Section 2. (0NZ9) · ScalingStacks

2. Preliminaries

In this section, we summarize relevant technical foundations in the theory of $\infty$ -categories and derived algebraic geometry. It is impossible to describe this entire edifice in a few pages, so we will aim to highlight the particular concepts, results and their references that will play a role in what follows.

2.1. $\infty$ -categories

We are interested in studying algebraic operations on categories of a homotopy-theoretic nature such as derived categories of sheaves or spectra. It is well established that the theory of triangulated categories, through which derived categories are usually viewed, is inadequate to handle many basic algebraic and geometric operations. Examples include the absence of a good theory of gluing or of descent, of functor categories, or of generators and relations. The essential problem is that passing to homotopy categories discards essential information (in particular, homotopy coherent structures, homotopy limits and homotopy colimits).

This information can be captured in many alternative ways, the most common of which is the theory of model categories. Model structures keep weakly equivalent objects distinct but retain the extra structure of resolutions which enables the formulation of homotopy coherence. This extra structure can be very useful for calculations but makes some functorial operations difficult. In particular, it can be hard to construct certain derived functors because the given resolutions are inadequate. There are also fundamental difficulties with the consideration of functor categories between model categories. However, much of the essential information encoded in model categories can be captured by the Dwyer-Kan simplicial localization. This construction uses the weak equivalences to construct simplicial sets (or alternatively, topological spaces) of maps between objects, refining the sets of morphisms in the underlying homotopy category (which are recovered by passing to $\pi_{0}$ ).

This intermediate regime between model categories and homotopy categories is encoded by the theory of $(\infty,1)$ -categories, or simply $\infty$ -categories. The notion of $\infty$ -category captures (roughly speaking) the notion of a category whose morphisms form topological spaces and whose compositions and associativity properties are defined up to coherent homotopies. Thus an important distinction between $\infty$ -categories and model categories or homotopy categories is that coherent homotopies are naturally built in to all the definitions. Thus for example all functors are naturally derived and the natural notions of limits and colimits in the $\infty$ -categorical context correspond to homotopy limits and colimits in more traditional formulations.

The theory of $\infty$ -categories has many alternative formulations (as topological categories, Segal categories, quasi-categories, etc; see [Ber] for a comparison between the different versions). We will follow the conventions of [L2], which is based on Joyal’s quasi-categories [Jo]. Namely, an $\infty$ -category is a simplicial set, satisfying a weak version of the Kan condition guaranteeing the fillability of certain horns. The underlying simplicial set plays the role of the set of objects while the fillable horns correspond to sequences of composable morphisms. The book [L2] presents a detailed study of $\infty$ -categories, developing analogues of many of the common notions of category theory (an overview of the $\infty$ -categorical language, including limits and colimits, appears in [L2, Chapter 1.2]).

Among the structures we will depend on are the $\infty$ -category of $\infty$ -categories [L2, 3], adjoint functors [L2, 5.2], and ind-categories and compact objects [L2, 5.3] (see also Section 3.1). Most of the objects we encounter form presentable $\infty$ -categories [L2, 5.5]. Presentable $\infty$ -categories are $\infty$ -categories which are closed under all (small) colimits (as well as limits, [L2, Proposition 5.5.2.4]), and moreover are generated in a weak sense by a small category. In particular, by a result of Simpson [L2, Theorem 5.5.1.1], they are given by suitable localizations of $\infty$ -categories of presheaves on a small $\infty$ -category. Presentable $\infty$ -categories form an $\infty$ -category ${\mathcal{P}r}^{\rm L}$ whose morphisms are continuous functors, that is, functors that preserve all colimits [L2, 5.5.3]. Note that since presentable categories are closed under all (co)products, categories with a finiteness condition (like compact spaces, coherent sheaves, etc.) do not fall under this rubric. A typical example is the $\infty$ -category of spaces.

Algebra (and algebraic geometry, see below) in the $\infty$ -categorical (or derived) setting has been developed in recent years by Toën-Vezzosi [ToVe1, ToVe2] and Lurie [L3, L4, L5]. This has resulted in a very powerful and readily applicable formalism, complete with $\infty$ -analogues of many of the common tools of ordinary category theory. We single out two powerful tools that are crucial for this paper and available in the $\infty$ -context thanks to [L4] (but are not available in a suitable form in the triangulated or model contexts):

$\bullet$

The general theory of descent, as embodied by the $\infty$ -categorical version of the Barr-Beck theorem ([L4, Theorem 3.4.5]).
$\bullet$

The theory of tensor products of and functor categories between presentable $\infty$ -categories (reviewed in Section 2.2).

2.1.1. Enhancing triangulated categories

The $\infty$ -categorical analogue of the additive setting of homological algebra is the setting of stable $\infty$ -categories [L3]. A stable $\infty$ -category can be defined as an $\infty$ -category with a zero-object, closed under finite limits and colimits, and in which pushouts and pullbacks coincide [L3, 2,4]. The result of [L3, 3] is that stable categories are enhanced versions of triangulated categories, in the sense that the homotopy category of a stable $\infty$ -category has the canonical structure of a triangulated category. We will mostly be concerned with $\infty$ -categories that are both presentable and stable, as studied in [L3, 17]. Typical examples are the $\infty$ -categorical enhancements of the derived categories of modules over a ring, quasi-coherent sheaves on a scheme, and the $\infty$ -category of spectra.

Given a triangulated category which is linear over a ring $k$ , we may consider enhancing its structure in three different ways, promoting it to

$\bullet$

a differential graded (dg) category,
$\bullet$

an $A_{\infty}$ -category, or
$\bullet$

a stable $\infty$ -category.

Among the many excellent references for dg and $A_{\infty}$ -categories, we recommend the survey [Ke]. Relative a ring of characteristic zero $k$ , all three formalisms become equivalent: $k$ -linear stable $\infty$ -categories are equivalent to $k$ -linear pre-triangulated dg categories (that is, those whose homotopy category is triangulated). Thus we recommend the reader interested in characteristic zero applications substitute the term “pre-triangulated $k$ -linear dg category” for “stable $\infty$ -category” throughout the present paper. The distinction between $k$ -linear stable $\infty$ -categories, dg- and $A_{\infty}$ -categories becomes important when considering commutative algebra away from characteristic zero, for which purpose we will only consider the former.

2.2. Monoidal $\infty$ -categories

The definitions and results of this paper depend in an essential way on noncommutative and commutative algebra for $\infty$ -categories, as developed by Lurie in [L4] and [L5]. We briefly summarize this theory in this section, giving detailed references for the benefit of the reader.

The definition of a monoidal $\infty$ -category is given in [L4, 1.1]. The homotopy category of a monoidal $\infty$ -category is an ordinary monoidal category. The $\infty$ -categorical notion incorporates not only the naive notion of multiplication and unit on an $\infty$ -category $\mathcal{C}$ but also all of the higher coherences for associativity, which are packaged in the data of a fibration over $\Delta^{\rm op}$ whose fiber over $[n]$ is $\mathcal{C}^{\times n}$ . (Alternatively, it can be captured concretely by a bisimplicial set with compatibilities, or as a monoid object in $\mathcal{C}$ [L4, Remark 1.2.15], that is, a simplicial object in $\mathcal{C}$ mimicking the classifying space of a monoid.) An algebra object in $\mathcal{C}$ can then be defined [L4, 1.1.14] as an appropriate section of this fibration. The $\infty$ -categorical notion of algebra reduces to the more familiar notions of $A_{\infty}$ $k$ -algebra or $A_{\infty}$ -ring spectrum when the ambient monoidal $\infty$ -category is that of differential graded $k$ -modules or that of spectra [L4, 4.3]. In other words, it encodes a multiplication associative up to coherent homotopies. Likewise, (left) modules over an algebra are defined by the same simplicial diagrams as algebras, except with an additional marked vertex at which we place the module [L4, 2.1]. There is also a pairing to $\mathcal{C}$ between left and right modules over an algebra object $A$ in $\mathcal{C}$ , namely, the relative tensor product $\cdot\otimes_{A}\cdot$ defined by the two-sided bar construction [L4, 4.5]. Monoidal $\infty$ -categories, algebra objects in a monoidal category, and module objects over an algebra object themselves form $\infty$ -categories, some of whose properties (in particular behavior of limits and colimits) are worked out in [L4, 1,2]. In particular, limits of algebra objects are calculated on the underlying objects [L4, 1.5] and module categories are stable [L4, Proposition 4.4.3].

The definition of a symmetric monoidal $\infty$ -category is given in [L5, 1], modeled on the Segal machine for infinite loop spaces. Namely, we replace $\Delta^{\rm op}$ in the definition of monoidal $\infty$ -categories by the category of pointed finite sets, thus encoding all the higher compatibilities of commutativity. Likewise, commutative algebra objects are defined as suitable sections of the defining fibration. On the level of homotopy categories, we recover the notion of commutative algebra object in a symmetric monoidal category, but on the level of chain complexes or spectra this notion generalizes the notion of $\mathcal{E}_{\infty}$ -algebra or $\mathcal{E}_{\infty}$ -ring spectrum (as developed in [EKMM, HSS]). An important feature of the $\infty$ -category of commutative algebra objects is that coproducts of commutative algebra objects are calculated by the underlying monoidal structure [L5, Proposition 4.7]. Section [L5, 5] introduces commutative modules over commutative algebra objects $A$ , which are identified with both left and right modules over the underlying algebras. The key feature of the $\infty$ -category of $A$ -modules is that it has a canonical symmetric monoidal structure [L5, Proposition 5.7], extending the relative tensor product of modules. Moreover, commutative algebras for this structure are simply commutative algebras over $A$ [L5, Proposition 5.9].

One of the key developments of [L4] is the $\infty$ -categorical version of tensor products of abelian categories [De]. Namely, in [L4, 4.1] it is shown that the $\infty$ -category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$ -categories has a natural monoidal structure. In this structure, the tensor product $\mathcal{C}\otimes\mathcal{D}$ of presentable $\mathcal{C},\mathcal{D}$ is a recipient of a universal functor from the Cartesian product $\mathcal{C}\times\mathcal{D}$ which is “bilinear” (commutes with colimits in each variable separately). Moreover, [L5, Proposition 6.18] lifts this to a symmetric monoidal structure in which the unit object is the $\infty$ -category of spaces. This structure is in fact closed, in the sense that ${\mathcal{P}r}^{\rm L}$ has an internal hom functor compatible with the tensor structure, see [L2, Remark 5.5.3.9] and [L4, Remark 4.1.6]. The internal hom assigns to presentable $\infty$ -categories $\mathcal{C}$ and $\mathcal{D}$ the $\infty$ -category of colimit-preserving functors $\Fun^{\rm L}(\mathcal{C},\mathcal{D})$ , which is presentable by [L2, Proposition 5.5.3.8]. In Section 5.1 below, we use the monoidal structure on ${\mathcal{P}r}^{\rm L}$ to define an analogue for $\infty$ -categories of the Hochschild cohomology of algebras or topological Hochschild cohomology of ring spectra.

The symmetric monoidal structure on the $\infty$ -category $\mathcal{P}r^{\rm L}$ of presentable $\infty$ -categories restricts to one on the full $\infty$ -subcategory of stable presentable $\infty$ -categories ([L4, 4.2] and [L5, 6.22]). The unit of the restricted monoidal structure is the stable category of spectra. In particular, this induces a symmetric monoidal structure on spectra and exhibits presentable stable categories as tensored over spectra. Thus if $\mathcal{C}$ is a symmetric monoidal stable $\infty$ -category (that is, a stable commutative ring object in stable $\infty$ -categories), we may consider module categories over $\mathcal{C}$ . These modules themselves will form a symmetric monoidal $\infty$ -category under the operation $\otimes_{\mathcal{C}}$ , which is characterized by the two-sided bar construction with respect to $\mathcal{C}$ . In our applications, $\mathcal{C}$ will be the $\infty$ -category $\qc(Y)$ of quasi-coherent sheaves on a derived stack $Y$ , and we will consider the tensor products of module categories of the form $\qc(X)$ for derived stacks $X\to Y$ .

In Section 5.1, we use this general formalism to define the center (or Hochschild cohomology) and universal trace (or Hochschild homology) for algebra objects in any symmetric monoidal $\infty$ -category. The case of spectra recovers topological Hochschild (co)homology, while the case of presentable $\infty$ -categories ${\mathcal{P}r}^{\rm L}$ provides a derived generalization of the Drinfeld center and will be the focus of our applications in Section 5.

In Section 4.1, we discuss basic properties of $\infty$ -categories of modules, and the tensor product of small stable $\infty$ -categories.

2.3. Derived algebraic geometry

Algebraic geometry provides a wealth of examples of symmetric monoidal categories, as well as powerful tools to study such categories. To a scheme or stack, we may assign its category of quasi-coherent sheaves with its commutative multiplication given by tensor product. This construction generalizes the category of $R$ -modules for $R$ a commutative ring (the case of $\operatorname{Spec}R$ ), and the category of representations of an algebraic group $G$ (the case of classifying stacks $BG$ ). Conversely, Tannakian formalism often allows us to reverse this process and assign a stack to a symmetric monoidal category.

Schemes and stacks are also natural sources of symmetric monoidal $\infty$ -categories, which refine the familiar symmetric monoidal derived categories of quasi-coherent sheaves. For example, to any stack in characteristic zero, we can assign the differential graded enhancement of its derived category, constructed by taking the differential graded category of complexes with quasi-coherent cohomology and localizing the quasi-isomorphisms (following [Ke, D, To1]). In general, we can consider the stable symmetric monoidal $\infty$ -category of quasi-coherent sheaves on any stack whose homotopy category is the familiar derived category.

Trying to geometrically describe algebraic operations on $\infty$ -categories of quasi-coherent sheaves quickly takes us from ordinary algebraic geometry to an enhanced version which has been developed over the last few years known as derived algebraic geometry [L1, L3, L4, L5, ToVe1, ToVe2]. In fact, derived algebraic geometry also provides a far greater abundance of examples of stable symmetric monoidal $\infty$ -categories. Thus it is most natural to both ask and answer algebraic questions about stable symmetric monoidal $\infty$ -categories in this context.

Derived algebraic geometry generalizes the world of schemes simultaneously in two directions. Regarding schemes in terms of their functors of points, which are functors from rings to sets (satisfying a sheaf axiom with respect to a Grothendieck topology), we want to replace both the source and the target categories by suitable $\infty$ -categories. First, we may consider functors from rings to spaces (or equivalently, simplicial sets) considered as an $\infty$ -category (with weak homotopy equivalences inverted). If we consider only $1$ -truncated spaces, or equivalently their fundamental groupoids, we recover the theory of stacks. This naturally leads to the introduction of higher stacks where we consider sheaves of (not necessarily $1$ -truncated) spaces on the category of rings. For example, we can take any space and consider it as a higher stack by taking (the sheafification of) the corresponding constant functor on rings.

To pass from higher stacks to derived stacks, we replace the source category of commutative rings by an $\infty$ -category of commutative ring objects in a symmetric monoidal $\infty$ -category. There are at least three natural candidates that are commonly considered:

(1)

connective (that is, homological or non-positively graded) commutative differential graded $k$ -algebras over a commutative ring $k$ of characteristic zero;
(2)

simplicial commutative rings, or simplicial commutative $k$ -algebras over a commutative ring $k$ ;
(3)

connective $\mathcal{E}_{\infty}$ -ring spectra, or connective $\mathcal{E}_{\infty}$ -algebras over the Eilenberg-MacLane spectrum $H(k)$ of a commutative ring $k$ .¹¹ 1 Note that by the results of [L5] reviewed above, an $\mathcal{E}_{\infty}$ -algebra over $H(k)$ is equivalently a commutative algebra object in the symmetric monoidal $\infty$ -category of $H(k)$ -module spectra.

When $k$ is a $\mathbb{Q}$ -algebra, the $\infty$ -categories of connective differential graded $k$ -algebras, simplicial commutative $k$ -algebras, and connective $\mathcal{E}_{\infty}$ -algebras over $H(k)$ are all equivalent. For a general commutative ring $k$ , simplicial commutative rings provide a setting for importing notions of homotopy theory into algebraic geometry over $k$ (for example, for the aim of describing centers of $\infty$ -categories of sheaves on schemes or stacks). Connective $\mathcal{E}_{\infty}$ -ring spectra are more subtle and provide the setting for importing algebro-geometric notions back into stable homotopy theory. General $\mathcal{E}_{\infty}$ -ring spectra provide a radical generalization which is at the heart of stable homotopy theory. (See [Sh] and references therein for the relation of algebra over rings and over the corresponding Eilernberg-MacLane spectra.)

The techniques and results of this paper apply equally in any of the three settings, and we will refer to any of the three as commutative derived rings without further comment.

Roughly speaking, a derived stack is a functor from the $\infty$ -category of commutative derived rings to the $\infty$ -category of topological spaces. It should satisfy a sheaf property with respect to a Grothendieck topology on commutative derived rings. Examples of derived stacks include:

(1)

spaces (constant functors),
(2)

ordinary schemes and stacks,
(3)

the spectrum of a derived commutative ring (the corresponding representable functor),
(4)

objects obtained by various gluings or quotients (colimits) and intersections or fiber products (limits) of the above examples.

We now recall the precise definitions of derived stacks, following [ToVe1, ToVe2, To2].

The opposite of the $\infty$ -category $\mathcal{A}lg_{k}$ of derived commutative $k$ -algebras admits a Grothendieck topology with respect to étale morphisms. For $A,B\in\mathcal{A}lg_{k}$ , a morphism $A\to B$ is étale if the induced morphism on connected component $\pi_{0}(A)\to\pi_{0}(B)$ is étale, and for $i>0$ , the induced map on higher homotopy groups is an isomorphism

A finite family of morphisms $\{f_{i}:A\to B_{i}\}$ is a étale covering if each $f_{i}$ is étale and the induced morphism is surjective

This induces a Grothendieck topology on the (opposite of the) homotopy category of $\mathcal{A}lg_{k}$ .

Now a derived stack $X$ is a covariant functor from $\mathcal{A}lg_{k}$ to the $\infty$ -category $\mathcal{T}op$ of topological spaces which is a sheaf with respect to the étale topology. In particular, for any étale cover $A\to B$ , the induced morphism

is an equivalence, where $B^{*}$ is the standard cosimplicial resolution of $A$ with $i$ th simplex $B^{\otimes_{A}i+1}$ .

We will work exclusively with derived stacks whose diagonal morphism $\Delta:X\to X\times X$ is representable and affine. Given a quasi-compact derived stack $X$ with affine diagonal, we can choose a cover $U\to X$ by an affine derived scheme, and obtain a C̆ech simplicial affine derived scheme $U_{*}\to X$ with $k$ -simplices given by the $k$ -fold fiber product $U\times_{X}\cdots\times_{X}U$ and whose geometric realization is equivalent to $X$ .

2.4. Derived loop spaces

Even if we are interested in studying primarily schemes, the world of derived stacks is a necessary setting in which to calculate homotopically correct quotients, fiber products and mapping spaces. We illustrate this with an important geometric operation on derived stacks: the formation of the derived loop space. This operation is one of our motivations for considering derived stacks in the first place, since the derived loop space of an ordinary scheme or stack is already a nontrivial derived stack. (See [BN1] for a different appearance of derived loop spaces in relation to cyclic homology, ${\mathcal{D}}$ -modules and representation theory.)

The free loop space of a derived stack $X$ is the internal hom $\mathcal{L}X=X^{S^{1}}=\Map(S^{1},X)$ of maps from the constant stack given by the circle $S^{1}$ . As a derived stack, the loop space may be described explicitly as the collection of pairs of points in $X$ with two paths between them, or in other words, as the derived self-intersection of the diagonal

\mathcal{L}X\simeq X\times_{X\times X}X.

Let us illustrate this notion in a few examples.

For $X$ a topological space (constant stack), $\mathcal{L}X$ is of course the free loop space of $X$ (again considered as a constant stack).

For $X=BG$ the classifying space of an algebraic group, $\mathcal{L}X=G/G$ is the adjoint quotient of $G$ , or in other words, the adjoint group for the universal bundle $EG={\rm pt}\to BG$ . Note that this agrees with the underived inertia stack of $BG$ .

For $X$ a smooth variety over a field of characteristic zero, the derived self-intersection of the diagonal can be calculated by a Koszul complex to be the spectrum of the complex of differential forms on $X$ (with zero differential) placed in homological degrees. We thus obtain that the loop space of a smooth variety $X$ can be identified with the relative spectrum of the symmetric algebra of the shifted tangent bundle $T_{X}[-1]$ .

Similarly, for any $n\geq 0$ , we may consider the derived stack of maps $X^{S^{n}}=\Map(S^{n},X)$ from the $n$ -sphere $S^{n}$ into $X$ . Concretely, the presentation of $S^{n}$ by two cells glued along $S^{n-1}$ leads to an iterated description of $X^{S^{n}}$ as the self-intersection

X^{S^{n}}=X\times_{X^{S^{n-1}}}X

along two copies of constant maps over $X^{S^{n-1}}$ .

More generally, for any topological space $\Sigma$ and stack $X$ we can consider the derived mapping stack $X^{\Sigma}=\Map(\Sigma,X)$ . As we will demonstrate, this construction suggests the world of derived algebraic geometry is a natural setting to construct topological $\sigma$ -models.

2.5. $\mathcal{E}_{n}$ -structures

When considering monoidal structures on vector spaces, we have the option of considering associative or commutative multiplications. When considering monoidal structures on categories, we are faced with three levels of increasing commutativity: “plain” monoidal categories carrying an associative multiplication; braided monoidal categories, in which there is a functorial isomorphism exchanging the order of multiplication $A\otimes B\to B\otimes A$ and satisfying the braid relations; and symmetric monoidal categories, in which the square of the braiding is the identity.

In homotopy theory, there is an infinite sequence of types of algebraic structures interpolating between associativity and commutativity, modeled on the increasing commutativity of $n$ -fold iterated loop spaces. These algebraic structures can be encoded by the little $n$ -disk or $\mathcal{E}_{n}$ operad for $n\geq 1$ or $n=\infty$ , which is an operad in the category of topological spaces. Recall that an operad $F$ (in spaces) is a sequence of spaces $F(k)$ , which parametrize $k$ -fold multiplication operations in the algebraic structures we are encoding, together with actions of the symmetric group permuting the entries and equivariant composition maps. The $k$ -th space of the $\mathcal{E}_{n}$ operad parametrizes disjoint collections of $k$ small balls in the $n$ -ball, with the natural “picture-in-picture” composition maps. An $\mathcal{E}_{n}$ -vector space carries operations labelled by components of the $\mathcal{E}_{n}$ operad, and is an associative algebra for $n=1$ and a commutative algebra for $n>1$ . The notion of $\mathcal{E}_{n}$ -category for a (usual discrete) category is sensitive to the fundamental groupoid of the $\mathcal{E}_{n}$ operad, leading to the three different notions of monoidal category ( $n=1$ ), braided monoidal category ( $n=2$ ) and symmetric monoidal category ( $n>2$ , since for $n>2$ the spaces in the $\mathcal{E}_{n}$ operad are simply connected). However, even on the level of graded vector spaces, with operations labelled by the homology of the $\mathcal{E}_{n}$ operad, we obtain different notions for every $n$ , with $n=2$ giving the notion of a Gerstenhaber algebra familiar from the study of Hochschild cohomology.

We have already encountered the notions of algebra object (corresponding to the case $\mathcal{E}_{1}$ , or equivalently $A_{\infty}$ ) and commutative algebra object (corresponding to $\mathcal{E}_{\infty}$ ) in the context of a symmetric monoidal $\infty$ -category, such as dg modules over a ring $R$ , spectra or presentable $\infty$ -categories. In [F1], the general theory of algebras over operads in $\infty$ -categories is developed and applied to algebra and geometry in the $\mathcal{E}_{n}$ case. Roughly speaking, in the $\infty$ -categorical context, we consider the operadic operations and compositions in a homotopy coherent fashion (see Section 5.3 for more details). Thus for example, an $\mathcal{E}_{2}$ -category is a homotopy-theoretic analogue of a braided monoidal category.

The notions of $\mathcal{E}_{n}$ -algebras and categories pervade homotopy theory, but have also become prominent in algebra and topological field theory. We briefly mention two such applications.

Deligne’s Hochschild cohomology conjecture asserts that the Hochschild cochain complex of an associative algebra is an $\mathcal{E}_{2}$ -algebra, lifting the Gerstenhaber algebra structure on Hochschild cohomology. Kontsevich’s conjecture generalizes this to assert that the Hochschild cohomology of an $\mathcal{E}_{n}$ -algebra is an $\mathcal{E}_{n+1}$ -algebra.

The space of states associated to an $n-1$ -sphere by an $n$ -dimensional topological field theory has a natural $\mathcal{E}_{n}$ -structure, given by tree-level field theory operations, independent of where the field theory takes its values (vector spaces, chain complexes, categories, etc.)

0NWA

Remark 2.1. After the completion of this paper, the paper [L5] was revised to include a thorough treatment of $\infty$ -categorical operads and their algebras. Furthermore, the paper [L7] studies in great detail the specific case of the $\mathcal{E}_{n}$ -operads (including a proof of general versions of the Deligne-Kontsevich conjecture). We refer the reader to these preprints for details on these topics.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5