Therefore P ∞ { X } can be identified with the homotopy fiber P ∞ { X } / / (cid:15) (cid:15) diag N f wCh P ⊗ ∆ • K (cid:15) (cid:15) { X } / / N f wCh K ∼ N wCh K . . The main goal of this section is to prove the following:
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 48
Proposition 4.8.
Let X be a cochain complex and φ : P ∞ → End X be a propmorphism. There exists a commutative square Ehaut K ( X ) × haut K ( X ) P ∞ { X } ∼ / / π (cid:15) (cid:15) (cid:15) (cid:15) diag N f wCh P ⊗ ∆ • K (cid:15) (cid:15) Bhaut K ( X ) ∼ / / N wCh K | X where π is a Kan fibration obtained by simplicial Borel construction and the hori-zontal maps are weak equivalences of simplicial sets, inducing an equivalence Ω [ ϕ ] (cid:0) Ehaut K ( X ) × haut K ( X ) P ∞ { X } (cid:1) ∼ → Ω ( X,ϕ ) (cid:16) diag N f wCh P ⊗ ∆ • K (cid:17) ≃ Ω ( X,ϕ ) N wCh P ∞ K of derived prestack groups. We first need to define the action of haut K ( X ) on P ∞ { X } : Lemma 4.9.
There is an ∞ -action of the derived prestack group haut ( X ) on P ∞ { X } .Proof. We have an equivalence haut ( X ) ≃ f haut ( X ), where f haut ( X ) is the sim-plicial submonoid of haut ( X ) whose vertices are the self acyclic fibrations X ∼ → X .This equivalence is simply given by the functorial factorization properties of theunderlying model category (which replace functorially any weak equivalence by aweakly equivalent acyclic fibration), implying that every self-weak equivalence of X is in the connected component of a self acyclic fibration of X . We then definethe map haut ( X ) × P ∞ { X } → P ∞ { X } ( f, ψ ) Rf ∗ ψ as follows. We associate to f its equivalent acyclic fibration Rf , and use it to get anew P ∞ -algebra structure on X . We let End Rf be the (dg-)Prop associated to themorphism R f . Indeed for any acyclic fibration g : X → Y , we have the coreflexiveequalizer End g ( n, m ) = Eq (cid:16) Hom ( X ⊗ n , X ⊗ m ) × Hom ( Y ⊗ n , Y ⊗ m ) ⇒ Hom ( X ⊗ n , Y ⊗ m ) (cid:17) where the maps are given by either postcomposition or precomposition by g . Notethat End g has a natural Prop structures and two canonical Prop maps g ∗ , g ∗ to End X and End Y . Now taking g = Rf : X → X , we have a lifting0 / / (cid:15) (cid:15) End
RfRf ∗ (cid:15) (cid:15) Rf ∗ / / End X P ∞ Ψ / / ; ; ✇✇✇✇ End X . since, by [32, Lemma 7.2], the right vertical morphism is an acyclic fibration inthe model category of dg properads, and P ∞ is cofibrant. Moreover, given twohomotopy automorphisms f and g , the lifts obtained by R ( g ◦ f ) ∗ ψ and Rg ∗ ( Rf ∗ ψ )are homotopy equivalent by contractibility of the space of lifts in the commutativesquare above. Note that for any cdga A , the category M od A is a cofibrantly gen-erated symmetric monoidal model category satisfying the limit monoid axioms [32, ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 49
Section 6.6], so that such lifting properties still holds for A -linear P ∞ -structures in M od cofA . Moreover, the naturality with respect to functors ( − ) ⊗ A B induced bymorphisms of cdgas A → B holds up to coherent homotopies again by contractibil-ity of the space of lifts against an acyclic fibration. This yields us a (left) action of G on P ∞ { X } . (cid:3) With this ∞ -action we start the proof of Proposition 4.8. Proof of Propoposition 4.8.
First we construct the commutative diagram
Ehaut K ( X ) × haut K ( X ) P ∞ { X } / / π (cid:15) (cid:15) (cid:15) (cid:15) diag N f wCh P ⊗ ∆ • K | X (cid:15) (cid:15) Bhaut K ( X ) / / N wCh K | X in the following way:(( f k , ..., f ) , ϕ : P ∞ ⊗ ∆ k → End X ) / / π = p ∗ haut K ( X ) (cid:15) (cid:15) (( Xφ ) ∼ → ( X, f k .ϕ ) ... ∼ → ( X, ( f k ◦ ... ◦ f ) .ϕ )) forget (cid:15) (cid:15) ( f k − , ..., f ) / / ( X f k − → ... f → X )where the left vertical map is the projection associated to the Borel construction andthe right vertical map forgets the P ∞ ⊗ ∆ k -algebra structure. The top horizontalmap transfers the P ∞ ⊗ ∆ k -algebra structure on X along the sequence of quasi-isomorphisms given by f k , ..., f and the bottom horizontal map is just an inclusion.It is clear by definition of faces and degeneracies in the simplicial structures involvedthat these four maps are simplicial.It remains to prove that the two horizontal maps are weak equivalences. Forthe bottom arrow, it follows from the work of Dwyer-Kan [20] which identifiesthe connected components of the classification space of a model category with theclassifying complexes of homotopy automorphisms Bhaut ( X ).For the top arrow, we have a morphism of homotopy fibers over XP ∞ { X } / / = (cid:15) (cid:15) Ehaut K ( X ) × haut K ( X ) P ∞ { X } (cid:15) (cid:15) / / Bhaut K ( X ) ∼ (cid:15) (cid:15) P ∞ { X } / / diag N f wCh P ⊗ ∆ • K / / N wCh K | X inducing another morphism of homotopy fibersΩ [ ϕ ] (cid:0) Ehaut K ( X ) × haut K ( X ) P ∞ { X } (cid:1) / / (cid:15) (cid:15) haut K ( X ) ∼ (cid:15) (cid:15) / / P ∞ { X } = (cid:15) (cid:15) Ω ( X,ϕ ) (cid:16) diag N f wCh P ⊗ ∆ • K (cid:17) / / Ω X N wCh K | X / / P ∞ { X } taken over the base point ϕ . (cid:3) Proposition 4.10.
For any chain complex X , the tangent Lie algebra Lie ( haut ( X )) of haut ( X ) is equivalent to End ( X ) = Hom Ch K ( X, X ) equipped with the commu-tator of the composition product as Lie bracket. ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 50
Proof.
This follows from Lurie-Pridham correspondence applied to the formal mod-uli problem
Bhaut ( X ). Recall (Definition 3.13) that Lie ( haut ( X )) = Lie ( \ haut ( X ) id )Further, \ haut ( X ) id is the based loop space (i.e. the automorphsims) of ObjDef o X : dgArt aug K → sSet , the deformation object functor of X of [64, Section 5.2]. Thelatter is a 1-proximate formal moduli problem in the sense of [64, Section 5.1] withassociated formal moduli problem denoted L ( ObjDef o X ). By [8, Lemma 2.11],there is an equivalence of formal moduli problems(4.3) \ haut ( X ) id ∼ = Ω (cid:0) ObjDef o X (cid:1) ∼ = −→ Ω (cid:0) L ( ObjDef o X ) (cid:1) . Using this equivalence (4.3) with Proposition 3.14 then shows that
Lie ( haut ( X )) ∼ = L L ( ObjDefo X ) . By [64, Theorem 5.2.8, Theorem 3.3.1], the Lie algebra associated to L ( ObjDef o X )is precisely Hom Ch K ( X, X ) with its (dg-)Lie algebra structure. (cid:3)
We deduce:
Proposition 4.11.
There is an equivalence of homotopy fiber sequences of Liealgebras g ϕP,X / / ∼ (cid:15) (cid:15) Lie ( haut P ∞ − Alg ( X, ψ )) / / ∼ (cid:15) (cid:15) End ( X ) ∼ (cid:15) (cid:15) g ϕP,X / / Lie (Ω [ ϕ ] ( P ∞ { X } //haut ( X ))) / / End ( X ) Proof.
By Proposition 4.8, we have a morphism of homotopy fiber sequences P ∞ { X } / / = (cid:15) (cid:15) Ehaut K ( X ) × haut K ( X ) P ∞ { X } (cid:15) (cid:15) / / Bhaut K ( X ) ∼ (cid:15) (cid:15) P ∞ { X } / / diag N f wCh P ⊗ ∆ • K / / N wCh K | X . Applying the based loop functor we get an equivalence of homotopy fiber sequencesof derived groups, since we also have, by Proposition 4.8, that the mapΩ [ ϕ ] (cid:0) Ehaut K ( X ) × haut K ( X ) P ∞ { X } (cid:1) ∼ → Ω ( X,ϕ ) (cid:16) diag N f wCh P ⊗ ∆ • K (cid:17) ≃ Ω ( X,ϕ ) N wCh P ∞ K is an equivalence. Taking the formal completions at the appropriate base pointsand applying the Lie -algebra ∞ -functor combined with Proposition 4.10, we obtainthe desired equivalence of homotopy fiber sequences of Lie algebras. (cid:3) As we have already seen, the forgetfull functor mapping P ∞ -algebras to theirunderlying complexes induces a morphism of the homotopy automorphisms derivedprestack groups of both categories. Lemma 4.12.
Let ( X, ψ ) be a P ∞ -algebra. The forgetful derived formal groupmorphism \ haut P ∞ − Alg ( X, Ψ) id −→ d haut ( X ) id has a section in derived formal groups. ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 51
Proof.
By Lemma 3.18, since being group-like is a property, it is sufficient to con-struct a E -monoid morphism(4.4) Ω X ⊗ R (cid:16) N wCh R (cid:17) −→ Ω ( X ⊗ R,ψ ⊗ R ) (cid:16) N wP ∞ − Alg ( M od cofR ) (cid:17) of simplicial presheaves. Since we are restricting to algebras (respectively chaincomplexes) weakly equivalent to the objects ( X ⊗ R, ψ ⊗ R ) (resp. X ⊗ R ) in-duced by our fixed object ( X, ψ ), we can restrict these ∞ -categories (zhich are ∞ -groupoids) to those spanned by these objects. We denote respectively N wP ∞ − Alg ( M od cofR ) ( X ⊗ R,ψ ⊗ R ) and ( N wCh R ) X ⊗ R those sugroupoids. By definition the ∞ -category of P ∞ -algebra structures on X ⊗ R is equivalent to the ∞ -category N wM ap P rop R ( P ∞ , End X ⊗ R ) of R -linear Prop morphisms.In order to conclude, it is thus sufficient to have an ∞ -category morphism(4.5) ( N wCh R ) X ⊗ R −→ N wM ap P rop R ( End X ⊗ R , End X ⊗ R )such that its essential image lies in the connected component of the identity id and its composition with the forgetful functor N wP ∞ − Alg ( M od cofR ) ( X ⊗ R,ψ ⊗ R ) → ( N wCh R ) X ⊗ R is the identity. Indeed, by precomposition with ψ ⊗ R and takingthe base loops (i.e. self-morphims) at respectively X ⊗ R and ( X ⊗ R, ψ ⊗ R ), sucha morphism (4.5) precisely gives an E -monoid morphismΩ X ⊗ R (cid:16) N wCh R (cid:17) −→ Ω ( X ⊗ R,ψ ⊗ R ) (cid:0) M ap
P rop R ( P ∞ , End X ⊗ R ) (cid:1) which is functorial in R ; and therefore induces the seeked for morphism (4.4).The proof of the existence of the morphism (4.5) is similar to the one of Lemma 4.9.Indeed, we apply to the acyclic fibration g = id X ⊗ R construction of the canonicalzigzags of weak-equivalences End ( X ⊗ R ) ≃ ←− End g ≃ −→ End ( X ⊗ R ) . Since we take g = id , this zigzag of weak-equivalences preserves the underlying P ∞ -structure of X ⊗ R and further, its composition with the forgetful functor isthe identity. (cid:3) Now we can state properly our result:
Theorem 4.13.
There is an equivalence of L ∞ -algebras Lie ( haut P ∞ − Alg ( X, ψ )) ∼ = g ψP,X ⋊ h End ( X ) . Proof.
Using the haut K ( X ) action (Lemma 4.9) and proposition 4.10, we can iden-tify the fiber sequence of Theorem 3.24 with g ϕP,X / / Lie ( haut P ∞ − Alg ( X, ψ )) / / End ( X ) . The result is then a direct consequence of Proposition 4.11 once we identify the(homotopy) Lie algebra
Lie (Ω [ ϕ ] ( P ∞ { X } //haut ( X ))) with the semi-direct product g ψP,X ⋊ h End ( X ). By Lemma 4.7, lemma 3.18 and the commutativity of the rightsquare of the diagram of Proposition 4.8, we only need to find a section of theformal group morphism associated to the derived prestack group map defined, onany artinian R , by haut P ∞ − Alg ( X, ψ )( R ) ∼ = Ω ( X ⊗ R,ϕ ) N wCh P ∞ R −→ Ω X ( N wCh R | X ) ∼ = haut ( X )( R ) . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 52
This is given by Lemma 4.12. (cid:3)
The operad of differentials.
Theorem 4.13 express the Lie algebra structureof the homotopy automorphisms of a P ∞ -algebra as a semi-direct product involv-ing the standard operadic deformation complex of P ∞ -algebras. In the next twosubsections we actually express the semi-direct product explicitly as an L ∞ -algebra g ψ + P + ,X , obtained as a Maurer-Cartan twisting of a convolution L ∞ -algebra involv-ing a “plus construction” for properads. The use of the + construction to deformusual deformation complex of morphism of properads also appear as a crucial partin Merkulov-Willwacher study of quantization functors [74].We start by recalling the following definition of Merkulov [73]. Definition 4.14.
Let P be any dg properad with presentation P = F ( E ) / ( R ) anddifferential δ . We define P + to be the dg-properad with presentation F ( E + ) / ( R )and differential δ + where the Σ-biobject E + is defined by E + (1 ,
1) = E (1 , ⊕ K [1] and E + ( m, n ) = E ( m, n ) . In other word we add to E a generating operation u of degree −
1, with one inputand one output. The differential δ + is modified so that its restriction to E is still δ and further δ + ( u ) = u ⊗ u ∈ E (1 , ⊗ E (1 , . The role of the generator u is thus to twist of a complex X when we consider a P + -algebra structure on X . The following is proved in [73] (and also follows fromthe argument of 4.17). Lemma 4.15.
The construction P ( P ) + is an endofunctor ( − ) + : P rop → P rop of the category of dg-properads.Further, properad morphisms ϕ + : P + → End ( X,d ) for a given complex X with differential d corresponds to properad morphisms P → End ( X,d − ϕ + ( u )) for X equipped with the twisted differential d − ϕ + ( u ) . In particular, if X is a graded vector space then P + -algebra structures on X equip X simultaneously with a P -algebra structure and a compatible differential.Let us reinterpret this construction by defining the following operad: Definition 4.16.
The operad of differentials Di is the quasi-free operad Di =( F ( E ) , ∂ ), where E (1) = K δ with δ a generator of degree − E ( n ) = 0 for n = 1and ∂ ( δ ) = δ ◦ δ is the operadic composition ◦ : Di (1) ⊗ Di (1) → Di (1).We will do an abuse of notation and still note Di the properad freely generatedby this operad. Lemma 4.17.
Let ( V, d V ) be a complex.(1) A Di -algebra structure φ : Di → End V on V is a twisted complex ( V, d V − δ V ) where δ V is the image of the operadic generator δ under φ .(2) A morphism of Di -algebras f : ( V, d V − δ V ) → ( W, d W − δ W ) is a chainmorphism f : ( V, d V ) → ( W, d W ) which satifies moreover f ◦ ( d V − δ V ) = ( d W − δ W ) ◦ f (it is a morphism of twisted complexes).Proof. (1) The morphism φ is entirely determined by the image of the generator δ .Since Di (1) → Hom ( V, V ) ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 53 is a morphism of complexes, its compatibility with the differentials reads φ ( ∂ ( δ )) = d V ◦ δ V + δ V ◦ d V which gives the equation of twisting cochains δ V = d V ◦ δ V + δ V ◦ d V , hence ( d V − δ V ) = d V + δ V − d V ◦ δ V − δ V ◦ d V = 0 . (2) A Di -algebra structure on V is given by a morphism Di ( V ) → V , and a Di -algebra morphism f : V → W is a chain morphism fitting in the commutativesquare Di ( V ) Di ( f ) / / (cid:15) (cid:15) Di ( W ) (cid:15) (cid:15) V f / / W .
Since a Di -algebra structure is determined by the image of the generator δ via Di (1) ⊗ V → V , this amounts to the commutativity of the square Di (1) ⊗ V Di (1) ⊗ f / / (cid:15) (cid:15) Di (1) ⊗ W (cid:15) (cid:15) V f / / W , which is exactly saying that f is a morphism of twisted complexes. (cid:3) Remark . Let us note that Di is a non-negatively graded quasi-free operad,hence a cofibrant operad. In particular, one do not need to take a resolution of itto consider the associated simplicial presheaf of Di -algebra structures.The only effect of the plus construction on the cohomology of a properad P is toadd a new generator of arity (1 ,
1) to H ∗ P whose square is zero. That will implythe next lemma. Lemma 4.19. If ϕ : P → Q is a quasi-isomorphism of properads, then the inducedmap ϕ + : P + → Q + is a quasi-isomorphism. In other words,the endofunctor ( − ) + : P rop → P rop preserves weak equivalencesof (dg-)properads.
Proof.
Let ϕ : P ∼ → Q be a quasi-isomorphism of dg props whose collections ofgenerators are respectively E P and E Q , such that E + P (1 ,
1) = E (1 , ⊕ K u P and E + Q (1 ,
1) = E (1 , ⊕ K u Q . Then H ∗ ( ϕ + ) sends [ u P ] to [ u Q ] (where [ − ] denotesthe cohomology class) and coincides with H ∗ ( ϕ ) on the other generators. The onlyrelations satisfied by [ u P ] and [ u Q ] are that they are both of square zero so H ∗ ( ϕ + )is still a prop isomorphism, hence ϕ + is a quasi-isomorphism. (cid:3) This operad Di is a model for the moduli problem associated to derived ho-motopy sel-equivalences haut ( X ). Indeed, the operadic moduli space Di { X } of a Di -algebra X controls the homotopy automorphism of the underlying complex: ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 54
Proposition 4.20.
There is an isomorphism of dg Lie algebras g trivDi,X ≃ Lie ( haut ( X )) . Proof.
The operad Di is a minimal model in the sense of Theorem 2.19. It is of theform Di = ( F ( s − C ) , ∂ ), where C is a cooperad generated by a single generator u ofdegree 0 with a coproduct determined by ∆ C ( u ) = u ⊗ u . Moreover, the trivial Di -algebra structure triv sends u to 0, so the Lie bracket on g trivDi,X = Hom Σ ( C, End X )is just the convolution Lie bracket obtained by taking the graded commutator ofthe convolution product. At the level of complexes, we have g trivDi,X = Hom Σ ( C, End X )= Hom Ch K ( K u, Hom ( X, X )) ∼ = End ( X ) . It remains to compare the Lie structures. Since both Lie brackets are gradedcommutators of associative products, we just have to compare these products. Theproduct on
End ( X ) is the composition of homomorphisms. The product on g trivDi,X is the convolution product, obtained on two elements f, g : K u → End ( X ) byapplying first the infinitesimal cooperadic coproduct ∆ (1) to u , then replacing thevertices by f ( u ) and g ( u ), and finally composing these maps in End ( X ). Underthe identification of Hom Ch K ( K u, End ( X )) with End ( X ), this gives exactly thecomposition product on End ( X ) so the two structures agree. (cid:3) Computing the tangent Lie algebra of homotopy automorphims.
Inthis section, we relate
Lie (cid:0) haut P ∞ ( X, ψ ) (cid:1) with the plus construction.Let I be the initial properad. Lemma 4.21.
Let P be a properad. (1) There is a cofibrant quasi-free resolution P ∞ of P such that the inducedmap P + ∞ −→ P + is a cofibrant quasi-free resolution of P + . (2) There is a commutative square of properads (4.6) Di / / (cid:15) (cid:15) P + ∞ (cid:15) (cid:15) I / / P ∞ where Di → I and P + ∞ → P ∞ are the forgetful maps (sending the gener-ator of Di to ), the upper horizontal map is the inclusion and the lowerhorizontal map is the initial morphism. (3) The commutative square (4.6) is a homotopy pushout.Proof. (1) By [71, Corollary 40], quasi-free properads ( F ( E ) , ∂ ) endowed with aSullivan filtration (that is an exhaustive filtration ( E i ) i ≥ such that E i → E i +1 aresplit dg-monomorphisms and δ ( X i ) ⊂ F ( X i − )) are cofibrant, and any properadadmits a resolution by such. The functor ( − ) + takes Sullivan quasi-free properadsto Sullivan quasi-free properads. Lemma 4.19 thus implies P + ∞ is a quasi freecofibrant resolution of P + if P ∞ is a resolution of P by a Sullivan properad.(2) and (3) We compare first P + ∞ and P ∞ ∨ Di , where ∨ stands for the coproductof properads (see [71, Appendix A.3] for its definition). Since the free properadfunctor F is a left adjoint, it preserves coproducts and thus comes with natural ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 55 isomorphisms F ( M ⊕ N ) ∼ = F ( M ) ∨ F ( N ). If we take the coproduct P ∞ ∨ Q ∞ of two quasi-free properads P ∞ = ( F ( M ) , ∂ P ) and Q ∞ = ( F ( M ) , ∂ Q ), then viathe previous isomorphism we can define a differential on F ( M ⊕ N ) by taking thederivation associated to ∂ P | M ⊕ ∂ Q | N : M ⊕ N → F ( M ) ⊕ F ( N ) ֒ → F ( M ⊕ N )by universal property of derivations and the fact that this morphism satisfies thetwisting cochain equation. In the case where Q = Di , it turns out that the freeproperad underlying P + ∞ is F ( M ⊕ K d ) and the differential on P + ∞ (see [73]) coincideswith the one above, yielding a properad isomorphism P + ∞ ∼ = P ∞ ∨ Di.
Remind that the coproduct is defined as the following pushout diagram over theinitial object I / / (cid:15) (cid:15) Di (cid:15) (cid:15) P ∞ / / P ∞ ∨ Di and let us consider the following pushout diagram Di / / (cid:15) (cid:15) I (cid:15) (cid:15) P ∞ ∨ Di / / g P ∞ . We already know that P ∞ ∨ Di ∼ = P + ∞ , so to conclude the proof of this Lemma wehave to show that g P ∞ = P ∞ . For this, let us remark that concatenating these twopushout diagrams I / / (cid:15) (cid:15) Di / / (cid:15) (cid:15) I (cid:15) (cid:15) P ∞ / / P ∞ ∨ Di / / g P ∞ gives a new pushout diagram I / / (cid:15) (cid:15) I (cid:15) (cid:15) P ∞ / / g P ∞ where the upper horizontal map is Id I , so that the pushout is indeed P ∞ . For themoment, we got a strict pushout, and we still have to explain why it is a homotopypushout. For this, we just have to notice that the three properads considered inthe diagram I ← Di ֒ → P + ∞ are cofibrant and that the map going to the right is a cofibration. This is sufficientto state that the limit of this diagram is equivalent to its homotopy limit, see forexample [12, Exemple 4.2]. (cid:3) ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 56
Remark . Let us note that, for any P ∞ -algebra ( X, ψ ), we have a commutativediagram of properad morphisms Di (cid:31) (cid:127) / / triv " " ❊❊❊❊❊❊❊❊❊ P + ∞ / / / / ψ + (cid:15) (cid:15) P ∞ ψ { { ①①①①①①①①① End X where the maps relating Di , P ∞ and P + ∞ are the ones defined in the Lemma (4.21). Lemma 4.23.
Let X be a chain complex. The diagram (4.6) induces an homotopypullback P ∞ { X } / / (cid:15) (cid:15) P + ∞ { X } (cid:15) (cid:15) I { X } / / Di { X } . of simplicial presheaves.Proof. For any cdga A , the functor M ap
P rop ( − , End X ⊗ A ) is a simplicial mappingspace with fibrant target in a model category, so it sends homotopy colimits tohomotopy limits. In particular, the homotopy pushout of the Lemma 4.23 inducesa homotopy pullback. (cid:3) Proposition 4.24.
The maps of simplicial presheaves of lemma 4.23 induces afiber sequence of L ∞ -algebras g ψP,X → g ψ + P + ,X → g trivDi,X . Proof.
Since I is the initial properad, the simplicial presheaf I { X } is nothing butthe constant presheaf sending everything to the point. Therefore, the homotopypullback of lemma 4.23 is a homotopy fiber sequence of simplicial sets which canbe pointed by the sequence of base points ψ ψ + triv. Taking the corresponding based loops and using proposition 3.14, we deduce afiber sequence of derived groups whose corresponding homotopy fiber sequence of L ∞ -algebras is the one of the proposition. (cid:3) Remark . An alternate way to get this fiber sequence, starting from Lemma4.21, is to observe that this pushout induces a pullback of convolution L ∞ -algebras g P,X / / (cid:15) (cid:15) g P + ,X (cid:15) (cid:15) g I,X / / g Di,X and that g I,X = 0, so that this a fiber sequence of L ∞ -algebras. Along this fibersequence, the Maurer-Cartan element ψ of g P,X is sent to ψ + , which is in turn sentto triv . Twisting our L ∞ -algebras by these Maurer-Cartan elements produces anew fiber sequence g ψP,X → g ψ + P + ,X → g trivDi,X . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 57
Moreover, the second arrow is a surjection, hence a fibration in the model categoryof L ∞ -algebras, and all objects are fibrant, so this fiber sequence is a homotopyfiber sequence.To conclude, we compare this fiber sequence with the fiber sequence g ψP,X → Lie ( haut P ∞ − Alg ( X, ψ )) → Lie ( haut ( X )) . of Theorem 3.24 to obtain: Theorem 4.26.
There is a quasi-isomorphism of L ∞ -algebras g ψ + P + ,X ≃ g ϕP,X ⋉ f End ( X ) ≃ Lie ( haut P ∞ ( X, ψ )) . Proof.
We already have by Proposition 4.8 an equivalence of homotopy fiber se-quences P ∞ { X } / / ∼ (cid:15) (cid:15) N wP ∞ − Alg | X / / ∼ (cid:15) (cid:15) Bhaut ( X ) ∼ (cid:15) (cid:15) P ∞ { X } / / P ∞ { X } //haut ( X ) / / Bhaut ( X ) . . To conclude the proof, we have to compare the lower fiber sequence with the fibersequence P ∞ { X } → P + ∞ { X } → Di { X } . Precisely, we apply twice the décalage construction to get two fiber sequences(4.7) Ω [ ϕ ] P ∞ { X } //haut ( X ) → haut ( X ) → P ∞ { X } and(4.8) Ω ϕ + P + ∞ { X } → Ω triv Di { X } → P ∞ { X } . We do not expect to get directly an equivalence at this level, however, our strategyis to define a commutative square(4.9) Ω triv Di { X } / / (cid:15) (cid:15) P ∞ { X } (cid:15) (cid:15) haut ( X ) / / P ∞ { X } hence inducing a morphism between the corresponding homotopy fiber sequences,so that the two vertical arrows induce equivalences of L ∞ -algebras at the tangentlevel, after completion of the derived groups at the appropriate base points.The right hand vertical arrow of (4.9) is just the identity morphism. Note that Di is the properad freely generated by a cofibrant operad . If O ∞ is a cofibrantdg operad, then the homotopies between two morphisms ϕ, ψ : O ∞ → End X are in bijection with ∞ -quasi-isotopies in O ∞ − Alg between the corresponding O ∞ -algebras, that is, ∞ -quasi-isomorphisms whose first level lies in the connectedcomponent of the identity in haut ( X ) [31, Theorem 5.2.1].In particular, a loop in Ω ϕ O ∞ { X } , that is, a self-homotopy of ϕ , induces aself- ∞ -isotopy of ( X, ϕ ). In the particular case where the operad is augmentedand ϕ is the trivial O ∞ -algebra structure on X (that is, it factorizes through the ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 58 augmentation O ∞ → I ), then such a self- ∞ -isotopy is just a self quasi-isomorphismin the connected component of the identity. Consequently, there is a natural mapΩ triv Di { X } → haut ( X )which makes the commutative square (4.9) above commutes, and becomes an iso-morphism when restricting the target to the connected component of id X . Thismeans that, even though this map is not an equivalence, taking the tangent Lie al-gebras of the derived formal groups obtained after completions at the appropriatebase points (trivial loop on the left, id X on the right) leads to a quasi-isomorphismof Lie algebras g trivDi,X ≃ → Lie ( \ haut ( X ) id ) = End ( X ) . Now, this commutative square (4.9) induces a morphism between the homotopyfibers given by (4.7) and (4.8)Ω ϕ + P + ∞ { X } → Ω [ ϕ ] P ∞ { X } //haut ( X ) . Since this square becomes a square of quasi-isomorphisms of Lie algebras at thetangent the level, the induced morphism between the Lie algebras of the fibers is aquasi-isomorphim as well g ϕ + P + ,X = Lie ( \ Ω ϕ + P + ∞ { X } ) ∼ → Lie ( \ Ω [ ϕ ] P ∞ { X } //haut ( X )) . Therefore, Proposition 4.11 gives us the equivalence of the later Lie algebra with
Lie ( haut P ∞ ( X, ϕ )). Moreover
Lie ( \ Ω [ ϕ ] P ∞ { X } //haut ( X )) is canonically equiva-lent to the homotopy semi-direct product g ϕP,X ⋉ f End ( X ) by Theorem 4.13. (cid:3) Remark . Although it is interesting to see the role of the Borel constructionhere, there is an alternate proof of Theorem 4.26 which makes no use of it. Let ussketch it; for this, we compare the fiber sequencesΩ [ ϕ ] P ∞ { X } //haut ( X ) → haut ( X ) → P ∞ { X } and haut P ∞ ( X, ϕ ) → haut ( X ) → P ∞ { X } by checking that actually, in both cases we are considering the fibers of the samemap from haut ( X ) to P ∞ { X } , which is the map sending a homotopy automorphismto its action on ϕ . Hence an equivalence of Lie algebras Lie ( haut P ∞ ( X, ϕ )) ≃ Lie (Ω [ ϕ ] P ∞ { X } //haut ( X )) . Then, the argument line of the proof above provides the equivalence
Lie (Ω [ ϕ ] P ∞ { X } //haut ( X )) ≃ g ϕ + P + ,X . Theorem 4.26 shows that the + construction is crucial to study deformation ofdg-algebras and not just deformations of algebraic structures on a fixed complex.
Example 4.28 (Strict associative algebras) . Let (
A, ψ ) be a (necessarily strict)associative algebra concentrated in degree 0. Then by proposition 4.10, one has haut ( A ) ∼ = Hom ( A, A ) the Lie algebra (concentrated in degree 0) of endomorphismsof the underlying vector space of A .It is a standard computation ([72, 73]) that the Lie algebra g ψAss = L Ass ∞ { A } ψ isisomorphic to the sucomplex C •≥ ( A, A )[1] = L n ≥ Hom ( A ⊗ n , A ) of the shiftedHochschild cochain complex C • ( A, A )[1] with Lie bracket given by the restriction
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 59 of the standard Gerstenhaber complex. Using these equivalences, the action givenby lemma 4.9 of
Hom ( A, A ) on C •≥ ( A, A )[1] is given by the usual action of
Hom ( A, A ) = C ( A, A )[1] on the Hochschild complex given by the Lie bracket.Therefore, we have that
Lie ( haut Ass ∞ ( A, m )) ∼ = C •≥ ( A, A )[1]) ∼ = Hom ( A ⊗ > , A )[1] ⋊ Hom ( A, A ) . The latter can be deduced by an immediate computation mimicking [73] from theoperad
Ass + ∞ obtained by the usual Koszul resolution of Ass , using theorem 4.26.In particular, the moduli space
Ass ∞ { A } ψ ( K [[ t ]]) controls the algebra structureson A [[ t ]] whose reduction modulo t is the given one.However, the moduli space haut Ass ∞ ( A, ψ )( K [[ t ]]) controls the algebra structureson A [[ t ]] whose reduction modulo t is the given one, up to isomorphism of algebraswhich are the identity modulo t .In other words, the set of connected components of such deformations in thefirst case is the set of all possible deformations, while the connected component ofthe derived prestack group haut Ass ∞ ( A, ψ ) are the set of all possible deformationsmodulo the standard gauge equivalences.Example 4.28 can be generalized to algebras concentrated in degree 0 over otheroperads, see Section 6.1.Note that this result extends for dg-algebras (and A ∞ -algebras as well), see 5.1. Corollary 4.29.
Let P ∞ = ( F ( s − C ) , ∂ ) ∼ → P be a cofibrant quasi-free resolutionof an operad P where C is a cooperad and ( X, ψ ) be a P ∞ -algebra. One has anequivalence of Lie algebras Lie ( haut P ∞ ( X, ψ )) ∼ = Hom Σ ( C ⊕ I, Q ) ∼ = Coder ( C ( X [1])) D Ψ ⋊ End ( X, X ) where the last term is the L ∞ -algebra of coderivations of the cofree coalgebra on X [1] twisted by the Maurer Cartan element D Ψ (the coderivation of square zerocorresponding to the P ∞ -algebra structure Ψ ) and the action of End ( X, X ) is givenby the composition of coderivations of C ( X [1]) .Proof. The isomorphism of Lie algebras given in [61, Proposition 10.1.17] inducesan isomorphism of Lie algebras g P,X ∼ = Coder ( C ( X [1])) . Then, the P ∞ -algebra structure ψ is a Maurer-Cartan element in g ψP,X , whoseimage under the Lie algebra isomorphism above gives a Maurer-Cartan element D ψ in Coder ( C ( X [1])), that is, a degree 1 coderivation of square zero. Twistingthis isomorphism by those Maurer-Cartan elements gives an isomorphism g ψP,X ∼ = Coder ( C ( X [1])) D ψ . Therefore the equivalence between the r.h.s and l.h.s in the theorem follows fromTheorem 4.26. The tangent action of
End ( X ) on g ψP,X induced by the action of haut P ∞ ( X, ψ ) on P ∞ { X } (lifting, for any f ∈ haut P ∞ ( X, ψ ), the P ∞ -structuresalong End f → End X by Lemma 4.9) gives under this isomorphism an action of End ( X ) on Coder ( C ( X [1])) D ψ defined by the composition of coderivations. Theequivalence between the middle term and the r.h.s of the equivalences now followsfrom proposition 4.20 and the fact that the dg-operad Di is of the form Di =( F ( K δ ) , ∂ ) ∼ = ( F ( s − I ) , ∂ )(Definition 4.16). (cid:3) ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 60
Example 4.30.
Let A be a dg Poisson algebra. If we see A as a Poisson algebra,we can consider the truncated Poisson complex CH • > P ( A ) and the full Poissoncomplex CH ∗ P ( A ), see § 5.2. If we see A as an associative algebra, we can con-sider the truncated and full Hochschild complexes of A , respectively CH • > ( A )and CH ∗ ( A ). Poisson complexes are associated to deformations of A as a Poissonalgebra, and Hochschild complexes are associated to deformations of A as an asso-ciative algebra (hence forgetting its Poisson bracket). All these complexes form Liealgebras up to a degree shift by one (in the Hochschild case, the Lie structure isknown as the Gerstenhaber bracket). According to what we proved above, the Liealgebras CH • > P ( A )[1] and CH • > ( A )[1] controls the deformation problems of A inthe ∞ -category of homotopy Poisson algebras and in the ∞ -category of homotopyassociative algebras respectively, in a precise sense given by the theory of derivedformal moduli problems (Corollaries 5.3 and 5.7). Their full version controls thedeformation theory of the category of A -modules (seen respectively as modules overa Poisson algebra or as modules over an associative algebra), which is in generalanother kind of formal moduli problem. Equivalently, the full Hochschild complexcontrols the deformations of A as a curved A ∞ -algebra [77].However, in the case where A is concentrated in degree zero, we observe that,first, deformations of A are deformations as a strict Poisson algebra or as a strictassociative algebra, and second, curved and uncurved deformations are equivalent.Consequently, the space of Maurer Cartan elements are the same for the truncatedand the untruncated versions of Poisson and Hochschild complexes.This observation is crucial in the study of formality theorems for Poisson al-gebras and deformation quantization of Poisson structures on manifolds [57, 83].Let us fix A = C ∞ ( R d ) the algebra of smooth functions on R d , and consider twocomplexes. First, the full Hochschild complex CH ∗ ( A, A ), second, the complex ofpolyvector fields T poly ( A ) = (cid:16)L k ≥ V k Der ( A )[ − k ] (cid:17) [1] (recall that vector fieldsare derivations of the ring of smooth functions). The complex of polyvector fieldsalso forms a (shifted) Lie algebra with a Lie structure induced by the bracket ofvector fields. The classical Hochschild-Kostant-Rosenberg theorem (HKR for short)states that the cohomology of CH ∗ ( A, A ) is precisely T poly ( A ). However, the HKRquasi-isomorphism is not compatible with their respective Lie algebra structures.In [57], Kontsevich proved that the HKR quasi-isomorphism lifts to an L ∞ -quasi-isomorphism T poly ( A )[1] ∼ → CH ∗ ( A, A )[1]by building an explicit formality morphism. An alternative proof of the formalitytheorem is due to Tamarkin [83] and provides a formality quasi-isomorphism ofhomotopy Gerstenhaber algebras (that is E -algebras). Here T poly ( A ) is actuallythe deformation complex of the trivial Poisson algebra structure. In general, the fullHochschild complex CH ∗ ( A, A ) controls deformations of A as a curved algebra, butsince A is in degree zero, the space of Maurer Cartan elements obtained from the fullHochschild complex is the same as the one from the truncated Hochschild complex.This is important, because the formality theorem holds for the full complex butnot for the truncated one. This formality theorem implies the equivalence of theassociated formal moduli problems. Then, applying these moduli problems to thering of formal power series K [[ t ]], one gets that the the set of isomorphism classesPoisson algebra structures on A [[ ~ ]] without constant term is in bijection with gauge ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 61 equivalence classes of ∗ ~ -products (that is, associative formal deformations of theproduct of A ).Deformation theory of Poisson algebras and more generally of shifted Poissonalgebras and their up to homotopy cousins are the main topic of the next section.5. Examples
Deformations of E n -algebras. we now generalize example 4.28 to the ho-motopy setting and to higher algebras, that is E n -algebras as well. The latterare higher generalizations of homotopy associative algebras. In fact the (symmet-ric monoidal) ∞ -category of E n -algebras can be described [62] as the (symmetricmonoidal) category of E -algebras with values in the (symmetric monoidal) ∞ -category of E n − -algebras; and thus ultimately (and informally) as the ∞ -categoryof complexes equipped with n -many compatible homotopy associate algebra struc-ture.To define E n -algebras, one first note that the configuration spaces of (rectilin-ear embeddings of) n -disks into a bigger n -disk gather into a topological operad D n , called the little n -disks operad. An E n -operad (in chain complexes) is a dgoperad quasi-isomorphic to the singular chains C ∗ ( D n ) of the little n -disks operad.There is an ∞ -functor from E n -algebras to L ∞ -algebras whose composition withthe forgetful functor to chain complexes is the shift X X [1 − n ].Given an ordinary associative (or E ) algebra A , its endomorphisms Hom biMod A ( A, A )in the category biM od A of A -bimodules form nothing but the center Z ( A ) of A . Deriving this hom object gives the Hochschild cochain complex C ∗ ( A, A ) ∼ = R Hom biMod A ( A, A ) of A , and the associated Hochschild cohomology HH ∗ ( A, A )of A satisfies HH ( A, A ) = Z ( A ). More generally, one has the following definition(see [27, 67, 43]). Definition 5.1.
The (full) Hochschild complex of an E n -algebra A , computing itshigher Hochschild cohomology, is the derived hom C ∗ E n ( A, A ) = R Hom E n A ( A, A ) inthe category of (operadic) A -modules over E n .The Deligne conjecture endows the Hochschild cochain complex with an E n +1 -algebra structure [43, Theorem 6.28] or [27, 67]. Associated to an E n -algebra A ,one also has its cotangent complex L A , which classifies square-zero extensions of A [27, 67]. Definition 5.2 ([27]) . The tangent complex T A of an E n -algebra A is the dual T A := Hom E n A ( L A , A ) ∼ = R Der ( A, A ).The latter isomorphism gives a L ∞ -structure to T A and Francis [27, 67] hasproved that T A [ − n ] has a canonical structure of E n +1 -algebra (lifting the L ∞ -structure). He further proved that there is a fiber sequence T A [ − n ] → CH ∗ E n ( A, A ) → A where the first map is a map of E n +1 -algebras.A corollary of our theorem 4.26 is the following operadic identification of thetangent complex T A of an E n -algebra (5.2): meaning each of this algebra structure is an (homotopy) algebra morphism with respect tothe other algebra structures note that the operadic E -module are precisely the bimodules ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 62
Corollary 5.3.
The E n -Hochschild tangent complex T A of an E n -algebra A isnaturally weakly equivalent as an L ∞ -algebra to g ψ + E + n ,A : T A ≃ Lie ( haut E n ( A, ψ )) ≃ g ψ + E + n ,A , where ψ + is the E + n -algebra structure on A trivially induced by its E n -algebra struc-ture ψ : E n → End A as above, and haut E n ( A ) is the derived prestack group ofhomotopy automorphisms of A as an E n -algebra.Proof. According to [27, Lemma 4.31], the homotopy Lie algebra of homotopyautomorphisms
Lie ( haut E n ( A, ψ )) is equivalent to the tangent complex T A of A .Hence Theorem 4.26 implies the corollary. (cid:3) Deformation complexes of
P ois n -algebras. We now introduce Tamarkindeformation complexes of a
P ois n -algebra [85] and prove that these complexes docontrol deformations of (dg-) P ois n -algebras.We denote by P ois n the operad of P ois n -algebras and uP ois n the operad ofunital P ois n -algebras.Let A be a dg P ois n -algebra, with structure morphism ψ : P ois n → End A . Wedenote by CH ∗ P ois n ( A, A ) its
P ois n -Hochschild cochain complex, also referred toas its P ois n -deformation complex as defined by Tamarkin [85] and Kontsevich [56].Following Calaque-Willwacher [10], we note that this complex is given by the sus-pension(5.1) CH ∗ P ois n ( A, A ) :=
Hom Σ ( uP ois n ∗ { n } , End A )[ − n ]of the underlying chain complex of the convolution Lie algebra. Here ( − ) ∗ is thelinear dual and { n } is the operadic n -iterated suspension. The inclusion of P ois n in uP ois n induces a splitting (as a graded space)(5.2) CH ∗ P ois n ( A, A ) ∼ = A ⊕ Hom Σ ( P ois n ∗ { n } , End A )[ − n ]and also gives rise to the truncated deformation complex(5.3) CH ( • > P ois n ( A, A ) =
Hom Σ ( P ois n ∗ { n } , End A )[ − n ]obtained by deleting the “unit part” A , which is more relevant to deformations of P ois n -algebras , see Lemma 5.6. Note that both complexes are naturally bigradedwith respect to the internal grading of A and the “operadic” grading coming from uP ois n ∗ . The notation CH ( • > P ois n ( A, A ) is there to suggest that we are taking thesubcomplex with positive weight with respect to the operadic grading.The suspensions CH ∗ P ois n ( A, A )[ n ] and CH ( • > P ois n ( A, A )[ n ] have canonical L ∞ -structures since they are convolution algebras, and CH ( • > P ois n ( A, A )[ n ] is canonicallya sub L ∞ -algebra of CH ∗ P ois n ( A, A )[ n ]. Tamarkin [85] (see also [56, 10]) provedthat the complex CH ∗ P ois n ( A, A ) actually inherits a (homotopy) P ois n +1 -algebrastructure lifting this L ∞ -structure. Further, by (5.2) we have an exact sequence ofcochain complexes(5.4) 0 −→ CH ( • > P ois n ( A, A ) −→ CH ∗ P ois n ( A, A ) −→ A −→ A [ n − ∂ Poisn [ n − −→ CH ( • > P ois n ( A, A )[ n ] −→ CH ∗ P ois n ( A, A )[ n ] . as opposed to deformation of categories of modules ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 63
Remark . The map ∂ P ois n : A ⊂ CH ∗ P ois n ( A, A ) → CH ( • > P ois n ( A, A ) is the partof the differential in the cochain complex CH ∗ P ois n ( A, A ) = A ⊕ CH ( • > P ois n ( A, A )which comes from the operadic structure. That is ∂ P ois n ( x ) ∈ Hom ( A, A ) is themap a
7→ ± [ x, a ] where the bracket is the bracket of the P ois n -algebra. The Jacobiidentity for the Lie algebra A [ n −
1] implies that the sequence (5.5) is a sequenceof L ∞ -algebras. Remark . The operad
P ois n is denoted e n in [10, 85] and the complex CH ∗ P ois n ( A, A )is simply denoted def ( A ) in Tamarkin [85]. We prefer to use the notations we haveintroduced by analogy with (operadic) Hochschild complexes.The next Lemma compares the L ∞ -algebra structure of the truncated P ois n Hochschild complex and the one associated to the derived prestack group of homo-topy automorphisms of a
P ois n -algebra: Lemma 5.6.
Let A be a dg P ois n -algebra with structure map ψ : P ois n → End A .There is an equality of dg Lie algebras g ψ + P ois + n ,A = CH ( • > P ois n ( A, A ) where the right hand side is the truncated cochain complex of a P ois n -algebra definedby Tamarkin as above.Proof. According to the definition of the plus construction ( − ) + given in Section 4,we have P ois + n ∞ = Ω( P ois ∗ n { n } ) + = ( F ( P ois ∗ n { n + 1 } ) + , ∂ + )where P ois n ∞ is the minimal model of P ois n , ( − ) ∗ is the linear dual, { n } is theoperadic n -iterated suspension, Ω is the operadic cobar construction and − is thecoaugmentation ideal of a coaugmented cooperad. Recall that the collection ofgenerators P ois ∗ n { n + 1 } + is given by P ois ∗ n { n + 1 } + (1) = P ois ∗ n { n + 1 } (1) ⊕ K [1] = P ois ∗ n { n + 1 } (1) ⊕ K d where d is a generator of degree 1 and P ois ∗ n { n + 1 } + ( r ) = P ois ∗ n { n + 1 } ( r )for r >
1. The restriction of the differential ∂ + on the generators decomposes into ∂ + = ∂ + δ where ∂ is the differential of the minimal model and δ is defined by δ ( d ) = d ⊗ d and zero when evaluated on the other generators (note that, by theKoszul sign rule and for degree reasons, we have δ ( d ) = 0 so we get a differentialindeed). Now let ψ + : P ois + n ∞ → End A be the operad morphism induced by ψ ,thus a Maurer-Cartan element of the convolution graded Lie algebra g P ois + n ,A . Wetwist this Lie algebra by ψ to get a dg Lie algebra g ψ + P ois + n ,A with the same Liebracket and whose differential is defined by ± ( d A ) ∗ + [ ψ, − ]where ( − ) ∗ denotes the post-composition, d A is the differential on End A inducedby the differential of A , the ± sign is defined according to the Koszul sign rule and[ − , − ] is the convolution Lie bracket. Note here that the Koszul dual cooperad hasno internal differential. We refer the reader to [61, Chapter 12] for more detailsabout such convolution Lie algebras. Now let us point out that P ois ∗ n { n + 1 } + (1) = P ois ∗ n { n + 1 } (1) ⊕ K [1] = ( P ois ∗ n { n } (1) ⊕ K )[1] , ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 64 which implies that g ψ + P ois + n ,A = Hom Σ ( P ois ∗ n { n } ⊕ I, End A ) ψ = Conv ( P ois ∗ n { n } , End A )where Conv ( P ois ∗ n { n } , End A ) is the convolution Lie algebra of [10, Section 2.2].This is an equality of dg Lie algebras, because the convolution bracket is defined bythe action of the infinitesimal cooperadic coproduct on the coaugmentation ideal,so is the same on both sides. (cid:3) Lemma 5.6 together with Theorem 4.26 implies that
Corollary 5.7.
The truncated Tamarkin deformation complex CH ( • > P ois n ( A, A ) con-trols deformations of A into the ∞ -category of dg P ois n -algebras, in other words isthe tangent Lie algebra of the derived prestack group haut P ois n ∞ ( A ) , where P ois n ∞ is a cofibrant resolution of P ois n .Remark . The proof of Lemma 5.6 also shows that the deformation complex g ψP ois n ,A of the formal moduli problem P ois n ∞ { A } ψ is given by the L ∞ -algebra CH ( • > P ois n ( A, A )[ n ], which is the kernel(5.6) CH ( • > P ois n ( A, A )[ n ] := ker (cid:16) CH ( • > P ois n ( A, A )[ n ] ։ Hom ( A, A )[ n ] (cid:17) and is thus a even further truncation of CH ∗ P ois n ( A, A ). The situation is thussimilar to what happens in deformation theory of associative algebras.One can also wonder which deformation problem controls the full complex CH ∗ P ois n ( A, A ).In view of our results and classical results on deformation theory of E n -algebras([55, 77, 27]), we can conjecture that CH ∗ P ois n ( A, A ) shall control deformations ofcategories of modules over
P ois n -algebras into E | n − | -monoidal dg-categories, withsome shift on the linear enrichment of the category when n ≤ Bialgebras.
Let us conclude our series of examples with one of properadicnature. Here we are interested in associative and coassociative bialgebras, andrefer the reader to Example 1.7 for a precise definition as well as the constructionof the corresponding properad
Bialg .What we call the
Gerstenhaber-Schack complex is the total complex of a bicom-plex, defined by(5.7) C ∗ GS ( B, B ) ∼ = Y m,n ≥ Hom dg ( B ⊗ m , B ⊗ n )[ − m − n ] . The horizontal differential is defined, for every n , by the Hochschild differentialassociated to the Hochschild complex of B seen as an associative algebra with coef-ficients in the B -bimodule B ⊗ n . The vertical differential is defined, for every m , bythe co-Hochschild differential associated to the co-Hochschild complex of B seen asa coassociative coalgebra with coefficients in the B -bicomodule B ⊗ m . The compat-ibility between these differentials, which gives us a well defined bicomplex, followsfrom the distributive law relating the product and the coproduct of the bialgebra B (see [37, 73] for details). Combining Theorem 4.26 with the computation of [73],we get: ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 65
Theorem 5.9.
The Gerstenhaber-Schack complex is quasi-isomorphic to the L ∞ -algebra controlling the deformations of dg bialgebras up to quasi-isomorphisms: C ∗ GS ( B, B ) ∼ = g ϕ + Bialg + ∞ ,B ≃ Lie ( haut Bialg ∞ ( B )) . Hence the Gerstenhaber-Schack complex is indeed the L ∞ -algebra controling thederived deformation theory of dg bialgebras in a precise meaning, something newsince the introduction of this complex by Gerstenhaber and Schack in their seminalpaper [37]. Moreover, as emphasized by the results of [73] and [45], this complexplays a crucial role in deformation quantization.6. Concluding remarks and perspectives
To conclude, let us give an overview of the various deformation complexes con-sidered in the litterature and their derived and underived formal moduli problems.6.1.
Algebras over operads in vector spaces.
Let X be a vector space and P an operad with Koszul dual C . Then the cohomological grading on the convolutionLie algebra g P,X = Hom Σ ( C, End X ) is entirely determined by the “weight grading”of operations in the cooperad C . In particular, in the case where X is of finitedimension n , this means that the degree 0 Lie subalgebra of g P,X is nothing but gl ( X ), whose associated Lie group is the general Lie group GL ( X ). This is thegauge group acting on the Maurer-Cartan elements of g P,X , so that the moduli setof Maurer-Cartan elements is MC ( g P,A ) =
M or ( P, End A ) /GL ( A ) . The deformation complex g ϕP,X of a P -algebra A = ( X, ϕ ) controls then the defor-mations of A as a P -algebra , up to linear automorphisms of A . If we replace C by C in the definition of g P,X , which is what we did in the present paper, then thereis no non trivial gauge group acting anymore, and the Maurer-Cartan moduli setis just MC ( g P,A ) =
M or ( P, End A ) . The corresponding underived formal moduli problem, or classical deformation func-tor, controls the deformations of A as a P -algebra in the category of vector spaces,up to isomorphisms.In the derived setting, one replaces Artinian algebras by their dg enhancement,so that the simple description above in terms of gauge group action does not existanymore (notice that, although V is in degree 0, we have to consider haut ( V ⊗ A )for any differential graded local Artinian algebra A , and V ⊗ A is not in degreezero anymore). The relevant theory, described in Section 4, defines the appropriatedeformation problems as loops over the homotopy quotient of the moduli spaceof P -algebra structures by a homotopy automorphisms group. Moreover, it turnsout, as we explained in Remark 3.23, that the corresponding derived formal moduliproblem is given by the formal completion \ N wP − Alg A at A of the n -geometricderived Artin stack of n -dimensional P -algebras.To clarify the link between our derived construction and the underived deforma-tion functor described above, let us restrict our derived moduli problem to localArtinian algebras. In this context, provided that P is an operad in vector spaces,the simplicial presheaves P ∞ { X } and haut ( X, ϕ ) are actually discrete. Indeed,
ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 66 given a local Artinian algebra R , each vertex of the Kan complex P ∞ { X ⊗ m R } factors uniquely through the composite P ∞ ։ P → End V ⊗ m R because of degree reasons, commutation with the differentials and the fact that End X ⊗ m R is concentrated in degree zero. Moreover, for the same reasons, a ho-motopy between two such maps cannot be anything else than the identity, so thatfinally P ∞ { X ⊗ m R } ∼ = M or prop ( P, End X ⊗ m R ) = M C ( g P,X )and the corresponding pointed functor over (
X, ϕ ) is P ∞ { X ⊗ m R } varphi ∼ = M C ( g ϕP,X ) . Note that this is coherent with the fact that left homotopies between such maps arein bijection with ∞ -isotopies of ( X, ϕ ), which boils down to Id ( X,ϕ ) when X is indegree zero. Also for degree reasons, the simplicial presheaf haut ( X, ϕ ) is equivalentto the discrete presheaf defined, for each local Artinian algebra R , by the strictautomorphism group Aut ( X ⊗ m R ), that is, the algebraic group of automorphismsof ( X, ϕ ). The homotopy action of haut ( X, ϕ ) on P ∞ { X } is then nothing butthe gauge action described above, so that the semi-direct product g varphi + P + ,X ≃≃ g ϕP,X ⋉ hol End ( X ) becomes the dg convolution Lie algebra considered in [61, Section12.2.22].6.2. Differential graded algebras over operads.
A differential graded struc-ture on the object X carries non trivial homotopies, and taking into account thisnew homotopy data that do not exist in the degree zero case involves deforma-tions of P -algebra structures into P -algebra structures up to homotopy , that is P ∞ -algebra structures, and therefore taking into account the non trivial homotopytype of the moduli space P ∞ { X } . Given a P ∞ -algebra A = ( X, ϕ ), there are apriori three possible variants of derived deformation problems one could look at:(1) Deformation theory of the operad morphism ϕ : P ∞ → End A ;(2) Deformation theory of A in the ∞ -category of P ∞ -algebras up to ∞ -isotopies ;(3) Deformation theory of A in the ∞ -category of P ∞ -algebras up to quasi-isomorphisms .Problem (1) is, as we saw before, controled by the derived formal moduli problem P ∞ { X } ( R ) = hof ib ϕ ( P ∞ { X ⊗ R } → P ∞ { X } )whose associated L ∞ -algebra is g ϕP,X (constructed with C ). Problem (2) is thesetting in which [61, Section 12.2.22] takes place: an R -deformation of a P -algebra A in the sense of (2) is a an R -linear P ∞ -algebra ˜ A ≃ A ⊗ R with a K -linear P ∞ -algebra ∞ -isomorphism ˜ A ⊗ R K ∼ → A , where ( − ) ⊗ R K is defined by the augmentation of R .Two deformations are equivalent if they are related by an R -linear ∞ -isomorphismwhose restriction modulo m R is the identity, that is ∞ -isotopies. It turns outthat, in the operadic case, problems (1) and (2) are equivalent: by [31, Theorem5.2.1] homotopies between morphisms from P ∞ to End X are in bijection with ∞ -isotopies between the corresponding P ∞ -algebras, and by [61, Section 12.2.22] thelater are also controled by the convolution L ∞ -algebra. Here the gauge group of thedeformation complex (for this moduli problem) of a P ∞ -algebra A is isomorphic tothe group of ∞ -isotopies of A . ERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES 67
We spent some time in this article to deal with Problem (3), which had previ-ously no known construction in the framework of derived deformation theory. Asexplained before, an R -deformation of a P -algebra A in the sense of (3) is a an R -linear P ∞ -algebra ˜ A ≃ A ⊗ R with a K -linear P ∞ -algebra quasi-isomorphism˜ A ⊗ R K ∼ → A , where ( − ) ⊗ R K is defined by the augmentation of R . We builta derived formal group \ haut P ∞ ( A ) id whose corresponding L ∞ -algebra admits twoequivalent descriptions Lie ( \ haut P ∞ ( A ) id ) ≃ g ϕP,X ⋉ hol End ( X ) ≃ g varphi + P + ,X where the middle one exhibits this moduli problem as originating from the homo-topy quotient of the space of P ∞ -algebra structures on X by the homotopy actionof self-quasi-isomorphisms haut ( X ), and the right one explains how one can en-code this explicitely as simultaneous compatible deformations of the P ∞ -algebrastructure and the differential of X .Another way to compare deformation problems (2) and (3) is to recall that thereare equivalences of ∞ -categories P ∞ − Alg [ W − qiso ] ≃ P − Alg [ W − qiso ] ≃ ∞ − P ∞ − Alg [ W − ∞− qiso ]where the first equivalence is induced by the operadic quasi-isomorphism P ∞ ∼ → P ,and the second equivalence is induced by the strictification theorem of [61, Chapter12], the later ∞ -category being the one of P ∞ -algebras with ∞ -morphisms, and with ∞ -quasi-isomorphisms as weak equivalences. Problem (3) concerns deformationtheory in the ∞ -category of P ∞ -algebras up to ∞ -quasi-isomorphisms , hence is arelaxed version of Problem (2) in this sense.6.3. Algebras over properads.
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