The tangent complex and Hochschild cohomology of E_n-rings
aa r X i v : . [ m a t h . A T ] A ug THE TANGENT COMPLEX AND HOCHSCHILD COHOMOLOGY OF E n -RINGS JOHN FRANCIS
Abstract.
In this work, we study the deformation theory of E n -rings and the E n analogue ofthe tangent complex, or topological Andr´e-Quillen cohomology. We prove a generalization of aconjecture of Kontsevich, that there is a fiber sequence A [ n − → T A → HH ∗E n ( A )[ n ], relatingthe E n -tangent complex and E n -Hochschild cohomology of an E n -ring A . We give two proofs:The first is direct, reducing the problem to certain stable splittings of configuration spaces ofpunctured Euclidean spaces; the second is more conceptual, where we identify the sequence asthe Lie algebras of a fiber sequence of derived algebraic groups, B n − A × → Aut A → Aut B n A .Here B n A is an enriched ( ∞ , n )-category constructed from A , and E n -Hochschild cohomologyis realized as the infinitesimal automorphisms of B n A . These groups are associated to moduliproblems in E n +1 -geometry, a less commutative form of derived algebraic geometry, in the senseof To¨en-Vezzosi and Lurie. Applying techniques of Koszul duality, this sequence consequentlyattains a nonunital E n +1 -algebra structure; in particular, the shifted tangent complex T A [ − n ] isa nonunital E n +1 -algebra. The E n +1 -algebra structure of this sequence extends the previouslyknown E n +1 -algebra structure on HH ∗E n ( A ), given in the higher Deligne conjecture. In order toestablish this moduli-theoretic interpretation, we make extensive use of factorization homology,a homology theory for framed n -manifolds with coefficients given by E n -algebras, constructed asa topological analogue of Beilinson-Drinfeld’s chiral homology. We give a separate exposition ofthis theory, developing the necessary results used in our proofs. Contents
1. Introduction 21.1. Acknowledgements 52. The Operadic Cotangent Complex 52.1. Stabilization of O -Algebras 102.2. The E n -Cotangent Complex 122.3. Structure of the Cotangent Complex and Koszul Duality 163. Factorization Homology and E n -Hochschild Theories 213.1. E n -Hochschild Cohomology 213.2. Factorization Homology and E n -Hochschild Homology 224. Moduli Problems 314.1. Automorphisms of Enriched ∞ -Categories 344.2. Infinitesimal Automorphisms of E n -algebras 364.3. Lie Algebras and the Higher Deligne Conjecture 40References 43 Mathematics Subject Classification.
Primary 14B12. Secondary 55N35, 18G55.
Key words and phrases. E n -algebras. Deformation theory. The tangent complex. Hochschild cohomology. Fac-torization homology. Topological chiral homology. Koszul duality. Operads. ∞ -Categories.The author was supported by the National Science Foundation under award number 0902974. . Introduction
In this paper, we study certain aspects of E n -algebra, that is, algebras with multiplication mapsparametrized by configuration spaces of n -dimensional disks inside a standard n -disk. We focus onthe deformation theory of E n -algebras, which is controlled by an operadic version of the tangentcomplex of Grothendieck and Illusie. One of our basic results is a relation between this E n -tangentcomplex and E n -Hochschild cohomology. This result generalizes a theorem of Quillen in the case of n = 1 in [Qu], and was first conjectured by Kontsevich in [Ko]. Before stating our main theorem,we first recall some important examples and motivations in the theory of E n -algebra.The E n operads interpolate between the E and E ∞ operads, and as a consequence, the categoriesof E n -algebras provide homotopy theoretic gradations of less commutative algebra, interpolatingbetween noncommutative and commutative algebra. Since the second space of the operad E n (2) ishomotopy equivalent to S n − with its antipodal action by Σ , one can intuitively imagine an E n -algebra as an associative algebra with multiplications parametrized by S n − as a Σ -space, in whichthe antipodal map on S n − exchanges an algebra structure with its opposite algebra structure. Thespaces S n − become more connected as n increases, and for this reason one may think that an E n -algebra is more commutative the larger the value of n .For the special case of n = 1, the space E (2) ≃ S has two components, which reflects thefact that an algebra and its opposite need not be isomorphic. The quotient E (2) Σ ≃ S = ∗ is equivalent to a point, and as a consequence the theory of E -algebras is equivalent to that ofstrictly associative algebras. For n = ∞ , the space E ∞ (2) ≃ S ∞ is contractible, corresponding to anessentially unique multiplication, but the quotient S ∞ Σ ∼ = RP ∞ ≃ B Σ is not contractible, and thisdistinguishes the theory of E ∞ -algebras from that of strictly commutative algebras in general. Therational homology H ∗ ( RP ∞ , F ) is trivial if F is a field of characteristic zero, in contrast, and this hasthe consequence that the theories of E ∞ -algebras and strictly commutative algebras agree over afield of characteristic zero. Otherwise, the homotopy theory of strictly commutative algebras is oftenill-behaved, so one might interpret this to mean that, away from characteristic zero, commutativitywants to be a structure, rather than a property.We now consider six occurrences of E n , each serving to motivate the study of E n -algebra: Iterated loop spaces : Historically, the theory of the E n operad and its algebras first developedin the setting of spaces, where Boardman and Vogt originally defined E n in order to describe thehomotopy theoretic structure inherent to an n -fold loop space, [BV]. E n -algebras were first used togive configuration space models of mapping spaces, and then May proved the more precise resultthat n -fold loop spaces form a full subcategory of E n -algebras in spaces, up to homotopy, [Ma]. Ring spectra : E n -structures next arose in the study of ring spectra in algebraic topology. Forinstance, various of the important spectra in topology do not support E ∞ -ring structures, but doadmit an E n -algebra structure for lesser n , which allows for certain advantageous manipulations(such as defining the smash product of A -module spectra). For example, the Morava K -theories K ( n ) admit a unique E -algebra structure, [Ang]; the Brown-Peterson spectra BP are presentlyonly known to admit an E -algebra structure, [BM2]; a Thom spectrum M f classified by a map f : X → BO obtains an E n -ring structure if the map f is an n -fold loop map, which is the case forthe spectra X ( n ) in Devinatz-Hopkins-Smith’s proof of Ravenel’s conjectures, Thom spectra for theBott map Ω SU ( n ) → BU , a 2-fold loop map. Quantum groups : A very different source of E n structures arose in the 1980s, with the adventof the theory of quantum groups. The Hopf algebras U q ( g ) of Drinfeld and Jimbo have an invertibleelement R in U q ( g ) ⊗ which satisfies the Yang-Baxter equation. This gives the category of U q ( g )-modules the structure of a braided monoidal category, or, equivalently, an E -algebra in categories,using the fact that the spaces E ( k ) are classifying spaces for the pure braid groups P k on k strands.This braided structure on the category gives rise to invariants of knots and 3-manifolds, such as theJones polynomial. Conformal and topological field theory : E n -algebras are topological analogues of Beilinson-Drinfeld’s chiral algebras, algebro-geometric objects encoding the operator product expansions in onformal field theory. That is, via the Riemann-Hilbert correspondence, E n -algebras bear the samerelation to chiral algebras as constructible sheaves bear to D-modules. Consequently, E n -algebrasplay a role in topological field theory analogous to that of chiral algebras in conformal field theory.For instance, if F is a topological field theory in dimension d + 1, i.e., a symmetric monoidal functoron the cobordism category of d -manifolds, F : Cob d +1 → C , then the value F ( S d ) on the d -sphere hasthe structure of a Frobenius E d +1 -algebra in C , and this encodes an important slice of the structureof the field theory. In the case d = 1 and C is vector spaces, this augmented E -algebra F ( S ) is astrictly commutative Frobenius algebra, and the field theory is determined by this algebraic object. Homology theories for n -manifolds : One can consider the notion of a homology theory forframed n -manifolds with coefficients in a symmetric monoidal ∞ -category C ⊗ . This can be defined asa symmetric monoidal functor H : Mflds fr n → C from framed n -manifolds, with framed embeddings asmorphisms, to C . A homology theory must additionally satisfy an analogue of excision: If a manifold M is decomposed along a trivialized neighborhood of a codimension-1 submanifold, M ∼ = M ∪ N M ,then the value H ( M ) should be equivalent to the two-sided tensor product H ( M ) ⊗ H ( N ) H ( M ). There is then an equivalence H (Mflds fr n , C ) ≃ E n -alg( C ) between homology theories with values in C and E n -algebras in C . A detailed discussion will be forthcoming in [Fra2]. Quantization : The deformation theory of E n -algebras is closely related to deformation quan-tization, going back to [Ko]. For instance, for a translation-invariant classical field theory F with A = O ( F ( R n )) the commutative algebra of observables, then certain E n -algebra deformations of A over a formal parameter ~ give rise to quantizations of the theory F , see [CG].This final example provides especial impetus to study the deformation theory of E n -algebras,our focus in the present work. In classical algebra, the cotangent complex and tangent complexplay a salient role in deformation theory: The cotangent complex classifies square-zero extensions;the tangent complex T A has a Lie algebra structure, and in characteristic zero the solutions to theMaurer-Cartan equation of this Lie algebra classify more general deformations. Consequently, ourstudy will be devoted the E n analogues of these algebraic structures.We now state the main theorem of this paper. Let A be an E n -algebra in a stable symmetricmonoidal ∞ -category C , such as chain complexes or spectra. T A denotes the E n -tangent complex of A , HH ∗E n ( A ) is the E n -Hochschild cohomology, A × is the derived algebraic group of units in A , and B n A is a C -enriched ( ∞ , n )-category constructed from A . B n A should be thought of as having asingle object and single k -morphism φ k for 1 ≤ k ≤ n −
1, and whose collection of n -morphisms isequivalent to A , Hom B n A ( φ n − , φ n − ) ≃ A ; this generalizes the construction of a category with asingle object from a monoid. Then we prove the following: Theorem 1.1.
There is a fiber sequence A [ n − −→ T A −→ HH ∗E n ( A )[ n ] of Lie algebras in C . This is the dual of a cofiber sequence of E n - A -modules Z S n − A [1 − n ] ←− L A ←− A [ − n ] where L A is the E n -cotangent complex of A and R S n − A is the factorization homology of the ( n − -sphere with coefficients in A . This sequence of Lie algebras may be also obtained from a fiber sequenceof derived algebraic groups B n − A × −→ Aut A −→ Aut B n A by passing to the associated Lie algebras. In particular, there are equivalences: Lie( B n − A × ) ≃ A [ n − , Lie(Aut A ) ≃ T A , Lie(Aut B n A ) ≃ HH ∗E n ( A )[ n ] . This is equivalent to the usual excision axiom if C is chain complexes and the monoidal structure is the coproduct,which can formulated as the assertion that C ∗ ( M ) is homotopy equivalent to C ∗ ( M ) ` C ∗ ( N ) C ∗ ( M ). his sequence, after desuspending by n , has the structure of a fiber sequence of nonunital E n +1 -algebras A [ − −→ T A [ − n ] −→ HH ∗E n ( A ) arising as the tangent spaces associated to a fiber sequence of E n +1 -moduli problems.Remark . In the case of n = 1, this theorem specializes to Quillen’s theorem in [Qu], which saysthat for A an associative algebra, there is a fiber sequence of Lie algebras A → T A → HH ∗ ( A )[1],where the Lie algebra structure, at the chain complex level, is given by [SS]. The final part of theresult above seems new even in the n = 1, where it says the there is an fiber sequence of nonunital E -algebras A [ − → T A [ − → HH ∗ ( A ). The existence of a nonunital E n +1 -algebra structure on A [ −
1] is perhaps surprising; see Conjecture 4.49 for a discussion of this structure. In general, the E n +1 -algebra structure on HH ∗E n ( A ) presented in the theorem is that given by the higher Deligneconjecture of [Ko]. In that paper, Kontsevich separately conjectured that A → T A → HH ∗E n ( A )[ n ] isa fiber sequence of Lie algebras and that HH ∗E n ( A ) admits an E n +1 -algebra structure; the statementthat A [ − → T A [ − n ] → HH ∗E n ( A ) is a fiber sequence of nonunital E n +1 -algebras is thus a commongeneralization of those two conjectures.We now summarize the primary contents of this work, section by section: Section 2 presents a general theory of the cotangent complex for algebras over an operad via stabilization : From this homotopy theoretic point of view, the assignment of the cotangent complexto a commutative ring is an algebraic analogue of the assignment of the suspension spectrum to atopological space. We begin with a brief review of the basic constructions in this subject, similar tothe presentations of Basterra-Mandell [BM1], Goerss-Hopkins [GH], and especially Lurie [Lu4]. Thefirst main result result of this section, Theorem 2.26, is a cofiber sequence describing the cotangentcomplex of an E n -algebra A as an extension of a shift of A itself and the associative envelopingalgebra of A (and this gives the fiber sequence in the statement of Theorem 1.1, but without anyalgebraic structure); the proof proceeds from a hands-on analysis of the cotangent complex in thecase of a free E n -algebra A , where the core of the result obtains from a stable splitting of configurationspaces due to McDuff, [Mc]. The second main focus of this section involves the algebraic structureobtained by the cotangent and tangent space of an augmented E n -algebra; after some standardgeneralities on Koszul duality in the operadic setting, `a la Ginzburg-Kapranov [GK], Theorem 2.26is then used to prove the next central result, Theorem 2.41, which states that the tangent space T A at the augmentation of an augmented E n -algebra A has the structure of a nonunital E n [ − n ]-algebra;i.e., T A [ − n ] is a nonunital E n -algebra. The idea that this result should hold dates to the work ofGetzler-Jones [GJ]; the result has been known in characteristic zero to experts for a long time dueto the formality of the E n operad, see [Ko] and [LV], which implies that the derived Koszul dual ofC ∗ ( E n , R ) can be calculated from the comparatively simple calculation of the classical Koszul dualof the Koszul operad H ∗ ( E n , R ), as in [GJ]. Section 3 , which can be read independently of the preceding section, gives a concise exposition ofthe factorization homology of topological n -manifolds, a homology theory whose coefficients are givenby E n -algebras (and, more generally, E B -algebras). This theory been recently developed in greatdetail by Lurie in [Lu6], though slightly differently from our construction. Factorization homology isa topological analogue of Beilinson-Drinfeld’s chiral homology theory, [BD], constructed using ideasfrom conformal field theory for applications in representation theory and the geometric Langlandsprogram. This topological analogue is of interest in manifold theory quite independent of the rest ofthe present work, a line of study we pursue in [Fra2]. A key result of Section 3 is Proposition 3.24,a gluing, or excision, property of factorization homology: This is used extensively in our work, bothretroactively in Section 2 (to calculate the relation of the n -fold iterated bar construction Bar ( n ) A of an augmented E n -algebra A and its cotangent space LA ) and later in Section 4. Section 4 studies O -moduli problems, or formal derived geometry over O -algebras, to then applyto E n -algebra. Using Gepner’s work on enriched ∞ -categories in [Gep], we obtain the natural fibersequence of derived algebraic groups B n − A × → Aut A → Aut B n A relating the automorphisms f A with the automorphisms of an enriched ( ∞ , n )-category B n A . The tangent complexes ofthese moduli problems are then calculated. The main result of this section, Theorem 4.34, is theidentification of the tangent complex of Aut B n A with a shift of the E n -Hochschild cohomology of A ; the proof hinges on an E n generalization of a theorem of [BFN], and it fundamentally reliesthe ⊗ -excision property of factorization homology. The proof of Theorem 1.1 is then completed byshowing that this moduli-theoretic construction of the fiber sequence A [ − → T A [ − n ] → HH ∗E n ( A )automatically imbues it with the stated E n +1 -algebraic structure: This is consequence of Proposition4.44, a general result in Koszul duality likely familiar to experts, which, together with Theorem2.41, shows that the tangent space of an E m -moduli problem satisfying a technical Schlessinger-typecondition obtains an E m [ − m ]-algebra structure. Remark . In this work, we use the quasicategory model of ∞ -category theory, first developedin detail by Joyal, [Jo], and then by Lurie in [Lu0], which is our primary reference. Most of thearguments made in this paper would work as well in a sufficiently nice model category or a topologicalcategory. For several, however, such as constructions involving categories of functors or monadicstructures, ∞ -categories offer substantial technical advantages. The reader uncomfortable with thislanguage can always substitute the words “topological category” for “ ∞ -category” wherever theyoccur in this paper to obtain the correct sense of the results, but with the proviso that technicaldifficulties may then abound in making the statements literally true. The reader only concernedwith algebra in chain complexes, rather than spectra, can likewise substitute “pre-triangulateddifferential graded category” for “stable ∞ -category” wherever those words appear, with the sameproviso. See the first chapter of [Lu0] or section 2.1 of [BFN] for a more motivated introduction tothis topic.1.1. Acknowledgements.
This work is based on my 2008 PhD thesis, [Fra1], and I thank myadvisor, Michael Hopkins, from whose guidance and insight I have benefitted enormously. I amindebted to Jacob Lurie for generously sharing his ideas, which have greatly shaped this work. Ithank David Gepner for his help with enriched ∞ -categories and for writing [Gep]. I am thankful toKevin Costello, Dennis Gaitsgory, Paul Goerss, Owen Gwilliam, David Nadler, Bertrand T¨oen, andXinwen Zhu for helpful conversations related to this paper. I thank Gr´egory Ginot, Owen Gwilliam,and Geoffroy Horel for finding numerous errors, typos, and expository faults in earlier drafts of thispaper. 2. The Operadic Cotangent Complex
An essential role in the classical study of a commutative ring is played by the module of K¨ahlerdifferentials, which detects important properties of ring maps and governs aspects of deformationtheory. The module Ω A of K¨ahler differentials of a commutative ring A is defined as quotient I/I , where I is the kernel of the multiplication A ⊗ A → A , and I is the ideal in I of elementsthat products of multiple elements. Ω A has the property that it corepresents derivations, i.e.,that there is a natural equivalence Hom A (Ω A , M ) ≃ Der(
A, M ). If A is not smooth, then theassignment M Der(
A, M ) is not right exact. Grothendieck had the insight that Ω A has aderived enhancement, the cotangent complex L A , which corepresents the right derived functor ofderivations. Quillen fitted this concept to a very general model category framework of taking the leftderived functor of abelianization. We first give a brief review of the rudiments of operadic algebrain ∞ -categories; for further details and proofs we refer to [Lu3] or [Fra1]. We will then discuss theappropriate version of the cotangent complex for algebras over an operad.For O a topological operad, we will also denote by O the symmetric monoidal ∞ -category whoseobjects are finite sets and whose morphism spaces are Map O ( J, I ) = ` π : J → I Q I O ( J i ), where π isa map of sets and J i = π − { i } is the inverse image of i . (This category is also known as the PROPassociated to the operad O .) Note that there is a natural projection of O →
Fin from O to the ∞ -category of finite sets (i.e., the nerve of the category of finite sets). Definition 2.1. An O -algebra structure on A , an object of a symmetric monoidal ∞ -category C ,is a symmetric monoidal functor ˜ A : O → C with an equivalence ˜ A ( { } ) ≃ A between A and value f A on the set with a single element. O -algebras in C , O -alg( C ), is the ∞ -category of symmetricmonoidal functors Fun ⊗ ( O , C ).There is an intrinsic notion of a module for an O -algebra, which we will use extensively. In orderto formulate this notion, we will need to use a version of the ∞ -category O using based sets. LetFin ∗ := Fin ∗ / denote the (nerve of the) category of based finite sets. Definition 2.2.
The ∞ -category O ∗ is the pullback in the following Cartesian diagram O ∗ / / (cid:15) (cid:15) O (cid:15) (cid:15) Fin ∗ / / Finwhere Fin ∗ → Fin is the forgetful functor, forgetting the distinguished nature of the basepoint ∗ .Note that O ∗ is acted on by O under disjoint union, where O × O ∗ → O ∗ sends ( I, J ∗ ) to ( I ⊔ J ) ∗ .Second, note that a symmetric monoidal functor A : E → F makes F an E -module. Thus, an O -algebra, A : O → C , makes C an O -module, with the action map O × C → C given by the intuitiveformula (
I, M ) A ⊗ I ⊗ M . Definition 2.3.
For an O -algebra A : O → C , the ∞ -category of O - A -modules is Mod O A ( C ) =Fun O ( O ∗ , C ), functors from O ∗ to C which are O -linear. Remark . If O has a specified map from the operad E , so that an O -algebra can be regardedas an E -algebra by restriction along this map, then for an O -algebra A we write Mod A ( C ) for the ∞ -category of left A -modules, with respect to this E -algebra structure on A . Note the distinctionfrom Mod O A ( C ); for instance, in the case O = E , Mod E A ( C ) is equivalent to A -bimodules in C , ratherthan left A -modules.Evaluation on the point ∗ defines a functor Mod O A ( C ) → C , which is the underlying object of an O - A -module M . We have a natural equivalence M ( J ∗ ) ≃ A ⊗ J ⊗ M . So, applying the functor M to the map J ∗ → ∗ produces a map O ( J ∗ ) → Map C ( A ⊗ J ⊗ M, M ), subject to certain compatibilityconditions, and this is the usual notion of an O - A -module, [GK].The collection of ∞ -categories Mod O A ( C ), as A varies, assembles to form an ∞ -category Mod O ( C )of all O -algebras and their operadic modules, see [Lu3] or [Fra1], so that the following is a pullbackdiagram: Mod O A ( C ) (cid:15) (cid:15) / / Mod O ( C ) (cid:15) (cid:15) { A } / / O -alg( C )The structure of an O - A -module is equivalent to the structure of a left module for a certain asso-ciative algebra U A in C , the enveloping algebra of A . That is, the forgetful functor G : Mod O A ( C ) → C preserves limits and consequently has a left adjoint, F : Definition 2.5. U A = F (1 C ) is the free O - A -module generated by the unit of C .Note that the monad structure on the composite functor GF gives U A an associative algebrastructure.If C is a stable symmetric monoidal ∞ -category whose monoidal structure distributes over directsums, then an O - A -module structure on an object M is exactly the structure necessary give thedirect sum A ⊕ M an O -algebra structure over A : This is the split square-zero extension of A by M , in which the restriction of the multiplication to M is trivial.Recall that, classically, a derivation d of a commutative ring A into an A -module M consists ofa map d : A → M satisfying the Leibniz rule, d ( ab ) = ad ( b ) + bd ( a ). This can be reformulated inan enlightening way: A map d : A → M is a derivation if and only if the map id + d : A → A ⊕ M , rom A to the split square-zero extension of A by M , is a map of commutative algebras. Thisreformulation allows for a general operadic notion of a derivation: Definition 2.6.
Let C be a stable presentable symmetric monoidal ∞ -category whose monoidalstructure distributes over colimits. For M an O - A -module in C , and B → A a map of O -algebras,then the module of A -derivations of B into M is the mapping objectDer( B, M ) := Map O -alg /A ( B, A ⊕ M ) . Since the monoidal structure of C is closed, then it is evident from the definition that derivationsdefines a bifunctor with values in C Der : ( O -alg( C ) /A ) op × Mod O A ( C ) / / C . Under modest hypotheses on the ∞ -category C , the functor of derivations out of A preserves smalllimits. Thus, one could ask that it be corepresented by a specific A -module. This allows us toformulate the definition of the cotangent complex. Definition 2.7.
The absolute cotangent complex of an O -algebra A ∈ O -alg( C ) consists of an O - A -module L A together with a derivation d : A → A ⊕ L A such that the induced natural transformationof functors Map Mod O A ( L A , − ) / / Der( A, − )is an equivalence, where the map Map Mod O A ( L A , M ) → Der(
A, M ) is defined by sending a map α : L A → M to the derivation α ◦ d .In other words, the absolute cotangent complex of A is the module corepresenting the functor of A -derivations Der( A, − ) : Mod O A ( C ) −→ C . From the definition, it is direct that if L A exists, thenit is unique up to a natural equivalence. We now describe this object more explicitly. Lemma 2.8.
The functor A ⊕ − that assigns to a module M the corresponding split square-zeroextension A ⊕ M , Mod O A ( C ) → O -alg( C ) /A , is conservative and preserves small limits.Proof. As established earlier, the forgetful functor G : O -alg( C ) → C preserves limits, and thereforethe functor G : O -alg( C ) /A → C /A is also limit preserving. This gives us the following commutativediagram Mod O A ( C ) (cid:15) (cid:15) A ⊕− / / O -alg( C ) /A (cid:15) (cid:15) C A ×− / / C /A Since the bottom and vertical arrows are all limit preserving and conservative, the functor on thetop must be limit preserving and conservative. (cid:3)
Proposition 2.9. If C is stable presentable symmetric monoidal ∞ -category whose monoidal struc-ture distributes over colimits, then the functor A ⊕ − : Mod O A ( C ) / / O -alg( C ) /A has a left ad-joint, which we will denote L A .Proof. Both Mod O A ( C ) and O -alg( C ) /A are presentable ∞ -categories under the hypotheses above,[Fra1] and [Lu3]. The functor A ⊕ − is therefore a limit preserving functor between presentable ∞ -categories. To apply the ∞ -categorical adjoint functor theorem, [Lu0], it suffices to show that A ⊕− additionally preserves filtered colimits. However, the forgetful functor O -alg( C ) → C preservesfiltered colimits, see [Fra1] or [Lu3], so in both the source and target of A ⊕ − filtered colimits arecomputed in C . (cid:3) s a consequence we obtain the existence of the cotangent complex of A as the value of the leftadjoint L on A . In other words, since there is an equivalence L A ≃ L A (id A ) and the functor L A exists, therefore the cotangent complex L A exists.We now consider the following picture that results from an O -algebra map f : B → A . O -alg( C ) /B f + + L B (cid:23) (cid:23) O -alg( C ) /AB × A − o o L A (cid:23) (cid:23) Mod O B ( C ) B ⊕− O O f ! + + Mod O A ( C ) f ! o o A ⊕− O O It is evident that the compositions of right adjoints commute, i.e., that for any O - A -module M there is an equivalence B × A ( A ⊕ M ) ≃ B ⊕ f ! M , where f ! denotes the forgetful functor from A -modules to B -modules, and f ! is its left adjoint, which can be computed by the relative tensorproduct f ! ≃ U A ⊗ U B ( − ).As a consequence, we obtain that the value of the L A on f ∈ O -alg( C ) /A can be computed interms of the absolute cotangent complex of B and the corresponding induction functor on modules.That is, for f : B → A an O -algebra over A , there is a natural equivalence of O - A -modules L A ( f ) ≃ f ! L B . This follows from the commutativity of the left adjoints in the above diagram, whichcommute because their right adjoints commute.We now consider a relative version of the cotangent complex L A | B for a map f : B → A , in whichwe view the O - A -module L A | B as a linear approximation to the difference between B and A . If L B is an analogue of the cotangent bundle of a smooth manifold M , then L A | B is analogous to thebundle of cotangent vectors along the fibers of a submersion M → N . This will reduce to the caseof the absolute cotangent complex already discussed when A is the unit k of C . Definition 2.10.
For B an O -algebra over A , the relative cotangent complex L A | B is an O - A -module corepresenting the functor of derivations Mod O A → Spaces sending M to the space of B -linear A -derivations from A to M , Der A | B ( A, M ) := Map O -alg B//A ( A, A ⊕ M ).As with the absolute cotangent complex, the relative cotangent complex L A | B is a value of alinearization functor L A | B on the ∞ -category of O -algebras over A and under B . L A | B is theleft adjoint to the functor Mod O A ( C ) → O -alg( C ) B//A that assigns to an O - A -module the square-zeroextension A ⊕ M , equipped with a map from B and a map to A . This obtains the following diagram. O -alg( C ) B//AA ∐ B − (cid:6) (cid:6) L A | B (cid:25) (cid:25) O -alg( C ) A//A L A | A . . qqqqqqqqqq Mod O A ( C ) f f ◆◆◆◆◆◆◆◆◆◆ A ⊕− O O where A ∐ B − denotes the coproduct in O -algebras under B . So for any C ∈ O -alg( C ) A//B , the valueof the relative cotangent complex on C is L A | B ( C ) ≃ L A | A ( A ∐ B C ). The O - A -module L A | B isobtained as the value L A | B ( A ). The curved arrows are left adjoint and straight arrows are right adjoints; we maintain this convention throughout. roposition 2.11. There is a cofiber sequence f ! L B → L A → L A | B in the ∞ -category of O - A -modules.Proof. To check that L A | B is the cofiber of the natural map f ! L B → L A , it suffices to check, for any M in O - A -modules, that Map Mod O A ( L A | B , M ) is the fiber of the natural map Map Mod O A ( L A , M ) → Map
Mod O A ( f ! L B , M ). Note that using that f ! is the left adjoint to the forgetful functor Mod O A ( C ) → Mod O B ( C ), we obtain the equivalence Map Mod O A ( f ! L B , M ) ≃ Map
Mod O B ( L B , M ) ≃ Map O -alg /B ( B, B ⊕ M ). We have thereby reduced the argument to evaluating the fiber ofMap O -alg /A ( A, A ⊕ M ) → Map O -alg /B ( B, B ⊕ M )which is exactly Map O -alg B//A ( A, A ⊕ M ). (cid:3) More generally, we have the following, known as the transitivity sequence: There is a naturalcofiber sequence f ! L B | C → L A | C → L A | B for any sequence of O -algebras C → B f −→ A .A particularly interesting case of the relative cotangent complex functor is that where both A and B are the unit k = 1 C of C , that is, the relative cotangent complex L k | k of augmented O -algebras in C . We will refer to the value value L k | k ( D ) ≃ L k | D [ −
1] of an augmented O -algebra D the cotangent space of the O -algebra D at the point of D given by the augmentation ǫ : D → k . This is equivalentto the case of the absolute cotangent complex of the non-unital O -algebra Ker( ǫ ), which is the O -indecomposables functor.We now turn to the question of describing more concretely what the cotangent complex L A actually looks like. For starters, the functor A ⊕ − : Mod O A → O -alg /A factors through the ∞ -category of augmented A -algebras. We thus obtain a corresponding factorization of L A through arelative cotangent complex L A | A . We will discuss relative cotangent complexes in more detail in thenext section, but in the meantime it suffices to say that L A | A is a functor from the ∞ -category of O -algebras augmented over A to O - A -modules fitting into the following picture. O -alg( C ) /A L A + + A ∐ O − - - O -alg( C ) A//A o o L A | A (cid:0) (cid:0) Mod O A ( C ) A ⊕− rrrrrrrrrr A ⊕− f f ▲▲▲▲▲▲▲▲▲▲ The functor L A | A is closely related to the notion of the indecomposables of a non-unital algebra.In the case of a discrete commutative non-unital ring J , the indecomposables Indec( J ) are defined asthe kernel of the multiplication map of J . Thus, there is a left exact sequence Indec( J ) → J ⊗ J → J .In the ∞ -categorical setting, it is just as convenient to define the functor of indecomposables in termsof the cotangent complex. I.e., the O -indecomposables Indec( J ) of a non-unital O - A -algebra J isgiven as Indec( J ) = L A | A ( A ⊕ J ), where A ⊕ J is the split extension of A by J .The formula L A ≃ L A | A ( A ∐ A ) ≃ Indec(Ker( A ∐ A → A )), however, is not an especiallyconvenient description. For instance, the coproduct A ∐ A in O -algebras is potentially wild. Althoughthe coproduct of E ∞ -algebras is very well-behaved, since it is just given by the tensor product, thecoproduct of associative or E n -algebras is more complicated. Further, the indecomposables functorIndec is similarly inconvenient, since it cannot be computed as just a kernel of a multiplication mapas in the associative case.However, in the case of E n -algebras we will see that the composition cancels out some of this extracomplication, and that for n finite the E n -cotangent complexes have a slightly simpler descriptionnot enjoyed by E ∞ -cotangent complexes.We will now give a more explicit description of the cotangent complex in the case of a free O -algebra A ≃ Free O X , which can expressed by the formula ` k ≥ O ( k ) ⊗ Σ k X ⊗ k , see [Lu3] or[Fra1]. emma 2.12. For A a free O -algebra on an object X in C , the cotangent complex of A is equivalentto U A ⊗ X .Proof. The proof is obtained by tracing the adjunctionsMap O -alg /A ( A, A ⊕ M ) ≃ Map C /A ( X, A ⊕ M ) ≃ Map C ( X, M ) ≃ Map
Mod O A ( U A ⊗ X, M ) . We obtain that U A ⊗ X corepresents derivations, implying the equivalence U A ⊗ X ≃ L A . (cid:3) This reduces the problem of describing the cotangent complex of a free algebra to that of de-scribing the enveloping algebra of a free algebra. Note that we have a functor ψ : O → O ∗ , whichadds a basepoint. On morphisms, ψ maps the space Hom O ( J, I ) → Hom O ∗ ( J ∗ , I ∗ ) to the subspaceof maps for which the preimage of ∗ is exactly ∗ .We now have the following description of the enveloping algebra U A of an O -algebra A . Lemma 2.13.
Let C be a presentable symmetric monoidal ∞ -category whose monoidal structuredistributes over colimits, and let A be an O -algebra in C , defined by a symmetric monoidal functor A : O → C . Then the enveloping algebra U A in C is equivalent to the value on ∗ of the left Kanextension of A along ψ , U A ≃ ψ ! A ( ∗ ) .Proof. There is a forgetful functor Mod O ( C ) (cid:15) (cid:15) Mod O A ( C ) o o (cid:15) (cid:15) O -alg( C ) × C U C C { A } × C o o C C and the left adjoint U sends the pair ( A, C ) to ( A, U A ) ∈ Mod O ( C ), where U A is the free O - A -modulegenerated by 1 C . This restriction functor is exactly that given by restriction along ψ : O ⊔{∗} → O ∗ .The left adjoint of this restriction is calculated by the Kan extension, which can be seen to linear,and therefore gives the enveloping algebra U A as the value on the basepoint. (cid:3) In the case where A is an O -algebra in vector spaces or chain complexes, the formula for the Kanextension recovers the pointwise description of the enveloping algebra U A given in [GK]. The previ-ous lemma allows a simple expression for the enveloping algebra in the special case of free O -algebras,see Fresse in [Fre1], by using the formula for a left Kan extension, i ! A ( ∗ ) ≃ colim J ∈O / ∗ A ⊗ J ∗ . Corollary 2.14.
Let A be the free O -algebra on X , as above, then the universal enveloping algebra U A is equivalent to ` n ≥ O ( n +1) ⊗ Σ n X ⊗ n , where Σ n acts on O ( n +1) by an inclusion Σ n ֒ → Σ n +1 .Proof. Let Σ denote the groupoid of finite sets and bijections. The free O -algebra generated by X is calculated by the coend O ⊗ Σ X , where X : Σ → C is the functor assigning J X ⊗ J , and O isregarded as a symmetric sequence assigning J O ( J ). Kan extending a coend is then computedas another coend, we obtain that the enveloping algebra of the free O -algebra on X is equivalent tothe coend O ∗ ⊗ Σ X , where O ∗ is regarded as a symmetric sequence assigning J O ( J ∗ ). Writingout the formula for the coend, O ∗ ⊗ Σ X ≃ colim Σ O ( J ∗ ) ⊗ X J ≃ ` j O ( j + 1) ⊗ Σ j X ⊗ j . (cid:3) Stabilization of O -Algebras. In this section, we will see that the O -algebra cotangent com-plex is part of a more general theory of stabilization. Stabilization and costabilization are ∞ -categorical analogues of passible to abelian group and abelian cogroup objects in ordinary cat-egories. Since Quillen realized Grothendieck-Illusie’s cotangent complex as a derived functor ofabelianization (i.e., Andr´e-Quillen homology), one would then hope that stabilization should havean analogous relation to the cotangent complex in the ∞ -categorical setting; this is the case, as wenext see. Definition 2.15.
Let C be a presentable ∞ -category, and let C ∗ = C ∗ / be the pointed envelope of C .The stabilization of C is a stable presentable ∞ -category Stab( C ) with a colimit-preserving functorΣ ∞ : C ∗ → Stab( C ) universal among colimit preserving functors from C ∗ to a stable ∞ -category. ∞∗ is the composite C → C ∗ → Stab( C ), given by first taking the coproduct with the final object, C C ⊔ ∗ , and then stabilizing. Example . If C is the ∞ -category of spaces, then C ∗ is pointed spaces, Stab( C ) is the ∞ -categoryof spectra, and Σ ∞ is the usual suspension spectrum functor. Remark . The ∞ -category Stab( C ) can be explicitly constructed as spectra in C , [Lu1].We denote the right adjoint of the stabilization functor by Ω ∞ ; objects in the image of Ω ∞ attainthe structure of infinite loop objects in C ∗ , hence the notation.The rest of this section will establish the following result on the stabilization of O -algebras. Ourdiscussion will mirror that of [Lu4], where these results are established in the commutative algebrasetting. Theorem 2.18.
Let C be a stable presentable symmetric monoidal ∞ -category whose monoidalstructure distributes over colimits. For A an O -algebra in C , the stabilization of the ∞ -categoryof O -algebras over A is equivalent to the ∞ -category of O - A -modules in C , i.e., there is a naturalequivalence Stab( O -alg( C ) /A ) ≃ Mod O A ( C ) and equivalences of functors Σ ∞∗ ≃ L A and Ω ∞ ≃ A ⊕ ( − ) .Remark . An equivalent result, in the case where C is spectra, was previously proved by Basterra-Mandell in [BM1].In proceeding, it will be useful to consider operadic algebras for more general operads, not inspaces. We complement our previous definition: Definition 2.20.
For O an operad in C , then O -alg( C ) is the full ∞ -subcategory of Mod O ( C Σ )consisting of left O -modules M which are concentrated in degree 0 as a symmetric sequence: M ( J ) =0 for J = Ø. Remark . Under the hypotheses above, C is tensored over the ∞ -category of spaces: There is anadjunction k ⊗ ( − ) : Spaces ⇆ C : Map C ( k, − ), where the left adjoint sends a space X to the tensorwith the unit, k ⊗ X . If E is an operad in spaces, then k ⊗ E defines an operad in C . There is thenan equivalence between our two resulting notions of E algebras in C : E -alg( C ) ≃ ( k ⊗ E ) -alg( C ).We will require the following lemma from the Goodwillie calculus, which is a familiar fact con-cerning derivatives of split analytic functors. See [Go] for a further discussion of Goodwillie calculus. Lemma 2.22.
Let T be a split analytic functor on a stable monoidal ∞ -category C defined bya symmetric sequence T ∈ C Σ with T (0) ≃ ∗ , so that T ( X ) = ` n ≥ T ( n ) ⊗ Σ n X ⊗ n . The firstGoodwillie derivative DT is equivalent to DT ( X ) ≃ T (1) ⊗ X .Proof. We calculate the following, DT ( X ) ≃ lim −→ Ω i T (Σ i X ) ≃ lim −→ Ω i (cid:16) a n ≥ T ( n ) ⊗ Σ n (Σ i X ) ⊗ n (cid:17) ≃ lim −→ Ω i ( T (1) ⊗ Σ i X ) ⊕ a n ≥ lim −→ Ω i ( T ( n ) ⊗ Σ n (Σ i X ) ⊗ n ) , using the commutation of Ω with the infinite coproduct and the commutation of filtered colimitsand infinite coproducts. However, we can now note that the higher terms are n -homogeneousfunctors for n >
1, and hence they have trivial first Goodwillie derivative. This obtains that DT ( X ) ≃ lim −→ Ω i ( T (1) ⊗ Σ i X ) ≃ T (1) ⊗ X . (cid:3) e will now prove the theorem above in the special case where A is just k , the unit of the monoidalstructure on C . In this case, O -algebras over A are literally the same as augmented O -algebras in C , O -alg aug ( C ) ≃ O -alg( C ) /k . There is an adjunction between augmented and non-unital O -algebras O -alg nu ( C ) k ⊕ ( − ) (cid:27) (cid:27) O -alg aug ( C ) I O O where I denotes the augmentation ideal functor, with left adjoint given by adjoining a unit. Theadjunction above is an equivalence of ∞ -categories, since the unit and counit of the adjunction areequivalences when C is stable. We now formulate a special case of the theorem above. First, recallthat the first term O (1) of an operad O has the structure of an associative algebra. Proposition 2.23.
There is a natural equivalence
Stab( O -alg nu ( C )) ≃ Mod O (1) ( C ) .Proof. Let T denote the monad associated to non-unital O -algebras, so that there is a naturalequivalence O -alg nu ( C ) ≃ Mod T ( C ). We may thus consider stabilizing this adjunction, to produceanother adjunction: Mod T ( C ) (cid:15) (cid:15) Σ ∞ . . Stab(Mod T C ) g (cid:15) (cid:15) Ω ∞ o o / / Mod gf ( C ) C D D ≃ / / Stab( C ) f C C The stabilization of Mod T ( C ) is monadic over C , [Lu4], since the right adjoint is conservative,preserves split geometric realizations, and hence satisfies the ∞ -categorical Barr-Beck theorem.The resulting monad g ◦ f on C is the first Goodwillie derivative of T , which by the above lemma iscomputed by O (1) ⊗ ( − ), with the monad structure of g ◦ f corresponding to the associative algebrastructure on O (1). Thus, the result follows. (cid:3) Note that if the operad O is such that O (1) is equivalent to the unit of the monoidal structure,then there is an equivalence Mod O (1) ( C ) ≃ C , so the theorem then reduces to the statement of theequivalence Stab( O -alg) ≃ C . In particular, the functor Ind η of induction along the augmentation η : O → ∞ .To complete the proof of the main theorem, we will reduce it to the proposition above. Consider O A , the universal enveloping operad of A , defined by the property that O A -alg( C ) is equivalentto O -algebras under A . The existence of O A is assured by the existence of the left adjoint tothe forgetful functor O -alg( C ) A/ → C ; O A can be explicitly constructed as the Boardman-Vogttensor product of O and U A . Likewise, we have that non-unital O - A -algebras is equivalent to non-unital O A -algebras. Since the ∞ -category of O -algebras augmented over A is again equivalent to O A -alg nu ( C ), we reduce the argument to considering this case.Thus, we obtain that Stab( O A -alg nu ( C )) is equivalent to Mod O A (1) ( C ). Since the first term of theenveloping operad O A (1) is equivalent to the enveloping algebra U A , and Mod U A ( C ) ≃ Mod O A ( C ),this implies that the equivalence Stab( O A -alg nu ( C )) ≃ Mod O A ( C ).By definition, the stabilization of an unpointed ∞ -category X is the stabilization of its pointedenvelope X ∗ , the ∞ -category of objects of X under ∗ , the final object. Thus the pointed envelopeof O -algebras over A is O -algebras augmented over and under A . This is the stabilization we havecomputed, which completes our proof of the theorem.2.2. The E n -Cotangent Complex. We now specialize to the case of O an E n operad, for n < ∞ ,in which case a certain splitting result further simplifies the description of the enveloping algebra ofa free algebra in Corollary 2.14. First, we briefly review some of the geometry of the configurationspaces E n ( k ). The map E n ( k + 1) → E n ( k ), given by forgetting a particular n -disk, is a fiber bundle ith fibers given by configurations of a disk in a standard disk with k punctures, which is homotopyequivalent to a wedge of k copies of the ( n − E n ( k + 1) ≃ Σ( E n ( k ) × W k S n − ). Iterating, there is then a stable equivalencebetween the space E n ( k + 1) and the product Q ≤ j ≤ k W j S n − . The map E n ( k + 1) → E n ( k ) isequivariant with respect to the action of Σ k on both sides, so one can ask that this splitting bearranged so as to be equivariant with respect to this action. The following lemma can be provedeither directly, by explicit analysis of the equivariant splittings of configuration spaces, or as aconsequence of McDuff’s theorem in [Mc]: Lemma 2.24.
There is a Σ k -equivariant splitting Σ ∞∗ E n ( k + 1) ≃ a i + j = k Ind Σ k Σ i (Σ ( n − j Σ ∞∗ E n ( i )) , where Ind Σ k Σ i is induction from Σ i -spectra to Σ k -spectra (cid:3) This has the following consequence. Again, assume C is a stable presentable symmetric monoidal ∞ -category whose monoidal structure distributes over colimits. Denote by Free E the free E -algebrafunctor. Proposition 2.25.
Let A be the free E n -algebra on an object X in C . There is a natural equivalence U A ≃ A ⊗ Free E ( X [ n − . Proof.
By Corollary 2.14, the enveloping algebra U A is equivalent to ` E n ( k + 1) ⊗ Σ k X ⊗ k . By thedescription of the spaces E n ( k + 1) in Lemma 2.24, we may rewrite this as a k E n ( k + 1) ⊗ Σ k X ⊗ k ≃ a k a i + j = k Ind Σ k Σ i Σ ∞∗ E n ( i )[( n − j ] ⊗ Σ k X ⊗ k ≃ a k a i + j = k ( E n ( i ) ⊗ Σ i X ⊗ i )[( n − j ] ⊗ X ⊗ j ≃ a k a i + j = k ( E n ( i ) ⊗ Σ i X ⊗ i ) ⊗ X [ n − ⊗ j ≃ (cid:18)a i E n ( i ) ⊗ Σ i X ⊗ i (cid:19) ⊗ (cid:18)a j X [ n − ⊗ j (cid:19) ≃ A ⊗ Free E ( X [ n − . (cid:3) This brings us to the main result of this section, which in the stable setting gives a descriptionof the absolute cotangent complex of an E n -algebra. Theorem 2.26.
Let C be a stable presentable symmetric monoidal ∞ -category whose monoidalstructure distributes over colimits. For any E n -algebra A in C , there is a natural cofiber sequence U A −→ A −→ L A [ n ] in the ∞ -category of E n - A -modules.Remark . This result has a more familiar form in the particular case of E -algebras, where theenveloping algebra U A is equivalent to A ⊗ A op . The statement above then becomes that there isa homotopy cofiber sequence L A → A ⊗ A op → A , which is a description of the associative algebracotangent complex dating back to Quillen for simplicial rings and Lazarev [La] for A ∞ -ring spectra. Proof.
We will prove the theorem as a consequence of an equivalent statement formulated in termsof the ∞ -category of all E n -algebras and their E n -modules, Mod E n ( C ). We first define the followingfunctors, L , U , and ı , from E n -alg( C ) to Mod E n ( C ): L is the cotangent complex functor, assigning thepair ( A, L A ) in Mod E n ( C ) to an E n -algebra A . U is the composite E n -alg( C ) ×{ } → E n -alg( C ) ×C → Mod E n ( C ), where the functor E n -alg( C ) × C → Mod E n ( C ) sends an object ( A, X ) ∈ E n -alg( C ) × C to( A, U A ⊗ X ), the free E n - A -module generated by X . Finally, the functor ı : E n -alg( C ) → Mod E n ( C ) ends A to the pair ( A, A ), where A is regarded as an E n - A -module in the canonical way. We willnow show that there is a cofiber sequence of functors U → ı → Σ n L .The first map in the sequence can be defined as follows. Denote Mod E n ,ℓ ( C ) the ∞ -category of E n -algebras with left modules in C : I.e., an object of Mod E n ,ℓ ( C ) roughly consists of a pair ( A, K )of an E n -algebra A and a left A -module K . Every E n -module has a left module structure by choiceof a 1-dimensional subspace of R n , so we have a forgetful functor Mod E n ( C ) → Mod E n ,ℓ ( C ). Thisfunctor has a left adjoint in which, for fixed E n -algebra A , there is an adjunction F : Mod A ( C ) ⇆ Mod O A ( C ) : G ; the left adjoint F is computed as the bar construction U A ⊗ A ( − ). Consequently,the counit of this adjunction, F G → id, applied to A ∈ Mod E n A ( C ), gives the desired map U A ≃ U A ⊗ A A ≃ F G ( A ) → A . The functoriality of the counit map thus defines a natural transformationof functors U → ı . We will identify Σ n L as the cokernel of this map. We first prove this inthe case that A is the free algebra on an object X , so that we have A ≃ ` E n ( i ) ⊗ Σ i X ⊗ i and U A ≃ ` E n ( i + 1) ⊗ Σ i X ⊗ i . The map U A → A defined above is concretely realized by the operadstructure maps E n ( i + 1) ◦ i +1 −−−→ E n ( i ) given by plugging the i + 1 input of E n ( i + 1) with the unit of C .The map ◦ i +1 is Σ i -equivariant, since it respects the permutations of the first i inputs of E n ( i + 1),so this gives an explicit description of the map U A ≃ a i ≥ E n ( i + 1) ⊗ Σ i X ⊗ i −→ a i ≥ E n ( i ) ⊗ Σ i X ⊗ i ≃ A. Using the previous result that U A ≃ A ⊗ Free E ( X [ n − (cid:18)a j ≥ E n ( j ) ⊗ Σ j X ⊗ j (cid:19) ⊗ (cid:18)a k ≥ X [ n − ⊗ k (cid:19) ≃ a i ≥ E n ( i + 1) ⊗ Σ i X ⊗ i −→ a i ≥ E n ( i ) ⊗ Σ i X ⊗ i . The kernel of this map exactly consists of the direct sum of all the terms E n ( j ) ⊗ Σ j X ⊗ j ⊗ X [ n − ⊗ k for which k is greater than zero. So we obtain a fiber sequence (cid:18)` j ≥ E n ( j ) ⊗ Σ j X ⊗ j (cid:19) ⊗ (cid:18)` k ≥ X [ n − ⊗ k (cid:19) (cid:15) (cid:15) / / (cid:18)` j ≥ E n ( j ) ⊗ Σ j X ⊗ j (cid:19) ⊗ (cid:18)` k ≥ X [ n − ⊗ k (cid:19) (cid:15) (cid:15) / / ` i ≥ E n ( i ) ⊗ Σ i X ⊗ i of E n - A -modules. It is now convenient to note the equivalence ` k ≥ X [ n − ⊗ k ≃ X [ n − ⊗ ` k ≥ X [ n − ⊗ k . That is, the fiber in the sequence above is equivalent to U A ⊗ X [ n − A is the free E n -algebra on an object X of C , we obtain a fiber sequence U A ⊗ X [ n − −→ U A −→ A. However, we can recognize the appearance of the cotangent complex, since we saw previously thatthe cotangent complex of a free algebra A is equivalent to U A ⊗ X . Thus, we now obtain thestatement of the theorem, that there is a fiber sequence L A [ n − → U A → A , in the special casewhere A is a free E n -algebra.We now turn to the general case. Denote the functor J : E n -alg( C ) → Mod E n ( C ) defined ob-jectwise as the cokernel of the map U → ı . We will show that the functor J is colimit preserving,a property which we will then use to construct a map from L to J . To show a functor preservesall colimits, it suffices to verify the preservation of geometric realizations and coproducts. Sincegeometric realizations commute with taking cokernels, we may show that J preserves geometricrealizations by showing that both the functor U and ı preserve them.First, consider the functor U : The inclusion E n -alg( C ) → E n -alg ×C preserves geometric real-izations; additionally, the free E n - A -module functor E n -alg ×C → Mod E n ( C ) is a left adjoint. U isthus the composite of a left adjoint and a functor that preserves geometric realizations, hence U preserves geometric realizations. Secondly, consider the functor ı . Given a simplicial object A • in E n -alg( C ), the realization of | ıA • | is equivalent to ( | A • | , | U A ⊗ U A • A • | ). We now use the general esult: For R • a simplicial algebra, M • an R • -module, and R • → S an algebra map, then there isan equivalence | S ⊗ R • M • | ≃ S ⊗ | R • | | M • | . Applying this in our example gives that | U A ⊗ U A • A • | is equivalent to U A ⊗ | U A • | | A • | . The geometric realization | U A • | is equivalent to U A , since by thedescription of U A as a left Kan extension it preseves these geometric realizations. Thus, we obtainthat ı does preserve geometric realizations and as a consequence J does as well.Now, we show that J preserves coproducts. First, if a functor F : E n -alg( C ) → D preservesgeometric realizations and coproducts of free E n -algebras, then F also preserves arbitrary coproducts.We see this as follows: Let A i , i ∈ I , be a collection of E n -algebras in C , and let C • A i be the functorialsimplicial resolution of A i by free E n -algebras, where C n A i := Free ◦ ( n +1) E n ( A i ). Since geometricrealizations commute with coproducts, there is a natural equivalence of F ( ` I A i ) ≃ F ( ` I | C • A i | )with F ( | ` I C • A i | ). Applying our assumption that F preserves coproducts of free algebras andgeometric realizations, we thus obtain equivalences F (cid:12)(cid:12)(cid:12)a I C • A i (cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)a I F ( C • A i ) (cid:12)(cid:12)(cid:12) ≃ a I F ( | C • A i | ) ≃ a I F ( A i )where the second equivalence again follows from F preserving geometric realizations. Thus, weobtain that F preserves arbitrary coproducts given the previous assumption. We now demonstratethat J preserves coproducts of free E n -algebras, which will consequently imply that J preservesall colimits. Note that the functor L is a left adjoint, hence it preserves all colimits. We showed,above, that for free algebras A = Free E n ( X ), there is an equivalence J ( A ) ≃ L A [ n ]. Let { A i } be acollection of free E n -algebras; since the coproduct of free algebras is again a free algebra, we obtainthat J ( ` I A i ) ≃ L ` I A i [ n ] ≃ ` I L A i [ n ] ≃ ` I J ( A i ). Thus, J preserves coproducts of free algebras,hence J preserves all colimits.The universal property of the cotangent complex functor L proved in Theorem 2.18 now appliesto produce a map from L to J : The stabilization functor L A : E n -alg( C ) /A → Mod E n A ( C ), fromTheorem 2.18, has the property that for any colimit preserving functor F from E n -alg( C ) /A to astable ∞ -category D , there exists an essentially unique functor F ′ : Mod E n A ( C ) → D factorizing F ′ ◦ L A ≃ F . Choose F to be the composite J A : E n -alg( C ) /A → Mod E n ( C ) /A → Mod E n A ( C ), wherethe first functor is J and the second functor sends a pair ( B f −→ A, M ), where M is an E n - B -module,to the E n - A -module U A ⊗ U B M . J A preserves colimits, since it is a composite of two functors eachof which preserve colimits. The universal property now applies to show that there is an equivalenceof functors ◦ L A ≃ J A , for some colimit preserving functor . However, we have shown there isalso an equivalence J A ( B ) ≃ L A ( B )[ n ] whenever B is a free E n -algebra. Since cotangent complexesof free algebras generate Mod E n A ( C ) under colimits, we may conclude that the functor is thereforethe n -fold suspension functor. Thus, we obtain the equivalence of functors J A ≃ Σ n L A . Since thisequivalence holds for every A , we finally have an equivalence of functors J ≃ Σ n L and a cofibersequence of functors U → ı → Σ n L . (cid:3) One may think of the result above as saying that the shifted E n - A -module A [ − n ] is very close tobeing the cotangent complex of A . There is an interesting interpretation of the difference betweenthe functors corepresented by A [ − n ] and L A , however, which we discuss later in this paper. Remark . Since the preceding proof was written in [Fra1], both Lurie and I separately realizedthat a more conceptual proof of this theorem is possible in terms of the theory of higher categories.Lurie’s proof is in [Lu6], and we present a closely related proof later in this paper. I have still includedthis proof, however, since its nuts-and-bolts character offers a complementary understanding.Let us apply the previous analysis of the absolute cotangent complex in the E n setting to obtaina similar description of the cotangent space of an augmented E n -algebra A . Corollary 2.29.
Let A be an augmented E n -algebra in C , as above, with augmentation f : A → k .Then there exists a cofiber sequence in C , k → k ⊗ U A A → L k | A [ n − , where L k | A is the relativecotangent complex of f . roof. Recall from the previous theorem the cofiber sequence U A → A → L A [ n ] of E n - A -modules.Given an E n -ring map f : A → B , we can apply the induction functor to obtain U B → f ! A → f ! L A [ n ], a cofiber sequence of E n - B -modules. Specializing to where f : A → k is the augmentationof A , this cofiber sequence becomes k → f ! A → f ! L A [ n ]. Note that since there is an equivalencebetween E n - k -modules in C and C itself, the enveloping algebra of the unit k is again equivalent to k . So we have an equivalence f ! A ≃ k ⊗ U A A .Finally, we can specialize the cofiber sequence f ! L A | k → L B | k → L B | A to the case of B = k , toobtain a cofiber sequence f ! L A → L k | k → L k | A . Since L k | k is contractible, this gives an equivalence L k | A [ − ≃ f ! L A . Substituting into k → k ⊗ U A A → f ! L A [ n ], we obtain a cofiber sequence k → k ⊗ U A A → L k | A [ n −
1] as desired. (cid:3)
Remark . The object k ⊗ U A A may be thought as the infinitesimal E n -Hochschild homology of A ,or the E n -Hochschild homology with coefficients in the augmentation, i.e., k ⊗ U A A =: HH E n ∗ ( A, k ).This result is then saying that, modulo the unit, the cotangent space is equivalent to a shift of theinfinitesimal E n -Hochschild homology. In the case n = 1 of usual algebra, the enveloping algebra U A is equivalent to A ⊗ A op , and we have the chain of equivalences HH E ∗ ( A, k ) = k ⊗ A ⊗ A op A ≃ k ⊗ A ⊗ A ⊗ A op k ≃ k ⊗ A k , so that the infinitesimal Hochschild homology of an algebra is given bythe bar construction: HH E ∗ ( A, k ) ≃ k ⊗ A k .2.3. Structure of the Cotangent Complex and Koszul Duality.
Until this point, we havestudied the cotangent complex solely as an object of the stable ∞ -category, such as Mod O A ( C ) or C . However, one may also ask what structure f ! L B obtains by the fact that it is born as a linearapproximation to a map f : A → B . For instance, taking as geometric motivation the case of asubmersion M → N , we have that the bundle of tangents along the fibers T M | N has the structureof a Lie algebroid on M . Before proceeding, we first provide the obvious notion of the O -tangentcomplex. Note that our conditions on C imply that Mod O A ( C ) is tensored and enriched over C , whichallows the following definition. Definition 2.31.
The relative tangent complex T A | B ∈ C of an O -algebra map B → A is the dualof L A | B as an O - A -module: T A | B = Hom Mod O A ( L A | B , A ).We will prove the following, to give a sense of a direction of this section: Proposition 2.32.
Let O be an augmented operad in C with O (1) ≃ k , the unit of C . Then thetangent space of an augmented O -algebra naturally defines a functor O -alg aug ( C ) op −→ O ! -alg nu ( C ) where O ! is the derived Koszul dual operad of O . This will be largely an application of ∞ -categorical Barr-Beck thinking, [Lu2]. For simplicity,we begin with the case of augmented O -algebras (and for which O (1) is the unit, which we willhenceforth assume). For an augmented O -algebra ǫ : A → k , we will denote the tangent space atthe augmentation by T A := Hom C ( ǫ ! L A , k ), and refer to T A simply as the tangent space of A at ǫ (i.e., at the k -point Spec k → Spec A , in the language of section 4).It is convenient to now use a slightly different description of operads and their algebras. Let C be a symmetric monoidal presentable ∞ -category for which the monoidal structure distributesover colimits. Recall the ∞ -category of symmetric sequences C Σ (i.e., functors from finite sets withbijections to C ): There is a functor C Σ → Fun( C , C ) given by assigning to a symmetric sequence X the endofunctor C ` i ≥ X ( i ) ⊗ Σ i C ⊗ i . There is a monoidal structure on C Σ agreeing withthe composition of the associated endofunctor, so that the preceding functor C Σ → Fun( C , C ) ismonoidal. Operads are exactly associative algebras in C Σ with respect to this monoidal structure.We refer to [Fra1] for the following, which relies on a description of free algebras in a monoidalcategory due to Rezk in [Re]. Let X be a monoidal ∞ -category for which the monoidal structuredistributes over geometric realizations and left distributes over colimits. roposition 2.33. For X as above, the bar construction defines a functor Alg aug ( X ) → Coalg aug ( X ) ,sending an augmented algebra A to Bar A = 1 X ⊗ A X , the geometric realization of the two-sidedbar construction Bar(1 X , A, X ) . The conditions on X are satisfied for symmetric sequences C Σ equipped with the compositionmonoidal structure (which, it is worthwhile to note, does not distribute over colimits on the right ).The following was first proved in the setting of model categories in [Ch] when C is spectra. Corollary 2.34.
Let C be a symmetric monoidal ∞ -category, which stable and presentable, and forwhich the monoidal structure distributes over colimits. Then the bar construction Bar : Alg aug ( C Σ ) → Coalg aug ( C Σ ) defines a functor from augmented operads in C to augmented cooperads in C , i.e.,coaugmented coalgebras for the composition monoidal structure on C Σ . Definition 2.35.
For an operad
O ∈
Alg aug ( C Σ ), as above, the derived Koszul dual operad O ! isthe dual of the cooperad given by the bar construction Bar O : I.e., O ! = (Bar O ) ∨ .We now apply this to our study of O -algebras. For a unital operad O , let O nu denote theassociated operad without degree zero operation, so that there is an equivalence O nu -alg( C ) ≃O -alg nu ( C ). Lemma 2.36.
For ǫ : A → k an augmented O -algebra in C , with augmentation ideal I A , there isan equivalence ǫ ! L A ≃ ◦ O nu A := | Bar(1 , O nu , I A ) | between the cotangent space of A at the k -point ǫ and the two-sided bar construction of O nu with coefficients in the left O -module I A and the unitsymmetric sequence 1, regarded as a right O nu -module.Proof. For a map of operads
P → Q , the bar construction
Q ◦ P ( − ) computes the left adjoint tothe restriction Q -alg( C ) → P -alg( C ). (See [Fra1] for a discussion of this fact in the ∞ -categorysetting.) Applying this to Q = 1 the unit symmetric sequence, we find that 1 ◦ O nu ( − ) computes theleft adjoint to the functor C → O nu -alg( C ) assigning an object of C the trivial O -algebra structure.Thus, the cotangent space functor and the bar construction are both left adjoints to equivalentfunctors, hence they are equivalent. (cid:3) We now have the following picture:
Corollary 2.37.
The cotangent space ǫ ! L A of an augmented O -algebra A naturally has the structureof an ◦ O -comodule in C . That is, there is a commutative diagram: O -alg aug ( C ) L % % ❏❏❏❏❏❏❏❏❏❏ ◦ O − / / Comod ◦ O ( C ) forget x x rrrrrrrrrrr C Proof.
The comonad underlying 1 ◦ O ◦ O A obtains a left 1 ◦ O (cid:3) Remark . Left comodules in C for the cooperad 1 ◦ O ◦ O
1. However, there are two important distinctions between these objects and usual coalgebras(i.e., 1 ◦ O C op ): These objects are automatically ind-nilpotent coalgebras, and theyhave an extra structure, analogous to divided power maps. Thus, 1 ◦ O ◦ O Lemma 2.39.
Dualizing defines a functor
Comod ◦ O ( C ) → O ! -alg nu ( C ) .Proof. For a 1 ◦ O C , there is a map C → ` i ≥ (1 ◦ O i ) ⊗ Σ i C ⊗ i . Dualizing gives amap Q i ≥ O ! ( i ) ⊗ Σ i ( C ⊗ i ) ∨ → C ∨ . We have a composite map a i ≥ O ! ( i ) ⊗ Σ i ( C ∨ ) ⊗ i −→ Y i ≥ O ! ( i ) ⊗ Σ i ( C ∨ ) ⊗ i −→ Y i ≥ O ! ( i ) ⊗ Σ i ( C ⊗ i ) ∨ −→ C ∨ here the map O ! ( i ) ⊗ Σ i ( C ∨ ) ⊗ i → O ! ( i ) ⊗ Σ i ( C ∨ ) ⊗ i , from the Σ i -invariants of the diagonal action tothe Σ i -coinvariants of the action, is the norm map. This gives C ∨ a nonunital O ! -algebra structure. (cid:3) Remark . The dual of a 1 ◦ O O ! -algebra: The factorization of the action maps O ! ( i ) ⊗ Σ i ( C ∨ ) ⊗ i → C ∨ through the norm mapis an O -analogue of a divided power structure on a commutative algebra, or a restricted structureon a Lie algebra. Thus, the tangent space of an O -algebra should be a pro-nilpotent restricted O ! -algebra. See [Fre1] for an extended treatment of this structure specific to simplicial algebra.However, in the particular case of the spaces E n ( i ), for n finite, the above norm map is actually ahomotopy equivalence: This is a consequence of the fact that E n ( i ) are finite CW complexes with afree action of Σ i . Thus, one does not obtain any extra restriction structure in the case of E n , ourcase of interest, and so we ignore this extra structure for the present work.We now restrict to the special case of E n , in which something special happens: The E n operad isKoszul self-dual, up to a shift. That is, there is an equivalence of operads in spectra, E ! n ≃ E n [ − n ].Unfortunately, a proof of this does exist in print. That this is true at the level of homology dates toGetzler-Jones, [GJ], and a proof at the chain level has recently been given by Fresse, [Fre2]: Fresseshows that there is an equivalence C ∗ ( E n , F )[ − n ] ≃ Tot[Cobar( C ∗ ( E n , F ))]. In chain complexes, wecan therefore apply Fresse’s theorem to obtain our next result. Lacking a direct calculation of theoperad structure on E ! n in full generality to feed into Corollary 2.37, we will produce the followingmore directly: Theorem 2.41.
The E n -tangent space T defines a functor E n -alg aug ( C ) op −→ E n [ − n ] -alg nu ( C ) . The key input to this construction will be a theorem of Dunn in [Du], upgraded by Lurie tothe ∞ -category context in [Lu6], roughly saying that E n -algebra is equivalent to n -times iterated E -algebra: Theorem 2.42 ([Du], [Lu6]) . Let X be a symmetric monoidal ∞ -category. Then there is a naturalequivalence E m +1 -alg( X ) ≃ E -alg( E m -alg( X )) . Iterating, there is an equivalence E -alg ( n ) ( C ) ≃E n -alg( C ) for all n and any symmetric monoidal ∞ -category C .Remark . One can derive intuition for this result from the theory of loop spaces. An ( n + 1)-fold loop space Ω n +1 X is precisely the same thing as a loop space in n -fold loop spaces, Ω n +1 X =Ω(Ω n X ). By May’s theorem in [Ma], ( n + 1)-fold loop spaces are equivalent to grouplike E n +1 -algebras in spaces. Applying May’s theorem twice, with the observation above, we obtain that ( n +1)-fold loop spaces are also equivalent to the subcategory of grouplike objects of E -alg( E n -alg(Spaces)).As a consequence, we have an equivalence E n +1 -alg(Spaces) gp ≃ E -alg( E n -alg(Spaces)) gp , due totheir shared equivalence to n -fold loop spaces. This is almost a proof of the theorem: If one couldremove the condition of being grouplike, then the result for spaces would imply it for general X .Let Bar ( n ) denote the n -times iterated bar construction, which defines a functor E -alg ( n )aug ( C ) →E -coalg ( n )aug ( C ) from n -times iterated E -algebra to n -times iterated E -coalgebas. Dunn’s theoremhas the following corollary: Corollary 2.44.
The dual of
Bar ( n ) defines a functor E n -alg aug ( C ) op → E n -alg aug ( C ) . To establish Theorem 2.41, we are now only required to do a calculation to show that the dualof the bar construction above calculates the tangent space (modulo the unit and after a shift). Ourcomputational input will be the following:
Lemma 2.45.
There is an equivalence between the n -times iterated bar construction and the infin-itesimal E n -Hochschild homology of an augmented E n -algebra: Bar ( n ) A ≃ HH E n ∗ ( A, k ) := k ⊗ U A A. he functor Bar is iterative, by definition: Bar ( m +1) A ≃ k ⊗ Bar ( m ) A k . In order to prove theabove assertion we will need a likewise iterative description of E n -Hochschild homology, which willrely on a similar equivalence U E m +1 A ≃ A ⊗ U E mA A , where U E i A is the associative enveloping algebraof A regarded as an E i -algebra.Postponing the proof of Lemma 2.45 to our treatment of factorization homology in the followingsection (where we will use the ⊗ -excision property of factorization homology in the proof), we cannow prove Theorem 2.41 quite succinctly: Proof of Theorem 2.41.
By Corollary 2.29, there is an cofiber sequence k → HH E n ∗ ( A, k ) → L k | A [ n − E n ∗ ( A, k ) ≃ Bar ( n ) A and L k | A ≃ ǫ ! L A [1], we obtain a sequence k → Bar ( n ) A → ǫ ! L A [ n ], and thus there is an equivalence ǫ ! L A [ n ] ≃ Ker(Bar ( n ) A → k ) betweenthe suspended cotangent space and the augmentation ideal of the augmented E n -coalgebra Bar ( n ) A .Dualizing, we obtain T A [ − n ] ≃ Ker(Bar ( n ) A → k ) ∨ : I.e., T A , after desuspending by n , has thestructure of a nonunital E n -algebra. Thus, we can define the lift of the functor T : E n -alg aug ( C ) op →E n [ − n ] -alg nu ( C ) by the equivalence T ≃ Ker(Bar ( n ) ( − ) → k ) ∨ [ n ]. (cid:3) The ∞ -categorical version of Dunn’s theorem has an important consequence, which we note herefor use later in Section 4. Let C be presentable symmetric monoidal ∞ -category whose monoidalstructure distributes over colimits. Theorem 2.46 ([Lu6]) . For A an E n +1 -algebra in C , the ∞ -category Mod A ( C ) of left A -modulesin C obtains the structure of an E n -monoidal ∞ -category. That is, Mod A ( C ) is an E n -algebra in Mod C (Cat Pr ∞ ) , the ∞ -category of presentable ∞ -categories tensored over C .Sketch proof. The functor of left modules Mod : E -alg( C ) → Mod C (Cat Pr ∞ ) is symmetric monoidal:There is a natural equivalence Mod A ⊗ B ( C ) ≃ Mod A ( C ) ⊗ Mod B ( C ). Since monoidal functors preserveall algebra structures, therefore Mod defines a functor E -alg( O -alg( C )) → O -alg(Cat Pr ∞ ) for anytopological operad O . Setting O = E n and applying Theorem 2.42 then gives the result. (cid:3) This concludes our discussion of the structure on the tangent space of an augmented O -algebra.A more subtle problem is to describe the exact structure on the absolute operadic cotangent andtangent complexes L A and T A ; these structures are, in some sense, global, rather than local. Wenow briefly discuss this issue, deferring a more involved discussion to a future work; the followingwill not be put to use in this work. The following discussion can be summarized as: • Local case – the cotangent space ǫ ! L R | A of an augmented O -algebra ǫ : R → A is a 1 ◦ O O A ; • Global case – the cotangent complex f ! L B of an O -algebra f : B → A over A is a 1 ◦ O O A , but with an additional structure of a coaction of L A , and this additionaldatum is equivalent to a generalization of a coalgebroid structure.Spelling this out, we have the following commutative diagram, obtained by describing the comon-ads of the adjunctions associated to the stabilization of O -algebras: O -alg /A L A / / A ∐ O − (cid:15) (cid:15) Comod L A (Comod ◦ O (Mod O A )) forget (cid:15) (cid:15) O -alg aug A L A | A $ $ ❏❏❏❏❏❏❏❏❏ L A | A / / Comod ◦ O (Mod O A ) forget u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Mod O A That is, taking the coproduct with A ∐ O A has the structure of a comonad in O -alg aug (Mod O A ),and this gives the functor given by taking the product with absolute cotangent complex L A the tructure of a comonad in Comod ◦ O (Mod O A ); moreover, every cotangent complex f ! L B obtains acoaction of L A , for f : B → A an O -algebra map, since there is a coaction of A ∐ O A on A ∐ O R .Reiterating, the stabilization of the natural map B → A ∐ O B in O -alg /A , which is the counit ofthe adjunction between O -alg /A and O -alg aug A , after stabilizing, gives rise to a map f ! L B −→ L A ⊕ f ! L B and which is part of a structure of a coaction on f ! L B of a comonad structure on the functor L A ⊕ − .It is tempting to then dualize to obtain an algebraic structure on T A , but the full resulting : Forinstance, the object T A no longer has an O - A -module structure, in general, though it should havea Lie algebra structure.There is one case where this works out quite cleanly, and in which dualizing is unproblematic:where O is the E ∞ operad. For simplicity, and to make the connection between this story andusual commutative/Lie theory, we shall assume C is of characteristic zero. The bar construction1 ◦ E ∞ ≃ Lie[1] ∨ produces the shifted Lie cooperad. The situation is summarizes by the followingcommutative diagram:CAlg /A L A / / A ⊗− (cid:15) (cid:15) Comod L A (Lie[ −
1] -coalg A ) forget (cid:15) (cid:15) / / Mod T A (Lie[ −
1] -alg A ) opforget (cid:15) (cid:15) CAlg aug A L A | A % % ❏❏❏❏❏❏❏❏❏ L A | A / / Lie[ −
1] -coalg A forget u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / Lie[ −
1] -alg op A Mod A Thus, T A obtains a Lie[1]-algebra structure in Mod A ( C ) from this construction; equivalently, T A [ − A ( C ), and this Lie algebra structure generalizes that given by the Atiyah classwhen A is a smooth commutative algebra. Further, the functor given by taking the product with T A [ −
1] is a monad in A -linear Lie algebras, and for every commutative algebra map f : R → A , thedual ( f ! L R ) ∨ [ − ≃ Hom R ( L A , R )[ −
1] is a Lie algebra, and has an action of the monad T A [ −
1] by A -linear Lie algebra maps.This ∞ -category Mod T A [ − (Lie -alg A ) is actually something very familiar, namely Lie A -algebroids,in a slightly altered guise. Let A be a commutative algebra over a field of characteristic zero, and letLie -algebroid A be the ∞ -category of Lie A -algebroids (obtained from the usual category of differen-tial graded Lie A -algebroids by taking the simplicial nerve of the Dwyer-Kan simplicial localization).Then we have the following comparison: Proposition 2.47.
For A a commutative algebra over a field of characteristic zero, there is anequivalence of ∞ -categories between Lie A -algebroids and A -linear Lie algebras with an action ofthe monad T A [ − , Lie -algebroid A −→ Mod T A [ − (Lie -alg A ) , given by taking kernel of the anchor map.Proof. We apply the Barr-Beck formalism, [Lu2]. For a Lie A -algebroid L with anchor map ρ : L → T A , the derived kernel of the anchor map Ker( ρ ) naturally has a Lie structure, and the bracketis A -linear. Thus, we obtain a functor Ker : Lie -algebroid A → Lie -alg A from Lie A -algebroidsto A -linear Lie algebras. This functor preserves limits and has a left adjoint, namely the functorLie -alg A → Lie -algebroid A that assigns to an A -linear Lie algebra g the Lie A -algebroid with zeroanchor map, g → → T A . The composite functor F on Lie -alg A , given by assigning to g the kernelof the zero anchor map, takes values F g ≃ T A [ − × g . F has the structure of a monad on Lie -alg A , nd this monad structure corresponds to that of T A [ − A Ker ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ / / Mod T A [ − (Lie -alg A ) v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ Lie -alg A in which the functor Ker : Lie -algebroid A → Mod T A [ − (Lie -alg A ) is an equivalence if and only ifthe functor Ker is conservative and preserves Ker-split geometric realizations. Firstly, Ker is clearlyconservative, since a map of complexes over T A is an equivalence if and only if it is an equivalenceon the kernel. Secondly, the forgetful functor to Mod A from both Lie -algebroid A and Lie -alg A preserves all geometric realizations, and, in particular, G -split ones. Thus, the Barr-Beck theoremapplies. (cid:3) Remark . The previous proposition generalizes to arbitrary ∞ -categories C , not of characteristiczero, with the appropriate adjustment in the definition of Lie A -algebroids. The proof is identical.We intend to study the homotopy theory of algebroids and the E n analogues in a later work.3. Factorization Homology and E n -Hochschild Theories E n -Hochschild Cohomology. We now consider the notion of the operadic Hochschild coho-mology of E n -algebras. The following definitions are sensible for general operads, but in this workwe will only be concerned with the E n operads. Definition 3.1.
Let A be an O -algebra in C , and let M be an O - A -module. Then the O -Hochschildcohomology of A with coefficients in M isHH ∗O ( A, M ) = Hom
Mod O A ( A, M ) . When the coefficient module M is the algebra A itself, we will abbreviate HH ∗O ( A, A ) by HH ∗O ( A ). Remark . When A is an associative algebra, the ∞ -category Mod E A is equivalent to A -bimodules,and thus E -Hochschild cohomology is equivalent to usual Hochschild cohomology. In constrast,when O is the E ∞ operad, and A is an E ∞ -algebra, the ∞ -category Mod E ∞ A is equivalent to usualleft (or right) A -modules. Consequently, the resulting notion of E ∞ -Hochschild cohomology is fairlyuninteresting: HH ∗E ∞ ( A, M ) = Hom
Mod A ( A, M ) is equivalent to M . Remark . The preceding definition does not require that C is stable. A particular case of inter-est in when C = Cat ∞ , the ∞ -category of ∞ -categories, in which case this notion of Hochschildcohomology categories offers derived analogues to the classical theory of Drinfeld centers, a topicdeveloped in [BFN].In the case that C is stable, the E n -Hochschild cohomology is closely related to our previouslydefined notion of E n -derivations and the cotangent complex. We have the following corollary ofTheorem 2.26. Corollary 3.4.
Let M be an E n - A -module in C , with A and C as above. There is then a naturalfiber sequence in C Der(
A, M )[ − n ] −→ HH ∗E n ( A, M ) −→ M. Proof.
Mapping the cofiber sequence U A → A → L A [ n ] into M , we obtain fiber sequencesHom Mod E nA ( U A , M ) ≃ (cid:15) (cid:15) Hom
Mod E nA ( A, M ) o o ≃ (cid:15) (cid:15) Hom
Mod E nA ( L A , M )[ − n ] o o ≃ (cid:15) (cid:15) M HH ∗E n ( A, M ) o o Der(
A, M )[ − n ] o o which obtains the stated result. (cid:3) particular case of the corollary above establishes a conjecture of Kontsevich in [Ko]. Kontsevichsuggested that for an E n -algebra A in chain complexes, there is an equivalence between the quotientof the tangent complex T A by A [ n −
1] and an E n version of Hochschild cohomology of A shifted by n −
1. This follows from the above by setting M = A , since the tangent complex of A is equivalentto Der( A, A ), we thus obtainHH ∗E n ( A )[ n − / / A [ n − / / Der(
A, A ) ≃ T A , implying the equivalence of complexes HH ∗E n ( A )[ n ] ≃ T A / ( A [ n − The infinitesimal analogue ofthis result was earlier proved by Hu in [Hu], under the assumption that k is a field of characteristiczero, which we now generalize: Corollary 3.5.
Let A be an augmented E n -algebra in C . Then there is a fiber sequence in C given by T k | A [1 − n ] → HH ∗E n ( A, k ) → k , where T k | A denotes the relative tangent complex at the augmentation f : A → k .Proof. As in the previous corollary, we obtain this result by dualizing a corresponding result for thecotangent space . From a previous proposition, we have a cofiber sequence k → f ! A → L k | A [ n − C ( L k | A [ n − , k ) → Hom C ( f ! A, k ) → Hom C ( k, k ). Since C is presentable and the monoidal structure distributes over colimits, C is closed,implying the equivalence k ≃ Hom C ( k, k ). Also, since f ! is the left adjoint to the functor C →
Mod E n A ( C ) given by restriction along the augmentation f , we have an equivalence Hom C ( f ! A, k ) ≃ Hom
Mod E nA ( A, k ). This is the infinitesimal E n -Hochschild cohomology of A , HH ∗E n ( A, k ), by definition.Thus, we can rewrite our sequence as Hom C ( L k | A , k )[1 − n ] → HH ∗E n ( A, k ) → k , which proves theresult. (cid:3) Factorization Homology and E n -Hochschild Homology. This preceding notion of theoperadic Hochschild cohomology of unital O -algebras is readily available for any operad O , and thissuggests one should look for a companion notion of Hochschild homology. Such a notion appearsunknown for a completely general O . However, there is a notion of Hochschild homology in thecase of the E n operad, given by factorization homology , a topological analogue of Beilinson-Drinfeld’shomology of factorization coalgebras, see [BD] and [FG]. This topic has also been developed in depthby Lurie in [Lu6], where he calls it topological chiral homology , and a closely related constructionwas given by Salvatore [Sa] in the example in which the target category C is topological spaces.We include the present treatment because a shorter discussion of the topic from a slightly simplerperspective might also be of use, and because we require specific results, such as Proposition 3.24and Proposition 3.33, for our proof of Theorem 1.1. A more involved treatment of this and relatedissues will be forthcoming in [Fra2] and [AFT]. More recent work on this subject includes [And],[CG], and [GTZ]. The notion of factorization homology appears very close to Morrison-Walker’sblob complex [MW] (at least for closed n -manifolds).Let B Top( n ) be the classifying space for the group of homeomorphisms of R n , and let B be aspace with a map B → B Top( n ). For M a topological manifold of dimension n , M has a topologicaltangent bundle classified by a map τ M : M → B Top( n ). A topological manifold M has a B -framinggiven a classifying map M → B lifting τ M . Definition 3.6.
Given a map B → B Top( n ), the B -framed (colored) operad, E B , is the symmet-ric monoidal ∞ -category whose objects are finite disjoint unions of B -framed n -disks and whosemorphisms are B -framed embeddings. This is the statement of the second claim of [Ko], where Kontsevich terms T A the deformation complex, whichhe denotes Def( A ). This statement was later called a conjecture by Kontsevich and Soibelman in their book ondeformation theory, [KS]. In the terminology of [Hu], the result says that the based E n -Hochschild cohomology is equivalent to a shift ofthe based Quillen cohomology of augmented E n -algebras. We offer conflicting terminology with reluctance. The term “chiral,” however, is potentially misleading, since therelation to the chiral sector of a conformal field theory, or other uses of the term, is quite tentative. f B is a connected space, then E B is equivalent to the PROP associated to an operad, de-fined as follows: First, choose a B -framing of the standard n -disk. Now, define the space E B ( I ) =Emb B ( ` I D n , D n ) of B -framed embeddings of n -disks: A point in this space consists of an embed-ding f : ` I D ni → D n , with a homotopy between the B -framing of T D ni and the B -framing on theisomorphic bundle f − i T D n , for each i . Given a surjection of finite sets J → I , the usual insertionmaps E B ( I ) × Q I E B ( J i ) → E B ( J ) give the collection of spaces {E B ( I ) } an operad structure. The ∞ -category E B is equivalent to the PROP associated to this operad. Lemma 3.7.
There is a homotopy equivalence
Emb B ( D n , D n ) ≃ Ω B of topological monoids, where Ω B is the based loop space of B .Proof. By definition, the space Emb B ( D n , D n ) sits in a homotopy pullback square:Emb B ( D n , D n ) / / (cid:15) (cid:15) Map /B ( D n , D n ) (cid:15) (cid:15) Emb
Top ( D n , D n ) / / Map / B Top( n ) ( D n , D n )There are evident homotopy equivalences Map / B Top( n ) ( D n , D n ) ≃ Ω B Top( n ) ≃ Top( n ) and like-wise Map /B ( D n , D n ) ≃ Ω B . By the Kister-Mazur theorem, [Ki], the inclusion of Top( n ) intoEmb Top ( D n , D n ) is a homotopy equivalence. The map defined by the bottom row is a homotopyinverse to this map, therefore it is a homotopy equivalence. The top map in the diagram is thereforethe pullback of a homotopy equivalence, and thus it is also a homotopy equivalence. (cid:3) Thus, a choice of basepoint in B similarly defines a map E n ( I ) × Ω B I → E B ( I ), which is ahomotopy equivalence for B connected. Remark . The connectedness assumption on B is not essential in what follows, but we will includeit for simplicity and because it holds in virtually all cases of interest, e.g., when B is one of ∗ , B O( n ), B Spin( n ), B PL( n ) or B = M , a connected topological n -manifold M . Example . Consider B = E Top( n ) → B Top( n ), a homotopy point of B Top( n ). An E Top( n )structure on an n -manifold M is then equivalent to a topological framing of τ M . The operad E E Top( n ) is homotopy equivalent to the usual E n operad, because there is a natural homotopy equivalence E n ( I ) ∼ −→ E E Top( n ) ( I ), sending a rectilinear embeddings to a framed embedding. By smoothingtheory, framed topological manifolds are essentially equivalent to framed smooth manifolds (exceptpossibly in dimension 4). Example . For B = B O( n ), with the usual map B O( n ) → B Top( n ), then E B O( n ) is equivalentto the ribbon E n operad. See [SW] for a treatment of this operad. Definition 3.11.
Let B → B Top( n ) be as above, and let C be a symmetric monoidal ∞ -category.Then E B -alg( C ) is ∞ -category Fun ⊗ ( E B , C ) of symmetric monoidal functors from E B to C .With this setting, we may give a construction of a topological version of factorization homology.Recall that an E B -algebra A in C is a symmetric monoidal functor A : E B → C , so that there is anequivalence A ( I ) ≃ A ⊗ I . Let M be a B -framed topological n -manifold. M defines a contravariantfunctor E M : E op B → Spaces, given by E M ( I ) = Emb B ( ` I D n , M ). That is, E M is the restriction ofthe Yoneda embedding of M to the ∞ -subcategory of n -disks. Definition 3.12.
As above, given B → B Top( n ), let A be an E B -algebra in C , and let M be atopological n -manifold with structure B . The factorization homology of M with coefficients in A isthe homotopy coend of the functor E M ⊗ A : E op B × E B → Spaces ×C ⊗ −→ C : Z M A := E M ⊗ E B A. emark . Although formulated slightly differently, this construction is equivalent to the con-struction of topological chiral homology by Lurie in [Lu6], which we will explain in [Fra2].The following example demonstrates how factorization homology specializes to the case of usualhomology:
Example . Let C ⊕ be the ∞ -category of chain complexes equipped with the direct sum monoidalstructure. Since every complex V has a canonical and essentially unique map V ⊕ V → V , thereis an equivalence E n -alg( C ⊕ ) ≃ C . The factorization homology of a framed n -manifold M withcoefficients in a complex V is then equivalent to R M V ≃ C ∗ ( M, V ), the complex of singular chainson M tensored with V .The B -framed manifold M , and hence the functor it defines, has an action of the group Top B ( M ),the group of B -framed homeomorphisms of M . Consequently, the factorization homology R M A inherits an action of Top B ( M ). More generally, factorization homology defines a functorMflds Bn × E B -alg( C ) R / / C where Mflds Bn is the ∞ -category of B -framed topological n -manifolds with morphisms given byembeddings, Hom( M, N ) := Emb B ( M, N ). If M is a topological k -manifold with a B -framingstructure on M × R n − k , then we will write R M ( − ) for R M × R n − k ( − ). Remark . There is alternative construction, which we briefly sketch. Let M , E B , and A beas above. The functor E M : E op n → Spaces defines a symmetric sequence with terms E M ( I ) :=Emb B ( ` I D n , M ), the space of B -framed topological embeddings of the disjoint union of disks ` I D n into M . There are natural maps E M ( I ) × Q I E B ( J i ) → E M ( J ) for every surjection of finitesets J → I . These maps give E M the structure of a right E B -module in symmetric sequences. Since A is an E B -algebra, it can be consider as a left E B -module in symmetric sequences (concentratedin sequence degree zero). Then one can define R M A = E M ◦ E B A , the geometric realization of thetwo-sided bar construction of E B with coefficients in the left module A and the right module E M .We now present several illustrative computations. Proposition 3.16.
Let A be an E n -algebra in C . Then there is a natural equivalence U A ≃ R S n − × R A between the enveloping algebra U A and the factorization homology of the ( n − -spherewith coefficients in A .Proof. By Lemma 2.13, the enveloping algebra U A is computed by the left Kan extension of the A : E n → C along the functor ψ : E n → E n ∗ , defined by adding a distinguished element ∗ to aset I ∈ E n . This Kan extension, i ! A ( ∗ ), is equivalent to the colimit of A over the overcategorycolim E n/ ∗ A . The colimit of a diagram A : X → C can be computed as the geometric realization ofthe simplicial diagram ` { K }∈X A ( K ) ` { J → I }∈X A ( J ) o o o o ` { F → E → D }∈X × X X A ( F ) . . . o o o o o o where X is the space of objects of X , X is the space of morphisms, and X × X X is the space ofcomposable morphisms. If C is tensored over the ∞ -category of spaces, then, e.g., the first objectin this simplicial diagram can be written as ` X A ≃ ` [ K ] ∈ π X Aut( K ) ⊗ A ( K ), where Aut( K ) isthe space of automorphisms of the object K in X , and π X is the collection of equivalence classesof objects of X . There is a similar description of the higher terms in the above simplicial object.Applying this to the case of the composite functor A : E n/ ∗ → E n → C , we can computecolim E n/ ∗ A as the geometric realization of a simplicial object ` K E n ( K ∗ ) ⊗ A ⊗ K ` Hom E n ( J,I ) E n ( I ∗ ) ⊗ A ⊗ J o o o o . . . o o o o o o he factorization homology R S n − × R A is defined as a coend, computed as the colimit of a simplicialobject ` K Emb fr ( ` K D n , S n − × R ) ⊗ A ⊗ K ` Hom E n ( J,I ) Emb fr ( ` I D n , S n − × R ) ⊗ A ⊗ J o o o o . . . o o o o o o We have a map E n ( K ∗ ) → Emb fr ( ` I D n , S n − × R ), given by translating the disks so that thedisk labeled by ∗ moves to the origin, and this map is a homotopy equivalence. Thus, the terms inthe two simplicial objects above are equivalent; it can be easily seen in addition that the maps aresame. We therefore obtain the equivalence U A ≃ R S n − A , since they are computed as the geometricrealizations of equivalent simplicial objects. (cid:3) Remark . This result confirms the intuition that since an E n - A -module structure and a leftaction of R S n − A both consist of an S n − family of left A -module structures, the two should beequivalent.We briefly note a corollary of Proposition 3.16: Corollary 3.18.
Let A = Free E n ( V ) be the E n -algebra freely generated by V in C . Then there is anequivalence Z S n − A ≃ A ⊗ Free E ( V [ n − . Proof.
Combining Proposition 3.16 and Proposition 2.25, we compose the equivalences R S n − A ≃ U A ≃ A ⊗ Free E ( V [ n − (cid:3) Remark . There is a generalization of the formula in the preceding corollary where one replaceseach occurrence of ‘1’ with ‘ i ’, to compute R S n − i A . This will be considered in [Fra2].Returning to our equivalence U A ≃ R S n − A , we see that there is, consequently, an E -algebrastructure on R S n − × R A corresponding to the canonical algebra structure of U A . This has a verysimple geometric construction, and generalization, which we now describe. Let M k be a k -manifoldwith a B -framing of M × R n − k . There is a space of embeddings ` I M × R n − ki → M × R n − k parametrized by E n − k ( I ): More precisely, there is a map of operads E n − k → E nd M × R n − k from the E n − k operad to the endomorphism operad of the object M × R n − k in Mflds Bn . As a consequence, M × R n − k attains the structure of an E n − k -algebra in the ∞ -category Mflds Bn . Passing to factorizationhomology, this induces a multiplication map m : Z ` I M × R n − k A ≃ (cid:16)Z M × R n − k A (cid:17) ⊗ I −→ Z M × R n − k A for each point of E n − k ( I ). This gives R M × R n − k A an E n − k -algebra structure. In other words, thefactorization homology functor R A : Mflds Bn → C is symmetric monoidal, so it defines a functor from E n − k -algebras in Mflds Bn to E n − k -algebras in C ; M × R n − k is an E n − k -algebra in Mflds Bn , therefore R M × R n − k A is an E n − k -algebra in C . In the particular case of M = S n − , this can be seen to beequivalent to the usual E -algebra structure of U A .We now turn to the problem of defining an analogue of Hochschild homology for E n -algebras.As we shall see shortly, in the case n = 1 there is an equivalence R S A ≃ HH ∗ ( A ), between thefactorization homology of the circle and Hochschild homology. It might be tempting to attempt todefine the E n -Hochschild homology of E n -algebra A as the factorization homology of the n -sphere S n with coefficients in A . However, unless n is 1 , , or 7, the n -sphere is not a parallelizable manifold,and so the construction requires some modification in order to be well-defined. Our modificationwill make use of the following basic observation. Lemma 3.20.
Let A be an E n -algebra in C . There is an equivalence of E -algebras R S n − A ≃ ( R S n − A ) op , between the factorization homology of S n − with coefficients in A and its oppositealgebra, induced by the product map τ : S n − → S n − × R of reflection in a hyperplane withreflection about the origin on R . roof. Let E ( I ) × ` I S n − × R m −→ S n − × R be our E ( I ) family of embeddings. The opposite E -algebra of ( R S n − × R A ) op is defined by the action of Σ on the operad E , which we now define.A configuration f : ` I D ֒ → D defines an ordering of the set I , e.g., the left-to-right order of thelabeled disks in D . For σ ∈ Σ , the nontrivial element, the map σ ( f ) has the same image, butreverses the ordering of the disks. The action of the element σ intertwines with reflection about ahyperplane τ : S n − × R → S n − × R as follows E ( I ) × ` I S n − × R σ × id (cid:15) (cid:15) id × τ / / E ( I ) × ` I S n − × R m (cid:15) (cid:15) E ( I ) × ` I S n − × R m (cid:15) (cid:15) (cid:15) (cid:15) S n − × R τ / / S n − × R where the diagram commutes up to a canonical homotopy (which translates and scales the concentricpunctured n -disks). Passage to factorization homology thereby gives a subsequent commutativediagram E ( I ) ⊗ R ` I S n − × R A σ ⊗ id (cid:15) (cid:15) m op ' ' id ⊗ τ / / E ( I ) ⊗ R ` I S n − × R A m (cid:15) (cid:15) E ( I ) ⊗ R ` I S n − × R A m (cid:15) (cid:15) (cid:15) (cid:15) R S n − × R A τ / / R S n − × R A in which the left hand vertical map m op defines the opposite E -algebra structure on R S n − × R A , andthe right hand vertical arrow m denotes its usual E -algebra structure. This is exactly the conditionthat the factorization homology map τ : ( R S n − × R A ) op → R S n − × R A is a map of E -algebras, andthe map is clearly an equivalence since the reflection map is a homeomorphism. (cid:3) Example . When A is an E -algebra, this result is quite familiar: R S A is equivalent as an E -algebra to A ⊗ A op , and its opposite is ( A ⊗ A op ) op ≃ A op ⊗ A . There is an obvious equivalenceof E -algebras τ : A op ⊗ A → A ⊗ A op switching the two factors, which is precisely the self map ofthe 0-sphere S given by reflection about the origin.As a consequence of this lemma, any left R S n − A -module M attains a canonical right R S n − A -module structure M τ obtained by the restriction, M τ := Res τ M . Definition 3.22.
Let A be an E n -algebra in C . Considering A as an E n - A -module in the usualfashion, and using Proposition 3.16, A has the structure of a left R S n − A -module with A τ thecorresponding right R S n − A -module. The E n -Hochschild homology of A is the tensor productHH E n ∗ ( A ) := A τ ⊗ R Sn − A A and, for M an E n - A -module, the E n -Hochschild homology of A with coefficients in M is the tensorproduct HH E n ∗ ( A, M ) := M τ ⊗ R Sn − A A. If A has an E n +1 -algebra refinement, then there is an equivalence HH E n ∗ ( A ) ≃ R S n A . This isbecause, for a A an E n +1 -algebra, there is an equivalence of E n - A -modules A ≃ A τ , from which weobtain HH E n ∗ ( A ) ≃ A ⊗ R Sn − A A ≃ R S n A . emark . The preceding construction of HH E n ∗ ( A ) actually applies somewhat more generally. If M is a topological n -manifold with a framing of the tangent microbundle τ M after restricting to thecomplement of a point x ∈ M and for which there is a framed homeomorphism U r x ∼ = S n − × R for a small neighborhood U of x and the standard framing on S n − × R , then one can construct anobject R M A for an E n -algebra A . This R M A can be defined as Z M A := A τ ⊗ R Sn − A Z M r { x } A. The group Top fr ∗ ( M ) of based homeomorphisms that preserve the framing away from M r { x } (equivalently, the pullback Top ∗ ( M ) × Top( M r { x } ) Top fr ( M r { x } )) acts naturally on R M A .Let N be a framed manifold with boundary ∂N . Then R N A has the structure of both a left andright R ∂N A -module, by the preceding constructions. We can now formulate the following gluing, or ⊗ -excision, property of factorization homology. Proposition 3.24.
Let M be a B -framed n -manifold expressed as a union M ∼ = U ∪ V U ′ by B -framed embeddings of B -framed submanifolds, and in which V ∼ = V × R is identified as the productof an ( n − -manifold with R . Then, for any E B -algebra A in C , there is a natural equivalence Z M A ≃ Z U A ⊗ R V A Z U ′ A. Proof.
Given a bisimplicial object X •∗ : ∆ op × ∆ op → C , one can compute the colimit of X •∗ inseveral steps. In one way, one can can first take the geometric realization in one of the horizontaldirection, which gives a simplicial object | X • | ∗ with n -simplices given by | X • | n = | X • ,n | , and thentake the geometric realization of the resulting simplicial object. In the other way, one can take thegeometric realization in the vertical simplicial direction to obtain a different simplicial object | X ∗ | • with n -simplices given by | X ∗ | n = | X n, ∗ | , and then take the geometric realization of | X ∗ | • . Theseboth compute colim ∆ op × ∆ op X •∗ .Both R U A ⊗ R V A R U ′ A and R M A are defined as geometric realizations of simplicial objects, thetwo-sided bar construction and the coend, respectively. To show their equivalence, therefore, wewill construct a bisimplicial object X •∗ such that the realization in the horizontal direction givesthe two-sided bar construction computing the relative tensor product, and the realization in thevertical direction gives the simplicial object computing the coend E M ⊗ E B A . To do so, observethat each of the terms in the two-sided bar construction, R U A ⊗ ( R V A ) ⊗ i ⊗ R U ′ A , is given as thegeometric realization of a simplicial object. That is, define X ij to be the i th term in the simplicialobject computing the tensor products of coends R U A ⊗ ( R V A ) ⊗ j ⊗ R U ′ A . For fixed a vertical degree j , X • ,j forms a simplicial object whose colimit is | X • ,j | ≃ R U A ⊗ ( R V A ) ⊗ j ⊗ R U ′ A . We will nowdefine the vertical maps. We begin with the case of the 0th column. Note the equivalence X ,j ≃ a J E U ( J ) ⊗ A ⊗ J ⊗ (cid:16)a I E V ( I ) ⊗ A ⊗ I (cid:17) ⊗ j ⊗ a K E U ′ ( K ) ⊗ A ⊗ K . We will show that the X ,j form a simplicial object as j varies, and that the realization | X , ∗ | is equivalent to ` I E M ( I ) ⊗ A ⊗ I . First, we can write the colimit M ∼ = U ∪ V U ′ as a geometricrealization of the simplicial object M U ⊔ U ′ o o U ⊔ V ⊔ U ′ o o o o U ⊔ V ⊔ V ⊔ U ′ . . . o o o o o o The degeneracy maps in this simplicial diagram, induced by U ⊔ V → U and V ⊔ U ′ → U ′ , arenot quite embeddings, and hence do not quite define maps of embedded disks E U ( J ) × E V ( I ) E U ( J ` I ), because the disks may intersect. However, this is easily rectified: Choose an embedding : U ֒ → U that contracts U into the complement of a closed neighborhood of the boundary ∂U .Replacing the identity map id U with , the map ⊔ f : U ⊔ V ֒ → U is now an embedding, henceinduces a map E U ( J ) × E V ( I ) → E U ( J ⊔ I ) for all finite sets J and I , and likewise for U ′ . Using hese maps, we can write E M ( I ), for each I , as a geometric realization of the embedding spaces ofthe pieces U , V , and U ′ : E M ( I ) ` I →{ , } E U ( I ) × E U ′ ( I ) o o ` I →{∗ , , } E U ( I ) × E V ( I ∗ ) × E U ′ ( I ) o o o o . . . o o o o o o By tensoring with A ⊗ I , which preserves geometric realizations, we obtain a simplicial objectcomputing E M ( I ) ⊗ A ⊗ I : ` E U ( I ) ⊗ A ⊗ I ⊗ E U ′ ( I ) ⊗ A ⊗ I ` E U ( I ) ⊗ A ⊗ I ⊗ E V ( I ∗ ) ⊗ A ⊗ I ∗ ⊗ E U ′ ( I ) ⊗ A ⊗ I o o o o . . . o o o o o o Taking the direct sum over all I , the j th term of the resulting simplicial object is equivalent to X ,j . Thus, the X , ∗ has the structure of a simplicial object, and the realization | X , ∗ | is equivalentto ` I E M ( I ) ⊗ A ⊗ I , the 0th term of the simplicial object computing R M A . An identical argumentgives each X i, ∗ a simplicial structure whose realization is the i th term of the simplicial objectcomputing R M A . (cid:3) The preceding proposition has several easy, but important, consequences:
Corollary 3.25.
For A an E -algebra, there is an equivalence R S A ≃ HH ∗ ( A ) between the factor-ization homology of the circle with coefficients in A and the Hochschild homology of A .Proof. We have the equivalences Z S A ≃ Z D A ⊗ R S A Z D A ≃ A ⊗ A ⊗ A op A ≃ HH ∗ ( A ) . (cid:3) Remark . With a more sophisticated proof, [Lu6], one can see further that the simplicial circleaction in the cyclic bar construction computing HH ∗ ( A ) agrees with the topological circle action byrotations on R S A .By exactly the same method of proof, we can obtain: Corollary 3.27.
Let A be an E ∞ -algebra, which obtains an E B -algebra structure by restriction alongthe map of operads E B → E ∞ , and let M be a B -manifold. Then there is a natural equivalence Z M A ≃ M ⊗ A between the factorization homology of M with coefficients in A and the tensor of the space M withthe E ∞ -algebra A . More generally, the following diagram commutes: Mflds Bn × E ∞ -alg( C ) / / (cid:15) (cid:15) Spaces ×E ∞ -alg( C ) ⊗ / / E ∞ -alg( C ) forget (cid:15) (cid:15) Mflds Bn × E B -alg( C ) R / / C Proof.
The proof is a standard induction on a handle decomposition of M , in the style of proofsof the h-principle. There is a slight complication in that if M ′ is a nonsmoothable topological 4-manifold, then M ′ will not admit a handle decomposition: However the 5-manifold M ′ × R can bedecomposed into handles. Since A is an E ∞ -algebra, it is an E -algebra, and R M ′ A can therefore becalculated as the factorization homology R M × R A of the 5-manifold M × R . Thus, in this case wecan instead perform induction on the handle decomposition of M := M ′ × R .The base case of the induction, M ∼ = R n , is immediately given by the equivalences R R n A ≃ A ≃ R n ⊗ A . Since both operations send disjoint unions to tensor products, we also have the equivalence R R n ⊔ R n A ≃ A ⊗ A ≃ ( R n ⊔ R n ) ⊗ A . Applying Proposition 3.24 to the presentation of S × R n − as a union of R n ∪ S × R n R n , we obtain the equivalence R S × R n − A ≃ A ⊗ A ⊗ A A ≃ S ⊗ A . We can ontinue the induction to show that for all thickened spheres, the factorization homology R S k × R n − k A is equivalent to the tensor S k ⊗ A .Now we show the inductive step: Let M be obtained from M by adding a handle of index q + 1. Therefore M can be expressed as the union M ∼ = M ∪ S q × R n − q R n , where R n is an openneighborhood of the ( q + 1)-handle in M . Again applying Proposition 3.24, we can compute thefactorization homology R M A as Z M A ≃ Z M A O R Sq × R n − q A Z R n A ≃ M ⊗ A O S q × R n − q ⊗ A R n ⊗ A ≃ M ⊗ A where the middle equivalence holds by the inductive hypothesis, the base case, and our previousexamination of the result in the special case of thickened spheres. (cid:3) The preceding equivalences between the enveloping algebra U A ≃ S n − ⊗ A in the case that A is an E ∞ -algebra, the tensor S n − ⊗ A , allow for some further interpretation of the sequence U A → A → L A [ n ] of Theorem 2.26: Remark . Let A be an E ∞ -algebra. Recall from our discussion of the first derivative in Goodwilliecalculus that the E ∞ -cotangent complex of A can be calculated as a sequential colimit A ⊕ L A ≃ lim −→ Ω nA Σ nA ( A ⊗ A ) ≃ lim −→ Ω nA ( S n ⊗ A )where A ⊗ A is regarded an an augmented E ∞ -algebra over A , and Ω nA and Σ nA are the iteratedloop an suspension functors in E ∞ -alg aug A . Using the equivalence, U E n A ≃ S n − ⊗ A , between the E n -enveloping algebra of A and the tensor of A with the ( n − L A as the sequential colimit lim −→ L E n A of the E n -cotangentcomplexes of A , using the description of L E n A as the kernel of ( S n − ⊗ A )[1 − n ] → A [1 − n ].We have already seen that the factorization homology R M × R k A has the structure of an E k -algebra;the preceding proposition allows us to see that that algebra structure can be made to be A -linear.Before stating the following corollary, we recall by Theorem 2.42 that the ∞ -category Mod A ( C ) isan E n − -monoidal ∞ -category; the definition of an O -algebra in a symmetric monoidal ∞ -categorycan be slightly modified to make sense in an O -monoidal ∞ -category, see [Lu3], [Fra1]. Then: Corollary 3.29.
For M n − k a framed manifold, k ≥ , and A an E n -algebra in C , then the factor-ization homology R M n − k × R k A has the structure of an E k -algebra in Mod A ( C ) .Proof. We describe the case k = 1. A framed embedding R n − ֒ → M gives R M × R A an R R n − × R A -module structure. The tensor product relative A of Z M × R A ⊗ A Z M × R A ≃ Z M × ( − , ∪ R n − M × [0 , A where the union on the right hand side is taken over the image R n − ֒ → M × { } , and is homotopicto the wedge M ∨ M . There is family of embedding of M × ( − , ∪ R n − M × [0 ,
1) into M × R parametrized by E (2) (which are homotopic to the coproduct map M ∨ M → M ), giving a space ofmaps from R M × R A ⊗ A R M × R A to R M × R A . Extending this construction to I -fold tensor productscan be seen to give R M × R A an E -algebra structure in A -modules. (cid:3) With some of the technical tools of factorization homology now at hand, we now return to anoutstanding problem from the previous section, the calculation relating the n -times iterated barconstruction and the infinitesimal Hochschild homology of an augmented E n -algebra. That is, wenow prove the equivalence Bar ( n ) A ≃ HH E n ∗ ( A, k ). This will be a basic argument that certainsimplicial objects defining tensor products calculate the same objects; it helpful to first state thefollowing trivial lemma. emma 3.30. For A an augmented E n -algebra in C and R an E -algebra in A -modules, then thereis an equivalence k ⊗ R M ≃ k O k ⊗ A R k ⊗ A M for every R -module M .Proof. Note that k ⊗ A R obtains an E -algebra structure in C , since the induction functor k ⊗ A − is monoidal hence preserves algebra structure. Letting the R -module input “ M ” vary, we obtaintwo linear functors Mod R ( C ) → C . To prove that two linear functors agree, it suffices to check ona generator for the ∞ -category, which is M = R . In this case, there is an obvious cancellation toboth sides, which are equivalent to k . (cid:3) Proof of Lemma 2.45.
We prove the result by induction. The case of n = 1 is already familiar, butwe restate to motivate the argument for higher n . The E n -Hochschild homology is calculated as k ⊗ R S A A = k ⊗ A ⊗ A A . We may then apply the reasoning of Lemma 3.30 to obtain k ⊗ A ⊗ A A ≃ k O k ⊗ A ( A ⊗ A ) k ⊗ A A ≃ k ⊗ A k = Bar (1) A. For the inductive step, we now assume the equivalence Bar ( i ) A ≃ HH E n ∗ ( A, k ) and show the equiv-alence for i + 1. By definition, Bar ( i +1) A is equivalent to the tensor product k ⊗ Bar ( i ) A k , and wenow show the same iteration produces the infinitesimal Hochschild homology. The essential inputis Proposition 3.16, which reduces the problem to factorization homology, and Proposition 3.24,which allows for induction by successively dividing spheres along their equators. This allows thecalculation HH E i +1 ∗ ( A, k ) = k ⊗ R Si A A ≃ k O k ⊗ A R Si A k ⊗ A A ≃ k O k ⊗ A R Si A k applying Lemma 3.30 for the middle equivalence. The algebra in the last term k ⊗ A R S i A is equivalentto k ⊗ R Si − A A , again using Proposition 3.24, which is equivalent to Bar ( i ) A , using the inductivehypthesis. We thus obtain the equivalence of the iterative simplicial objects calculate HH E n ∗ ( A, k )and Bar ( n ) A . (cid:3) Example . The preceding lemma has a clear interpretation when A is an E ∞ -algebra: In thiscase, there is an equivalence Bar ( n ) A ≃ Σ nk A between the n -fold bar construction and the n -foldsuspension of A in the ∞ -category of augmented E ∞ -algebras; likewise, there is an equivalencebetween the infinitesimal E n -Hochschild homology with k ⊗ S n − ⊗ A A ≃ k ⊗ A A ⊗ S n − ⊗ A A ≃ k ⊗ A ( S n ⊗ A ). The equivalence Σ nk A ≃ k ⊗ A ( S n ⊗ A ) is then implied by a basic observation forpointed topological spaces: The based n -fold loops Ω n X is equivalent to the fiber ∗ × X X S n of thespace of all maps over the base point in X . Remark . The equivalence of Lemma 2.45, Bar ( n ) A ≃ HH E n ∗ ( A, k ), may be thought of asinstance of the pushforward for factorization homology: R M × N A ≃ R M R N A . HH E n ∗ ( A, k ) calculatesthe factorization homology of the n -disk D n with coefficients in the pair ( A, k ). Likewise, Bar (1) A = k ⊗ A k calculates the factorization homology of the interval D with coefficients in the pair ( A, k ).Thus, Bar ( n ) A can be seen to equivalent to HH E n ∗ ( A, k ), by expressing the n -disk as a product D n ∼ = ( D ) n , and using the pushforward formula n − Proposition 3.33.
For a framed ( n − -manifold M , and an E n -algebra A in C , there is a naturalequivalence Mod R M × R A ( C ) ≃ Z M Mod A ( C ) where the framing of M × R is the product of the given framing on M and a framing of R . roof. We again prove the equivalence by induction on a handle decomposition of M . The twosides are equivalent in the case of M ∼ = R n − . By Proposition 3.24, the factorization homology R M Mod A ( C ) glues by tensor products, decomposing M into glued together Euclidean spaces. Theright hand side does as well, using the result, Mod A ⊗ B C ≃ Mod A ⊗ Mod B Mod C , a consequence of,e.g., Theorem 4.7 of [BFN] in the special case of algebras, i.e., affine stacks. (cid:3) Remark . The preceding proposition will be important for our purposes in the case M = S n − × R , in Proposition 4.36.We end with the following conceptual characterization of factorization homology. Note that asymmetric monoidal functor H : Mflds Bn → C gives H ( N n − × R ) the structure of an E -algebra in C . Let C be a presentable symmetric monoidal ∞ -category whose monoidal structure distributesover geometric realizations and filtered colimits. Definition 3.35. H (Mflds Bn , C ), the ∞ -category of homology theories for B -framed n -manifoldswith coefficients in C , is the full ∞ -category of symmetric monoidal functors H : Mflds Bn → C satisfying ⊗ -excision: H ( U ∪ V U ′ ) ≃ H ( U ) ⊗ H ( V ) H ( U ′ ) for every codimension-1 gluing, V ∼ = N n − × R .Induction on a handle decomposition (excepting dimension 4) allows the proof of the followingresult: Theorem 3.36 ([Fra2]) . There is an equivalence of H (Mflds Bn , C ) ≃ E B -alg( C ) between C -valuedhomology theories for B -framed n -manifolds and E B -algebras in C . The functor H (Mflds Bn , C ) →E B -alg( C ) is given by evaluation on R n , and the adjoint is given by factorization homology. We defer a proof to [Fra2], which focuses on the application of factorization homology to topology.4.
Moduli Problems
In this final section, we consider some moduli functors and algebraic groups defined by certainsymmetries of E n -algebras. Previously, we showed that for an E n -algebra A , there exists a fibersequence A [ n − → T A → HH ∗E n ( A )[ n ]. However, we did not exhibit any algebraic structure onthe sequence, and we gave no rhyme or reason as to why it existed at all. Our goal is to completethe proof of Theorem 1.1, by giving a moduli-theoretic interpretation of this sequence. Namely, wewill show that this sequence arises as the Lie algebras of a very natural fiber sequence of derivedalgebraic groups B n − A × → Aut A → B n A , and relatedly, a sequence of E n +1 -moduli problems. Asa consequence of this interpretation, the sequence A [ n − → T A → HH ∗E n ( A )[ n ] will obtain a Liealgebra structure and, relatedly, a nonunital E n +1 [ − n ]-algebra structure.Our construction of the group Aut B n A of automorphisms of an enriched ( ∞ , n )-category B n A will rely on a basic result, Corollary 4.21, for which we rely on a preprint of Gepner, [Gep]. intu-itively, B n A has a single object and a single k -morphism φ k for 1 ≤ k < n , and the hom-objectMor( φ n − , φ n − ) = A .We now begin our treatment. For the remainder of this work, C is a stable presentable symmetricmonoidal ∞ -category whose monoidal structure distributes over colimits. We assume further that C ≃
Ind( C ) is generated under filtered colimits by a small ∞ -subcategory C ⋄ ⊂ C of compact objects(i.e., C is compactly generated), and the compact objects coincide with the dualizable objects in C . We make use of the notions of the cotangent and tangent complexes of a moduli functor, forwhich we give an abbreviated summary. See [TV1], [TV2], [To], and [Lu5] for a general treatmentof derived algebraic geometry and [Lu7] and [Fra1] for derived algebraic geometry for E n -rings. Inthe following, O is an operad for which O (1) is the unit. Definition 4.1.
For X a functor from O -alg( C ) to the ∞ -category of spaces, the ∞ -category of O -quasicoherent sheaves on X isQC O X = lim R ∈ ( O -alg( C )) op /X Mod O R ( C ) ≃ Hom
Fun ( X, Mod) here X is regarded as a functor to ∞ -categories by composing with the inclusion of spaces into ∞ -categories, X : O -alg( C ) → Spaces → Cat ∞ , and Mod : O -alg( C ) → Cat ∞ is the covariantfunctor assigning to R the ∞ -category Mod O R ( C ), and to a morphism R → R ′ the induction functor U ′ R ⊗ U R ( − ).In other words, an O -quasicoherent sheaf M on X is an assignment of an R -module η ∗ M forevery R -point η ∈ X ( R ), compatible with base change. Example . Denote by Spec R , the functor of points Map O -alg ( R, − ) associated to R . In this case,the above limit is easy to compute: The ∞ -category O -alg( C ) op / Spec R has a final object, namelySpec R itself, and as a consequence, there is a natural equivalence QC O Spec R ≃ Mod O R ( C ).We can now make the following definition of the relative cotangent complex, which generalizesour previous notion of the cotangent complex of a map of O -algebras. Definition 4.3.
Let X and Y be functors from O -alg( C ) to the ∞ -category of spaces, and let f : X → Y be a map from X to Y . The relative cotangent complex L X | Y of f , if it exists, is the O -quasicoherent sheaf on X for which there is natural equivalenceMap R ( η ∗ L X | Y , M ) ≃ Fiber η ( X ( R ⊕ M ) → X ( R ) × Y ( R ) Y ( R ⊕ M )) . Remark . This definition is likely difficult to digest on first viewing: Intuitively, the relativecotangent complex is a linear approximation to the difference between X and Y , and it providesa linear method of calculating the value of X on a split square-zero extension X ( R ⊕ M ) givenknowledge of Y and X ( R ).If X and Y both admit absolute cotangent complexes (i.e., cotangent complexes relative toSpec k ), then there is a cofiber sequence f ∗ L Y → L X → L X | Y , known as the transitivity sequence.Our particular focus will be on the case of the tangent complex associated to a k -point of amoduli functor e : Spec k → X . It is convenient for our examples not to define the tangent space atthis point in terms of the cotangent complex, because it might be the case that the tangent complexexists while the cotangent complex fails to exist. That is, we can make sense of the notion of thetangent space at the following extra generality. Definition 4.5.
The ∞ -category M O ( C ) of infinitesimal O -moduli problems over C consists of thefull ∞ -subcategory of all functors F ∈
Fun( O -alg aug ( C ⋄ ) , Spaces) for which:(1) F ( k ) is equivalent to a point;(2) The restriction F ( k ⊕− ) : C ⋄ → Spaces to split square-zero extensions preserves finite limits;(3) F preserves products, i.e., the map F ( R × k R ′ ) → F ( R ) × F ( R ′ ) is an equivalence for all R , R ′ ∈ O -alg aug ( C ⋄ ).The first condition allows us to specify a single point Spec k → F to study; the second condition,we next show, implies the existence of the tangent space at that point. The third condition willallow the tangent space of the moduli problem to attain algebraic structure, as we shall see in thefinal section. Remark . Note that for a map of operads
O → Q , restriction along the forgetful functor onalgebras induces M O ( C ) → M Q ( C ). In particular, restriction along the forgetful functor defined by E n → E n + k induces functors M E n ( C ) → M E n + k ( C ), for all k ≥ F , we construct a functor T F : C ⋄ → Spectra, from the ∞ -category C ⋄ of compact objects of C to the ∞ -category of spectra. The i th space of the spectrum T F ( M ) is defined as T F ( M )( i ) = F ( k ⊕ M [ i ]) . By condition (2) of F being infinitesimal, this sequence of spaces forms an Ω-spectrum, which is a(typically nonconnective) delooping of the infinite loop space F ( k ⊕ M ). ntuitively, the functor T F assigns to an object M the spectrum of M -valued derivations of k on F . By the second assumption on F , T F can be seen to preserve finite limits, i.e., T F is an exactfunctor in the terminology of [Lu1]. We make use of the following result. See, for instance, [BFN]. Proposition 4.7.
The functor
C →
Fun ex ( C ⋄ , Spectra) , defined by sending an object M to the exactfunctor Map( k, M ⊗ − ) , is an equivalence. Thus, there exists an object T in C associated to the colimit preserving functor T F , with theproperty that there is an equivalence of spectra T F ( M ) ≃ Map C ( k, T ⊗ M ). We will abuse notationand refer to this functor and the object by the same symbols: Definition 4.8.
The tangent space T F of an infinitesimal moduli problem F is the object of C forwhich there is a natural equivalence Map C ( k, T F ⊗ M ) ≃ T F ( M ) for all M ∈ C .Given a moduli functor X : O -alg( C ) → Spaces with a map p : Spec k → X , we can reduce X at the point p to define a functor, X p : O -alg aug ( C ) → Spaces, as having values X p ( R ) =Fiber p ( X ( R ) → X ( k )), the homotopy fiber of the map X ( R ) → X ( k ) over the point p ∈ X ( k ). Definition 4.9.
For a moduli functor X , a map p : Spec k → X is a formally differentiable pointof X if the reduction X p , as defined above, is an infinitesimal moduli problem.In other words, a point is formally differentiable if it is possible to define the tangent space atthat point. Remark . If the functor T F also preserves infinite products and coproducts when restricted to C ,rather than C ⋄ , then, by Yoneda representability reasoning, there will additionally exist a cotangentobject L F that corepresents T F . If F is the reduction of a pointed moduli functor X p which has acotangent complex L X in QC O X , then there are equivalences T X p ≃ ( p ∗ L X ) ∨ and LX p ≃ p ∗ L X . Lemma 4.11.
For a moduli functor with a formally differentiable point p : Spec k → X , there is anatural equivalence T Ω X p ≃ T X p [ − , where Ω X is the pointed moduli functor whose R -points aregiven as the based loop space Ω p ( F ( R )) based at the point p : ∗ → F ( R ) .Proof. To validate this equivalence, it suffices to determine an equivalence of the functors T Ω X p and Ω T X p , which is immediately manifest. (cid:3) Now equipped with the requisite notions of a pointed moduli problem and its infinitesimal tangentcomplex, we turn to the particular moduli problems of interest.
Definition 4.12.
Let A and C be E n -algebras in C . The algebraic space of morphisms, Mor( A, C ),is a functor from E ∞ -algebras to spaces defined byMor( A, C )( R ) = Map E n -alg R ( R ⊗ A, R ⊗ C ) . This construction can be extended to make E n -alg( C ) enriched over Fun( E ∞ -alg( C ) , Spaces), inthe sense of [Gep]. Note that the space Mor(
A, A )( R ) naturally has a composition structure foreach R , whereby the functor Mor( A, A ) can be made to take values in E -algebras in spaces (or,equivalently, topological monoids). In the following, we will refer to a moduli functor valued intopological groups, or loop spaces, as an algebraic group. We may now define the algebraic groupof automorphisms of an E n -algebra. Definition 4.13.
The algebraic group Aut A , of automorphisms of an E n -algebra A , is the functor E ∞ -alg( C ) to loop spaces whose R -points consists of all maps Aut A ( R ) ⊂ Map E n -alg R ( R ⊗ A, R ⊗ A )that are homotopy equivalences. For more general purposes, further conditions are required, as a sheaf condition with respect to a Grothendiecktopology on E ∞ -alg( C ) op . For our restricted purpose in this work, which involves infinitesimal automorphisms andformal geometry, these extra conditions are not necessary (though they would typically be satisfied for the groups ofinterest). hat is, Aut A is the open subfunctor of Mor( A, A ) consisting of equivalences. The classify-ing functor B Aut A is the infinitesimal moduli functor associated to the composite B ◦ Aut A : E ∞ -alg( C ) → Ω– Spaces → Spaces.
Remark . For the functors F = B Aut A , the associated functor Fiber( F ( R ) → F ( k )), for R an E ∞ -algebra over k , has a familiar interpretation: It is infinitesimally equivalent to the functor Def A of deformations of A , in that there is a map that induces an equivalence on tangent spaces.We now turn to second type of algebraic group which will be of great interest for us, the groupof units of an E n -algebra. This definition first requires the following construction. For C any closedpresentable symmetric monoidal ∞ -category, there is functor ( − ) ⊗ C : Spaces → C , given byassigning to a space X the tensor X ⊗ C , where 1 C is the unit of the monoidal structure on C .This functor is symmetric monoidal, by assumption on C , and as a consequence its right adjointMap(1 C , − ) : C →
Spaces is right lax symmetric monoidal. In other words, if A is an O -algebra in C , for any topological operad O , then Map(1 C , A ) attains the structure of an O -algebra in spaces. Definition 4.15.
The functor GL : E n -alg( C ) → Ω n – Spaces is the composite of Map(1 C , − ) withthe functor E n -alg(Spaces) → E n -alg(Spaces) gp ≃ Ω n – Spaces which assigns to an E n -monoid itssubspace of invertible elements (which is equivalent to an n -fold loop space).This allows the formulation of the algebraic units of an E n -algebra. Definition 4.16.
The algebraic group A × of units of A is the functor E ∞ -alg( C ) → Ω n – Spaces as-signing to R the n -fold loop space A × ( R ) = GL ( R ⊗ A ), where GL is the functor E n -alg(Mod R ) → Ω n – Spaces of the previous definition. Remark . In the example where C is the ∞ -category of spectra, then this notion of GL clearlycoincides with the standard notion from algebraic topology. In particular, for a nonconnective E n -ring A with connective cover τ ≥ A , the spaces GL ( A ) and GL ( τ ≥ A ) will be equivalent. However,the algebraic groups A × and ( τ ≥ A ) × will differ despite their equivalence on k -points, due to thenonequivalence of τ ≥ ( R ⊗ A ) and τ ≥ ( R ⊗ τ ≥ A ) for general R .4.1. Automorphisms of Enriched ∞ -Categories. We now consider our final type of algebraicgroup, Aut B n A : It will take some preliminaries to finally arrive at the definition. First, note that ourdefinition of the algebraic group Aut A should apply verbatim to define an algebraic group structureon automorphisms of any object “ B n A ” as long as it can be suitably base-changed, i.e., so long as“ R ⊗ B n A ” can be defined for each R ∈ E ∞ -alg( C ).Let X be a monoidal ∞ -category. We will use the notion of ∞ -categories enriched in X , whichwe denote Cat ∞ ( X ), developed by Gepner in [Gep]. We will not give the technical definitions butinstead summarize the very rudimentary properties from [Gep] necessary for our purposes: Givena monoidal ∞ -category X , one constructs E -alg ⋆ ( X ), an ∞ -category of E -algebras in X withmany objects; Cat ∞ ( X ), ∞ -categories enriched in X , is a localization of E -alg ⋆ ( X ), obtained byinverting the enriched functors which are fully faithful and essentially surjective.There is a functor, which we will denote B , E -alg( X ) B / / Cat ∞ ( X )where for any A ∈ E -alg( X ), B A is an enriched ∞ -category with a single distinguished object ∗ ∈ B A , and such that there is an equivalence of the hom object Mor B A ( ∗ , ∗ ) = A as algebras in X . We denote by 1 the enriched ∞ -category B X . The functor B factors as E -alg( X ) / / Cat ∞ ( X ) / / / Cat ∞ ( X )through Cat ∞ ( X ) / , enriched ∞ -categories with a distinguished object 1. There is likewise a functorCat ∞ ( X ) / → E -alg( X ) sending an enriched ∞ -category A with a distinguished object 1 → A to he endomorphism algebra object End A (1). We have the following adjunction:Cat ∞ ( X ) / End(1) (cid:15) (cid:15) E -alg( X ) B C C We summarize these points in the following theorem:
Theorem 4.18 ([Gep]) . For a symmetric monoidal ∞ -category X , there is an symmetric monoidal ∞ -category Cat ∞ ( X ) with unit , with a symmetric monoidal functor B : E -alg( X ) → Cat ∞ ( X ) / which is fully faithful.Example . In the case that X is Spaces, the ∞ -category of spaces equipped with the Cartesianmonoidal structure, then E -alg(Spaces) is equivalent to the ∞ -category of topological monoids andCat ∞ (Spaces) is equivalent to Cat ∞ . The functor B is equivalent to the functor that assigns to atopological monoid G to its simplicial nerve N • G = B G , thought of as an ∞ -category with a singleobject whose endomorphisms equal G . (Since the usual classifying space BG is equivalent to thegeometric realization of the simplicial nerve | B G | , this is the motivation for the notation “ B ”.)This theorem has the following corollary. Corollary 4.20.
For any A and C in E -alg( X ) , there is a homotopy pullback diagram of spaces: Map E -alg( X ) ( A, C ) (cid:15) (cid:15) / / Map
Cat ∞ ( X ) ( B A, B C ) (cid:15) (cid:15) ∗ / / Map
Cat ∞ ( X ) (1 , B C ) Proof.
Since the functor B : E -alg( X ) → Cat ∞ ( X ) / has a right adjoint and the unit of theadjunction is an equivalence, we have that B is fully faithful. Therefore, Map E -alg ( A, C ) is homo-topy equivalent to Map
Cat ∞ ( X ) / ( B A, B C ). The result now follows from the standard formula formapping objects in an under category. (cid:3) For C a symmetric monoidal ∞ -category, the ∞ -category E -alg( C ) inherits the symmetricmonoidal structure of C . Thus, the construction above can be iterated to obtain a functor B : E -alg( E -alg( C )) → Cat ∞ (Cat ∞ ( C )), where ∞ -category Cat ∞ (Cat ∞ ( C )) is our definition forCat ( ∞ , ( C ), ( ∞ , C , [Gep]. Iterating, we obtain a fully faithful functor E -alg ( n ) -alg( C ) ֒ → Cat ( ∞ ,n ) ( C ) / where E -alg ( n ) ( C ) is the ∞ -category of n -times iterated E -algebras in C , and 1 = B n C is the unitof Cat ( ∞ ,n ) ( C ). As a consequence, we have an identical formula for mapping spaces as in Corollary4.20. That is, we may now use this result to describe mapping spaces of E n -algebras, using thetheorem of Dunn, and Lurie, that an E n -algebra is an n -times iterated E -algebra. By the samereasoning as for Corollary 4.20, we have the following corollary of Theorem 4.18 and Theorem 2.42: Corollary 4.21.
For any E n -algebras A and C in X , there is a homotopy pullback diagram: Map E n -alg( X ) ( A, C ) (cid:15) (cid:15) / / Map
Cat ( ∞ ,n ) ( X ) ( B n A, B n C ) (cid:15) (cid:15) ∗ / / Map
Cat ( ∞ ,n ) ( X ) (1 , B n C ) inally, we will need the following comparison between enriched ∞ -categories and tensored ∞ -categories. Let C be a presentable symmetric monoidal ∞ -category whose monoidal structure dis-tributes over colimits, and let Mod ( n ) A abbreviate the n -fold application of the Mod-functor, definedby Mod ( k +1) A := Mod Mod ( k ) A (Cat Pr( ∞ ,k ) ( C ))where Cat Pr( ∞ ,k ) ( C ) consists of those ( ∞ , k )-categories enriched in C which are presentable. An A -module in C is equivalent to an enriched functor B A → C . Likewise, there is an equivalenceMod ( k +1) A ≃ Fun( B k +1 A, Cat
Pr( ∞ ,k ) ( C )). Proposition 4.22 ([Gep]) . The map
Map( B n A, B n C ) → Map(Mod ( n ) A , Mod ( n ) C ) is full on com-ponents, and the essential image consists of those functors F for which there exists an equivalence F (Mod ( i ) A ) ≃ Mod ( i ) C , for each i < n . We now define the algebraic group Aut B n A : Definition 4.23.
For an E n -algebra in a symmetric monoidal ∞ -category C , the algebraic group ofautomorphisms of B n A is the functor Aut B n A : E ∞ -alg( C ) → Ω– Spaces whose R -points are givenby the subspace Aut B n A ( R ) ⊂ Map
Cat ( ∞ ,n ) (Mod R ( C )) ( B n ( R ⊗ A ) , B n ( R ⊗ A ))of those functors which are equivalences.In order to apply the theory of infinitesimal moduli problems to our algebraic groups of interest,it is necessary to make the following observation. Lemma 4.24.
The moduli functors
Mor(
A, C ) , Aut A , A × , and Aut B n A have formally differentiablepoints, i.e., the reduction of the moduli functors at their natural basepoints form infinitesimal moduliproblems.Proof. We give the proof for f ∈ Mor(
A, C ), the others cases being similar. Condition (1) isimmediate. Conditions (2) and (3) are implied by the fact the tensor products in C distribute overfinite limits, and therefore the reduction Mor( A, C ) f preserves finite limits. (cid:3) Infinitesimal Automorphisms of E n -algebras. The results of the rest of this section involvethe interrelation of these algebraic groups and the cohomology theories of E n -algebra studied earlierin this paper, and are summarized in the following theorem. Theorem 4.25.
There is a fiber sequence of algebraic groups B n − A × → Aut A → Aut B n A . Passingto the associated tangent spaces gives a fiber sequence A [ n − → T A → HH ∗E n ( A )[ n ] .Remark . This theorem elaborates on Kontsevich’s conjecture from [KS] and [Ko]. Kontsevichconjectured an equivalence of Lie algebras HH ∗E n ( A )[ n ] ≃ T A /A [ n − A defined over a field of characteristic zero, that the Maurer-Cartan elements of HH ∗E n ( A )[ n ] classifydeformations of the enriched ∞ -category B n A , or, equivalently, certain deformations of Mod ( n ) A .This is a generalization of the familiar result that Maurer-Cartan elements of the Lie bracket ofusual Hochschild cohomology HH ∗ ( A )[1] classify deformations of B A or, equivalently, deformationsof the category of A -modules, for A an associative algebra. This n = 1 case is very close to Keller’stheorem in [Ke]. Remark . By the Lie algebra of an algebraic group G in C , we mean its tangent space at theidentity map id : Spec k → G , which can be expressed as any of the equivalent tangent spaces,Lie( G ) := T G ≃ T B G [ − ≃ T k | B G . It remains to show that this tangent space indeed possesses aLie algebraic structure: We address this point in the final section.There is an analogue of the preceding theorem for a map f : A → C of E n -algebras: heorem 4.28. There is a fiber sequence of moduli problems
Mor(
A, C ) → Mor( B n A, B n C ) → B n C × , and given an E n -ring map f : A → C , looping gives a corresponding sequence of algebraicgroups Ω f Mor(
A, C ) → Ω f Mor( B n A, B n C ) → B n − C × . Passage to the tangent spaces at thedistinguished point gives a fiber sequence of Lie algebras Der(
A, C )[ − → HH ∗E n ( A, C )[ n − → C [ n − . We will prove this piecemeal, beginning with the fiber sequence of moduli functors.
Proposition 4.29.
There is a fiber sequence of algebraic groups B n − A × → Aut A → Aut B n A .Proof. By Corollary 4.21, there is a fiber sequence of spacesMor(
A, A )( R ) → Mor( B n A, B n A )( R ) → Mor( B n k, B n A )( R )for every R . Since limits in functor ∞ -categories are computed pointwise in the target, this impliesthat Mor( A, A ) → Mor( B n A, B n A ) → Mor( B n k, B n A ) is a fiber sequence of moduli functors.Restricting to equivalences in the first and second terms gives rise to an additional fiber sequenceAut A → Aut B n A → Mor( B n k, B n A ).Next, we identify the moduli functor Mor( B n k, B n A )( R ) with B n A × . It suffices to producea natural equivalence on their k -points, the spaces Map Cat ( ∞ ,n ) ( C ) ( B n k, B n A ) and B n GL ( A ),the argument for general R -points being identical. By the adjunction Map( k, − ) : E n -alg( C ) ⇆ E n -alg(Spaces) : ( − ) ⊗ k , a map out of B n k in Cat ( ∞ ,n ) ( C ) is equivalent to a map out of the con-tractible, trivial category ∗ in Cat ( ∞ ,n ) . Thus, we have the equivalence Map Cat ( ∞ ,n ) ( C ) ( B n k, B n A ) ≃ Map
Cat ( ∞ ,n ) ( ∗ , B n Map( k, A )). For the n = 1 case of Cat ∞ , there is an equivalence Map Cat ∞ ( ∗ , C ) = C ∼ , the subspace of C consisting of all invertible morphisms. Iterating this relation to obtain thesame result for all n , this implies the equivalenceMap Cat ( ∞ ,n ) ( ∗ , B n Map( k, A )) ≃ Map
Cat ( ∞ ,n ) ( ∗ , B n GL ( A )) ≃ B n GL ( A ) , which completes our argument that Mor( B n k, B n A ) and B n A × define the same moduli functor.Cumulatively, we may now identify a natural fiber sequence of functors Aut A → Aut B n A → B n A × .The homotopy fiber of the map Aut A → Aut B n A can thereby be identified as the looping of the base,Ω B n A × , which is equivalent to B n − A × . The map Aut A → Aut B n A is a map of algebraic groups,and limits of algebraic groups are calculated in the underlying ∞ -category of functors, therefore theinclusion of the fiber B n − A × → Aut A is a map of algebraic groups. (cid:3) Lemma 4.30.
For a fiber sequence of infinitesimal moduli functors X → Y → Z , passage to thetangent spaces results in a fiber sequence T X → T Y → T Z in C .Proof. To prove that
T X → T Y → T Z is a fiber sequence, it suffices to show that
T X → T Y → T Z is a fiber sequence of functors, i.e., that for every M ∈ C , that T X ( M ) → T Y ( M ) → T Z ( M ) isa fiber sequence of spectra. This, in turn, follows from the corresponding fact for the space-valuedfunctors: If this sequence forms a fiber sequence of spaces for every M , then the previous sequencewill be a fiber sequence of spectra, since the functor Ω ∞ preserves fibrations. (cid:3) The next step is the identification of the tangent spaces of the individual terms in the sequence B n − A × → Aut A → Aut B n A . Lemma 4.31.
There is a natural equivalence
Lie(Aut A ) ≃ T A and more generally, an equivalenceof functors T Aut A ≃ Hom
Mod E nA ( L A , − ⊗ A ) Proof.
First, there is an equivalence of tangent spaces T Aut A and T Mor(
A, A ), for the followinggeneral reason. Let X → Y is a map of moduli functors for which X ( R ) → Y ( R ) is an inclusion ofcomponents for every R , which can be thought of as a generalization of the notion of a map beingformally Zariski open. In this case, the fibers of the map X ( R ⊕ M ) → X ( R ) × Y ( R ) Y ( R ⊕ M )are trivial, for all R and M , and thus the relative cotangent and tangent complexes are trivial. Inparticular, the relative tangent complex of Aut A → Mor(
A, A ) is trivial, and from the transitivitysequence we obtain the natural equivalence T Aut A ≃ T Mor(
A, A ). ow, let N be an object of C . By definition, the space Map C ( L k | Mor(
A,A ) , N ) is the loop space ofthe fiber of the map Mor( A, A )( k ⊕ N ) → Mor(
A, A )( k ) induced by the projection map k ⊕ N → k .This fiber is the mapping space Map E n -alg /A ( A, A ⊕ A ⊗ N ), which is equivalent to Map Mod E nA ( L A , A ⊗ N ). Thus, we have an equivalence Map k ( L k | Mor(
A,A ) , N ) ≃ Ω Map
Mod E nA ( L A , A ⊗ N ) for all N ∈ C .By setting N = k , we obtain the equivalence T k | Aut A ≃ T k | Mor(
A,A ) ≃ T A [ − F = B Aut A completes the proof. (cid:3) Remark . This is an derived algebraic analogue of the familiar topological fact that the Liealgebra of the diffeomorphism group of a smooth manifold is equivalent to the Lie algebra of vectorfields.
Lemma 4.33.
There is a natural equivalence
Lie( B n − A × ) ≃ A [ n − .Proof. This is a consequence of the following. First, for a left A -module V , let End A ( V ) be thefunctor E ∞ -alg( C ) → Spaces that assigns to k ′ the space of maps End A ( V )( k ′ ) = Map k ′ ⊗ A ( k ′ ⊗ V, k ′ ⊗ V ). A standard calculation shows the equivalence T k | End A ( V ) [1] ≃ Hom A ( V, V ). Using that A × → End A ( A ) is formally Zariski open, we obtain that tangent space of A × at the identity is A ,which can delooped n − (cid:3) Having described Lie(Aut A ) and Lie( B n − A × ), we are left to identify the Lie algebra of infinites-imal automorphisms of the enriched ( ∞ , n )-category B n A . Let A be an E n -algebra in C , as before.The remainder of this section proves the following, which will complete the proof of the part ofTheorem 4.25 that our sequence relating the tangent complex and Hochschild cohomology is theinfinitesimal version of our sequence of algebraic groups relating automorphisms of A and those of B n A : Theorem 4.34.
There is an equivalence
Lie(Aut B n A ) ≃ HH ∗E n ( A )[ n ] between the Lie algebra of the algebraic group of automorphisms of the C -enriched ( ∞ , n ) -category B n A , and the n -fold suspension of the E n -Hochschild cohomology of A . We prove the theorem by applying the following proposition.
Proposition 4.35.
There is an equivalence Ω n Aut B n A ( k ) ≃ GL (HH ∗E n ( A )) .Proof of Theorem 4.34. To prove the theorem, it suffices to show an equivalence of spaces between T Ω n Aut B n A ( N ) and Map Mod E nA ( A, A ⊗ N ) for every N in C : By choosing N = k [ j ] to be shifts of k , this would then imply the equivalence of Theorem 4.34.The space Map Mod E nA ( A, A ⊗ N ) is the fiber, over the identity, of the mapMap Mod E nA ⊕ A ⊗ N ( A ⊕ A ⊗ N, A ⊕ A ⊗ N ) / / Map
Mod E nA ( A, A ) , which is equivalent to the fiber of the mapGL (HH ∗E n ( A ⊕ A ⊗ N )) / / GL (HH ∗E n ( A ))since the bottom row is a subspace, full on connected components, in the top row. We thereforehave a map of homotopy fiber sequences: T Ω n Aut B n A ( N ) (cid:15) (cid:15) / / Ω n Map( B n A ⊕ A ⊗ N, B n A ⊕ A ⊗ N ) ∼ (cid:15) (cid:15) / / Ω n Map( B n A, B n A ) ∼ (cid:15) (cid:15) Map
Mod E nA ( A, A ⊗ N ) / / GL (HH ∗E n ( A ⊕ A ⊗ N )) / / GL (HH ∗E n ( A ))Since the two right hand vertical arrows, on the base and total space, are homotopy equivalences,this implies that the left hand map, on fibers, is a homotopy equivalence: T Ω n Aut B n A ( N ) ≃ Map
Mod E nA ( A, A ⊗ N ). (cid:3) he rest of this section will be devoted to the proof of Proposition 4.35. See [Lu6] for a closelyrelated treatment of this result. The essential fact for the proof is the following: Proposition 4.36.
There is an equivalence HH ∗E n − (Mod A ( C )) ≃ Mod E n A ( C ) .Remark . Proposition 4.36 generalizes a result proved in [BFN], where this statement was provedin the case where A had an E ∞ -algebra refinement. The argument below is essentially identical,replacing the tensor S k ⊗ A , used in [BFN], by the factorization homology R S k A . Proof.
The proof is a calculation of the ∞ -category of functors Fun E n − –Mod A (Mod A , Mod A ), whichare E n − -Mod A -module functors. First, we have an equivalence between E n − -Mod A -module ∞ -categories and ( R S n − Mod A )-module ∞ -categories, as a consequence of Proposition 3.16. Thus, weobtain an equivalence HH ∗E n − (Mod A ) ≃ Fun R Sn − Mod A (Mod A , Mod A ) . Secondly, for M a stably parallelizable k -manifold of dimension less than n −
1, then R M Mod A ≃ Mod R M A , by Proposition 3.33. Applying this in the case of M = S n − produces the furtherequivalence HH ∗E n − (Mod A ) ≃ Fun
Mod R Sn − A (Mod A , Mod A ) . Finally, we apply the general equivalence of ∞ -categoriesFun Mod R (Mod A , Mod B ) ≃ Mod A op ⊗ R B in the case of R = R S n − A and A = B . Again using the basic features of factorization homology,the equivalences A op ⊗ R Sn − A A ≃ R S n − A and R S n − A ≃ U A give the promised conclusion ofHH ∗E n − (Mod A ) ≃ Mod R Sn − A ≃ Mod E n A . (cid:3) This has an immediate corollary, that the E n -Hochschild cohomology of A is equivalent to theendomorphisms of the unit of the tensor structure for the E n − -Hochschild cohomology of Mod A : Corollary 4.38.
For A as above, there is a natural equivalence HH ∗E n ( A ) ≃ Hom HH ∗E n − (Mod A ) (1 , . We have the following transparent lemma, which we will shortly apply.
Lemma 4.39.
For X an ∞ -category with a distinguished object , and X ∼ the underlying spaceconsisting of objects of X , then there is an equivalence GL (Map X (1 , ≃ Ω X ∼ . (cid:3) This gives the following:
Corollary 4.40.
For A an E n -algebra, there are equivalences GL (HH ∗E n ( A )) ≃ Ω GL (Mod E n A ) ≃ Ω GL (HH ∗E n − (Mod A )) . We now complete the proof of Proposition 4.35 (which, in turn, completes the proof of Theorem4.34).
Proof of Proposition 4.35.
By iterating the previous corollary, we obtainGL (HH ∗E n ( A )) ≃ Ω n GL (cid:16) HH ∗E (cid:16) Mod ( n ) A (cid:17)(cid:17) . For the case n = 0, the E -Hochschild cohomology is given simply by endomorphisms, HH ∗E ( R ) =Hom( R, R ). Hence we have the equivalence GL (HH ∗E n ( A )) ≃ Ω n id Map(Mod ( n ) A , Mod ( n ) A ). UsingProposition 4.22, we finally conclude that GL (HH ∗E n ( A )) ≃ Ω n id Map( B n A, B n A ), proving ourproposition. (cid:3) Remark . The proof of Proposition 4.35 extends to show that Lie(Ω f Mor( B n A, B n C )) is equiv-alent to HH ∗E n ( A, C )[ n − f : A → C an E n -algebra map. t remains to equate the two sequences constructed in the main theorems of this paper areequivalent: Proposition 4.42.
The two sequences HH ∗E n ( A ) → A → T A [1 − n ] constructed in Theorems 2.26and 4.25 are equivalent.Proof. It suffices to show that the two maps HH ∗E n ( A ) → A are equivalent to conclude the equiv-alence of these two sequences, up to an automorphism of T A . The map of Theorem 4.25 was thelinearization of a map Ω n Aut B n A → A × , which was the units of the usual map Map Mod E nA ( A, A ) → Map
Mod A ( A, A ), given by the forgetful functor Mod E n A → Mod A . The map in Theorem 2.26 is the E n - A -module dual of the map U A → A , defined by the counit of the adjunction between Mod E n A andMod A . Thus, we obtain that these two maps are equivalent. (cid:3) Lie Algebras and the Higher Deligne Conjecture.
In the previous section, we showedthat the fiber sequence A [ n − → T A → HH ∗E n ( A ) could obtained as the tangent spaces associatedto a fiber sequence of derived algebraic groups. The sole remaining point of discussion is to identifythe algebraic structure on this sequence. As tangent spaces of algebraic groups, one should expectthat this is a sequence of (restricted) Lie algebras, as Kontsevich conjectured in [Ko]: This is indeedthe case. This sequence has more structure, however: After shifting, it is is a sequence of nonunital E n +1 -algebras.We briefly explain in what sense this can be regarded as being more structured. An associativealgebra can be equipped with the commutator bracket, which gives it the structure of a Lie algebra.A similar fact is the case for general E n -algebras. If A is an E n -algebra in chain complexes over afield F , then there is a map E n (2) ⊗ A ⊗ → A . Passing to homology, and using that H ∗ ( E n (2) , F ) ∼ =H ∗ ( S n − , F ) ∼ = F ⊕ F [ n − F ⊕ F [ n − ⊗ H ∗ ( A ) ⊗ → H ∗ ( A ). We thus obtaintwo different maps. The degree 0 map defines an associative multiplication, which is quite familiar;the degree n − Lie bracket , as first proved by Cohen, [Co]. Thus,at least at the level of homology, one can think of an E n -algebra structure on A as consisting of aLie algebra on the shift A [ n −
1] together with some extra structure.We now show that these structures exist on the tangent spaces we have discussed. That is, thatthe tangent space of an infinitesimal moduli problem admits same structure as that afforded by thetangent space of an augmented algebra. In particular, for a moduli problem for O -algebras, thetangent space has an O ! -algebra structure. First, we show that our E ∞ -moduli problems of interestadmit refinements to E n +1 -moduli problems. Proposition 4.43.
There is a lift of the infinitesimal E ∞ -moduli problems associated to B Aut A ,B Aut B n , and B n − A × to the ∞ -category M E n +1 ( C ) of infinitesimal E n +1 -moduli problems.Proof. It suffices to show that there is a factorization of G : E ∞ -alg( C ) → E n +1 -alg( C ) → Ω– Spaces,for each G among the groups above. Recall Theorem 2.46, the consequence of the theorem of Dunn,[Du] and [Lu6], for an E n +1 -algebra R , the ∞ -category Mod R ( C ) has the structure of an E n -monoidal ∞ -category. Using this, we now define the above functors: Aut A : E n +1 -alg( C ) → Spaces takesvalues Aut A ( R ) ⊂ Map E n -alg(Mod R ( C )) ( R ⊗ A, R ⊗ A )consisting of those maps that are homotopy equivalences. Likewise, Aut B n A : E n +1 -alg( C ) → Spacesis defined by taking valuesAut B n A ( R ) ⊂ Map
Cat ∞ ,n (Mod R ( C )) ( B n ( R ⊗ A ) , B n ( R ⊗ A )) . And likewise the previous formula may be applied to define B n − A × . (cid:3) The argument that an infinitesimal O -moduli problem F has a tangent space in C is identical tothe E ∞ case: The assignment M F ( k ⊕ M ) can be delooped to form a spectrum-valued functor,which is equivalent to one of the form Map C ( k, T F ⊗ − ). We now show that the tangent space ofan O -moduli problem obtains the same algebraic structure that the tangent space of an augmented -algebra possesses. The essential idea is that infinitesimal moduli problems are expressible as geo-metric realizations and filtered colimits of affines (i.e., functors of the form Spec A = Map O ( A, − )):Geometric realizations and filtered colimits preserve algebraic structure, therefore the tangent spaceof the moduli problem retains the algebraic structure of the terms in the resolution. Proposition 4.44.
Assume that there exists a functorial O ∨ -algebra structure on the tangentspace of an augmented O -algebra, for some operad O ∨ . I.e., we are given a factorization of T : O -alg aug ( C ) op → C , through the forgetful functor O ∨ -alg nu ( C ) → C . Then the tangent space ofevery infinitesimal moduli problem F canonically obtains an O ∨ -algebra structure. I.e., there is alift: O -alg aug ( C ⋄ ) op e T / / Spec (cid:15) (cid:15) O ∨ -alg nu ( C ) M O ( C ) T ♠♠♠♠♠♠ Proof.
To economize, we abbreviate O -alg := O -alg aug ( C ⋄ ) for the duration of the proof. Con-sider the embedding Spec : O -alg op ⊂ P ( O -alg op ), the Yoneda embedding into presheaves. Since O -alg op is a small ∞ -category, this factors through the ∞ -category of ind-objects, Ind( O -alg op ) ⊂P ( O -alg op ), whose essential image in the ∞ -category of presheaves consists of those presheaves thatpreserve finite limits. Consider also the ∞ -subcategory of presheaves consisting of all those functorsthat preserve products, which we denote P Σ ( O -alg op ) ⊂ P ( O -alg op ). We thus have the followingsequence of fully faithful inclusions of ∞ -categories O -alg op ⊂ Ind( O -alg op ) ⊂ M O ( C ) ⊂ P Σ ( O -alg op ) ⊂ P ( O -alg op ) . First, we define the functor e T : Ind( O -alg op ) → O ∨ -alg nu ( C ). Any ind-object X can be realizedas a filtered colimit lim −→ Spec A i in the ∞ -category of presheaves, and this gives an equivalence ofthe tangent space T X ≃ lim −→ T A i , where the filtered colimit is computed in C . However, this isa filtered diagram of O ∨ -algebras, and the forgetful functor O ∨ -alg nu ( C ) → C preserves filteredcolimits, hence T X obtains the structure of an O ∨ -algebra.We now extend the functor e T to each moduli problem F ∈ M O ( C ). Since F preserves products,and O -alg op is a small ∞ -category, there exists a simplicial resolution of F by ind-representables,by Lemma 5.5.8.14 of [Lu0]. That is, there exists a simplicial presheaf F • → F mapping to F , witheach F i ≃ lim −→ Spec A l ind-representable, and such that the map |F • ( R ) | → F ( R ) is an equivalencefor every R ∈ O -alg. The tangent space T F is thereby equivalent to the geometric realization of T F • , the tangent spaces of the resolution. Since the forgetful functor O ∨ -alg( C ) → C preservesgeometric realizations, there is an equivalence in C between T F and | e T F • | , hence T F obtains thestructure of an O ∨ -algebra.To state this slightly more formally, we have the following pair of left Kan extensions: O -alg opSpec (cid:15) (cid:15) e T / / O ∨ -alg nu ( C ) / / CM O ( C ) Spec ! e T C C ❢ ❦ ♣ ✈ ⑦ ✆ Spec ! T : : ❝ ❢ ❤ ❧ ♦ ♣ r t By the preceding, both left Kan extensions can be calculated in terms of filtered colimits and geo-metric realizations that are preserved by the forgetful functor O ∨ -alg nu ( C ) → C . As a consequencewe obtain the equivalence Spec ! e T F ≃
Spec ! T F for each infinitesimal moduli problem F . Since theleft Kan extension Spec ! T F ≃ T F exactly calculates the usual tangent space of F , we obtain acanonical O ∨ -algebra structure on T F , for each F (cid:3) Remark . The condition of being product-preserving is essential in the definition of an infini-tesimal moduli problem: One can always left Kan extend the functor O -alg aug ( C ⋄ ) op → O ∨ -alg( C ) long the inclusion, Spec, but there would be no guarantee that the result would be the tangentspace of F , due to the difference between colimits in O ∨ -algebras and colimits in C . Condition (2)is a slight weakening of a Schlessinger-type formal representability condition, [Sc].Proposition 4.44, together with Proposition 2.37, implies the following: Corollary 4.46.
For O ! = (1 ◦ O ∨ the derived Koszul dual operad of the operad O , the tangentspace of an infinitesimal O -moduli problem has the structure of an O ! -algebra. That is, there is afunctor T : M O ( C ) → O ! -alg nu ( C ) . The next result follows from Proposition 4.44 and Proposition 2.41, coupled with Proposition4.43.
Corollary 4.47.
There is a commutative diagram M E ( C ) / / T (cid:15) (cid:15) . . . / / M E m ( C ) T (cid:15) (cid:15) / / . . . / / M E ∞ ( C ) T (cid:15) (cid:15) E [ −
1] -alg nu ( C ) forget + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ E m [ − m ] -alg nu ( C ) forget (cid:15) (cid:15) Lie[ −
1] -alg( C ) forget s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ C in which the vertical arrows are given the tangent spaces of the moduli problems, and the hori-zontal arrows are given by restriction of moduli problems along the functors E n + k -alg aug ( C ⋄ ) →E n -alg aug ( C ⋄ ) .Proof. By Proposition 2.41, we have functors E m -alg aug ( C ⋄ ) op → E m [ − m ] -alg nu ( C ) for each m . Thetheorem follows by applying Proposition 4.44 to obtain the corresponding picture for infinitesimalmoduli problems. (cid:3) Remark . If an additional hypothesis is placed on the infinitesimal moduli problems, and C = Mod F is chain complexes over a field F , then Lurie, in [Lu7], has outlined an argument thatthe functor from E n -moduli problems to nonunital E n -algebras is an equivalence. At the level ofgenerality of the present work, however, it seems very possible that such modification does notproduce an equivalence between moduli problems and their tangent spaces.This completes the proof of Theorem 1.1: Since the reduction of the functors B n A × → B Aut A → B Aut B n A defines a fiber sequence in M E ∞ ( C ), therefore their tangent spaces attain the structureof Lie algebras, after shifting by 1. Since this fiber sequence lifts to a sequence of infinitesimal E n +1 -moduli problems, their tangent spaces attain the structure of nonunital E n +1 -algebras, aftershifting by n . Applying Lemma 4.30, these tangent complexes form a fiber sequence, the termsof which are A [ − T A [ − n ] and HH ∗E n ( A ) by the calculations of Lemma 4.33, Lemma 4.31, andTheorem 4.34.Despite showing the existence of an interesting nonunital E n +1 -algebra structure on A [ − E n + i -algebra structure on A [ − i ], we have not offered a satisfying de-scription of it. We suggest the following. Conjecture 4.49.
Given the equivalence of operads E n [ − n ] ≃ E ! n , then the nonunital E n +1 -algebrastructure on A [ − given in Theorem 1.1 is equivalent, after suspending by n , to that defined byrestricting the E n -algebra structure of A along the map of operads E n +1 [ − n − ≃ E ! n +1 −→ E ! n ≃ E n [ − n ] , Koszul dual to the usual map of operads E n → E n +1 . Likewise, the Lie algebra structure constructedon the sequence A [ n − → T A → HH ∗E n ( A )[ n ] is equivalent to that obtained from the sequenceof nonunital E n +1 -algebras A [ − → T A [ − n ] → HH ∗E n ( A ) by restricting along a map of operads Lie → E n [1 − n ] , which is the Koszul dual of the usual operad map E n +1 → E ∞ . e conclude with several comments. Remark . This gives a new construction and interpretation of the E n +1 -algebra on E n -Hochschildcohomology HH ∗E n ( A ), the existence of which is known as the higher Deligne conjecture, [Ko], andwas previously proved in full generality in [Lu6] and [Th], in [HKV] over a field, and in [MS] for n = 1, among many other places. To summarize, when A is an E n -algebra in a stable ∞ -category,then HH ∗E n ( A ) has an E n +1 -algebra structure because it can be identified with the tangent complexto a E n +1 -moduli problem. This should not be thought of as a genuinely new proof, however, becauseit uses the same essential ingredient as [Lu6], [HKV], and possibly all of the other proofs: that E n -algebras are n -iterated E -algebras. The particular benefit of this construction of the E n +1 -algebrastructure is that it relates it to deformation theory, as well as to nonunital E n +1 -algebra structureson T A [ − n ] and A [ − E n +1 -algebra structure on T A [ − n ] waspreviously constructed in [Ta]. Remark . A consequence of Theorem 1.1 is, in characteristic zero, solutions to the Maurer-Cartan equation for the Lie bracket on HH ∗E n ( A )[ n ] classify deformations of B n A , hence Mod ( n ) A . Itwould interesting to have a direct construction of these deformations. References [And] Andrade, Ricardo. From manifolds to invariants of E n -algebras. Thesis (PhD) – Massachusetts Institute ofTechnology. 2010.[Ang] Angeltveit, Vigleik. Uniqueness of Morava K-theory. Vigleik Angeltveit (2011). Uniqueness of Morava K -theory.Compos. Math. 147 (2011), no. 2, 633–648,[AFT] Ayala, David; Francis, John; Tanaka, Hiro. Structured singular manifolds and factorization homology.Preprint. Available at http://arxiv.org/abs/1206.5164[Ba] Basterra, Maria. Andr´e-Quillen cohomology of commutative S-algebras. Journal of Pure and Applied Algebra144 (1999) no. 2, 111-143.[BM1] Basterra, Maria; Mandell, Michael. Homology and cohomology of E ∞ ring spectra, Math. Z. 249 (2005),903–944.[BM2] Basterra, Maria; Mandell, Michael. The Multiplication on BP . Preprint, 2010. arXiv:1101.0023.[BD] Beilinson, Alexander; Drinfeld, Vladimir. Chiral algebras. American Mathematical Society Colloquium Publi-cations, 51. American Mathematical Society, Providence, RI, 2004.[BFN] Ben-Zvi, David; Francis, John; Nadler, David. Integral transforms and Drinfeld centers in derived algebraicgeometry. J. Amer. Math. Soc. 23 (2010), no. 4, 909–966.[BV] Boardman, J. Michael; Vogt, Rainer. Homotopy invariant algebraic structures on topological spaces. LectureNotes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York, 1973. x+257 pp.[Ch] Ching, Michael. Bar constructions for topological operads and the Goodwillie derivatives of the identity. Geom.Topol. 9 (2005), 833–933.[Co] Cohen, Frederick. The homology of C n +1 -spaces, n ≥ ∼ costello/renormalization[Du] Dunn, Gerald. Tensor product of operads and iterated loop spaces. J. Pure Appl. Algebra 50 (1988), no. 3,237–258.[Fra1] Francis, John. Derived algebraic geometry over E n -rings. Thesis (PhD) – Massachusetts Institute of Technology.2008.[Fra2] Francis, John. Factorization homology of topological manifolds. Preprint. Available athttp://arxiv.org/abs/1206.5522[FG] Francis, John; Gaitsgory, Dennis. Chiral Koszul duality. Selecta Math. (N.S.) 18 (2012), no. 1, 27–87.[Fre1] Fresse, Benoit. On the homotopy of simplicial algebras over an operad. Trans. Amer. Math. Soc. 352 (2000),no. 9, 4113–4141.[Fre2] Fresse, Benoit. Koszul duality of E n -operads. Selecta Math. (N.S.) 17 (2011), no. 2, 363–434.[Gep] Gepner, David. Enriched ∞ -categories. In preparation.[GJ] Getzler, Ezra; Jones, J. Operads, homotopy algebra and iterated integrals for double loop spaces. Unpublishedwork, 1994. Available at arXiv:hep-th/9403055.[GTZ] Ginot, Gr´egory; Tradler, Thomas; Zeinalian, Mahmoud. Derived higher Hochschild homology, topologicalchiral homology and factorization algebras. Preprint.[GK] Ginzburg, Victor; Kapranov, Mikhail. Koszul duality for operads. Duke Math. J. 76 (1994), no. 1, 203–272. N -discs operad. Preprint.[La] Lazarev, Andrey. Homotopy theory of A ∞ ring spectra and applications to M U-modules. K -Theory 24 (2001),no. 3, 243–281.[Lu0] Lurie, Jacob. Higher topos theory. Annals of Mathematics Studies, 170. Princeton University Press, Princeton,NJ, 2009. xviii+925 pp.[Lu1] Lurie, Jacob. Derived Algebraic Geometry 1: Stable ∞ -categories. Preprint. Available atarXiv:math.CT/0608228.[Lu2] Lurie, Jacob. Derived Algebraic Geometry 2: Noncommutative Algebra. Preprint. Available atarXiv:math.CT/0702299.[Lu3] Lurie, Jacob. Derived Algebraic Geometry 3: Commutative algebra. Preprint. Available atarXiv:math.CT/0703204.[Lu4] Lurie, Jacob. Derived Algebraic Geometry 4: Deformation theory. Preprint. Available atarXiv:math.CT/0703204.[Lu5] Lurie, Jacob. Derived Algebraic Geometry 5: Structured spaces. Preprint. Available atarXiv:math.CT/0703204.[Lu6] Lurie, Jacob. Derived Algebraic Geometry 6: E k ∼ lurie/[Ma] May, J. Peter. The geometry of iterated loop spaces. Lectures Notes in Mathematics, Vol. 271. Springer-Verlag,Berlin-New York, 1972. viii+175 pp.[MS] McClure, James; Smith, Jeff. A solution of Deligne’s Hochschild cohomology conjecture. Recent progress inhomotopy theory. Contemporary Mathematics 293, 2002, 153-194.[Mc] McDuff, Dusa. Configuration spaces of positive and negative particles. Topology 14 (1975), 91–107.[MW] Morrison, Walker. Blob Complex. Preprint. Available at arXiv:1009.5025v2[Qu] Quillen, Daniel. On the (co-) homology of commutative rings. 1970 Applications of Categorical Algebra (Proc.Sympos. Pure Math., Vol. XVII, New York, 1968) pp. 65–87. Amer. Math. Soc., Providence, R.I.[Re] Rezk, Charles. Spaces of algebra structures and cohomology of operads. Thesis (PhD) – Massachusetts Instituteof Technology. 1996.[Sa] Salvatore, Paolo. Configuration spaces with summable labels. Cohomological methods in homotopy theory (Bel-laterra, 1998), 375–395, Progr. Math., 196, Birkh¨auser, Basel, 2001.[SW] Salvatore, Paolo; Wahl, Nathalie. Framed discs operads and Batalin-Vilkovisky algebras. Q. J. Math. 54 (2003),no. 2, 213–231.[Sc] Schlessinger, Michael. Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208–222.[SS] Schlessinger, Michael; Stasheff, James. The Lie algebra structure of tangent cohomology and deformation theory.Journal of Pure and Applied Algebra 38 (1985), 313-322.[Ta] Tamarkin, Dmitri. The deformation complex of a d -algebra is a ( d +1)-algebra. Unpublished work, 2000. Availableat arXiv:math/0010072v1[Th] Thomas, Justin. Kontsevich’s Swiss Cheese Conjecture. Thesis (PhD) – Northwestern University. 2010.[To] To¨en, Bertrand. Higher and derived stacks: a global overview. Algebraic geometry – Seattle 2005. Part 1, 435-487,Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009.[TV1] To¨en, Bertrand; Vezzosi, Gabriele. Homotopical Algebraic Geometry I: Topos theory. Adv. Math. 193 (2005),no. 2, 257–372.[TV2] To¨en, Bertrand; Vezzosi, Gabriele. Homotopical algebraic geometry. II. Geometric stacks and applications.Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224 pp. Department of Mathematics, Northwestern University, Evanston, IL 60208-2370
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