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Relative Calabi–Yau structures

Published online by Cambridge University Press:  12 February 2019

Christopher Brav
Affiliation:
National Research University Higher School of Economics, Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow, 119048, Russia email c.brav@hse.ru
Tobias Dyckerhoff
Affiliation:
Fachbereich Mathematik, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany email tobias.dyckerhoff@uni-hamburg.de
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Abstract

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

Type
Research Article
Copyright
© The Authors 2019 

1 Introduction

A fundamental insight in algebraic topology is that the geometric notion of orientation can be captured in purely algebraic terms: An orientation of a given compact $n$ -dimensional manifold $X$ corresponds to a fundamental class

(1.1) $$\begin{eqnarray}[X]\in H_{n}(X;\mathbb{Z})\end{eqnarray}$$

characterized by the requirement that, for every point $x\in X$ , it maps to a generator of the local homology group $H_{n}(X,X\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$ . Further, evaluation on a chosen fundamental class $[X]$ yields a map

(1.2) $$\begin{eqnarray}\operatorname{tr}:H^{\ast }(X;\mathbb{Z})\longrightarrow \mathbb{Z}[-n]\end{eqnarray}$$

such that the bilinear pairing $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\mapsto \operatorname{tr}(\unicode[STIX]{x1D6FC}\cup \unicode[STIX]{x1D6FD})$ on $H^{\ast }(X;\mathbb{Z})$ is nondegenerate.

A key observation from noncommutative geometry is that this algebraic perspective on orientations has the following noncommutative variant which we formulate for a differential graded $k$ -algebra $A$ where $k$ is a field: an $n$ -dimensional fundamental class for $A$ is a Hochschild homology class

(1.3) $$\begin{eqnarray}[A]\in \operatorname{HH}_{-n}(A)\end{eqnarray}$$

with the property that the induced map of $A$ -bimodules

$$\begin{eqnarray}A^{!}\longrightarrow A[-n]\end{eqnarray}$$

is a quasi-isomorphism. Here, using the notation $A^{e}=A\otimes _{k}A^{\operatorname{op}}$ , the module

$$\begin{eqnarray}A^{!}=R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A,A^{e})\end{eqnarray}$$

is called the inverse dualizing bimodule. The map $A^{!}\rightarrow A[-n]$ is then obtained by acting on the first factor of the fundamental class $[A]\in \operatorname{HH}(A)=A\otimes _{A^{e}}^{L}A$ . Under the assumption that $A$ is perfect as an $A$ -bimodule, it can further be shown that, for every $A$ -module $M$ whose underlying complex of vector spaces is perfect, the fundamental class $[A]$ induces a $k$ -linear map

(1.4) $$\begin{eqnarray}\operatorname{tr}_{M}:\operatorname{Ext}_{A}^{\ast }(M,M)\longrightarrow k[-n]\end{eqnarray}$$

such that the bilinear pairing $(f,g)\mapsto \operatorname{tr}(f\circ g)$ on $\operatorname{Ext}_{A}^{\ast }(M,M)$ is nondegenerate.

To provide some intuition for these noncommutative fundamental classes, we demonstrate how to interpret classical fundamental classes in noncommutative terms. Let $X$ be a connected $n$ -dimensional manifold and let $A=C_{\ast }(\unicode[STIX]{x1D6FA}X)$ be the differential graded algebra of chains on the space of Moore loops based at a chosen point in $X$ . Then, by [Reference JonesJon87], we have an isomorphism $\operatorname{HH}_{-\ast }(A)\simeq H_{\ast }(LX)$ where $LX$ denotes the free loop space of $X$ . A fundamental class $[X]$ in the classical sense (1.1) induces a noncommutative fundamental class in the sense of (1.3) via pushforward along the inclusion $X$ into $LX$ as the space of constant loops. Further, the trace map (1.2) can be recovered from (1.4) by setting $M$ to be the $A$ -module $k$ . More generally, $A$ -modules $M$ whose underlying complex is perfect can be interpreted as local systems of perfect complexes on $X$ , so that $k$ corresponds to the trivial rank $1$ local system.

The noncommutative perspective has the advantage that it is widely applicable and covers variations of the notion of orientation that arise in other contexts. For example, when $X$ is an algebraic variety, it is natural to ask whether the fundamental class can be represented by an algebraic volume form. This algebraic variant of orientation which, in light of the celebrated differential geometric results of Calabi and Yau, is often referred to as a Calabi–Yau structure, is captured noncommutatively by setting $A$ to be the differential graded algebra of derived endomorphisms of a generator of the derived category of coherent sheaves on $X$ . Similarly, noncommutative fundamental classes describe interesting versions of orientations in symplectic geometry, representation theory, and topological field theory.

When studying noncommutative fundamental classes in the various above contexts, it becomes apparent that, in essentially all examples, they admit a rather subtle additional circular symmetry which manifests itself in the form of a lift of the fundamental class from the Hochschild class (1.3) to a class in negative cyclic homology. In particular, this additional symmetry is necessary to interpret fundamental classes in terms of oriented topological field theories (see, e.g., [Reference LurieLur09b]). Further, to avoid the necessity to choose generators, it is convenient to generalize from differential graded (dg) algebras to dg categories, thus arriving at the following refinement of (1.3).

Definition 1.5. Let $A$ be a smooth $k$ -linear dg category (i.e., $A$ is Morita equivalent to a dg algebra which is perfect as a bimodule). An $n$ -dimensional left Calabi–Yau structure on $A$ consists of a cycle

(1.6) $$\begin{eqnarray}\widetilde{[A]}:k[n]\longrightarrow {\operatorname{CC}_{\bullet }(A)}^{S^{1}}\end{eqnarray}$$

in the negative cyclic complex such that the induced morphism of $A$ -bimodules

(1.7) $$\begin{eqnarray}A^{!}\longrightarrow A[-n]\end{eqnarray}$$

is a quasi-isomorphism where $A^{!}$ denotes the derived $A^{\operatorname{op}}\otimes A$ -linear dual of $A$ .

The smoothness hypothesis implies that $A^{!}$ is the Morita-theoretic left adjoint of the evaluation functor

$$\begin{eqnarray}A^{\operatorname{op}}\otimes A\rightarrow \operatorname{Ch}(k),(x,y)\mapsto A(x,y)\end{eqnarray}$$

which explains our choice of terminology. The idea to formulate Calabi–Yau structures in terms of the quasi-isomorphism (1.7) has appeared in [Reference GinzburgGin06]. The additional $S^{1}$ -equivariance data (1.6) has been proposed by Kontsevich and Vlassopoulos in [Reference Kontsevich and VlassopoulosKV13]. The result that, for a compact oriented manifold $X$ , the dg algebra $C_{\ast }(\unicode[STIX]{x1D6FA}X)$ carries a canonical left Calabi–Yau structure is stated in [Reference LurieLur09b] and proved in [Reference Cohen and GanatraCG15]. The natural refinement of (1.4) to the current context is captured by a different notion of Calabi–Yau structure, introduced in [Reference Kontsevich and SoibelmanKS06].

Definition 1.8. Let $A$ be a (locally) proper $k$ -linear dg category (i.e., all morphism complexes of $A$ are perfect). An $n$ -dimensional right Calabi–Yau structure on $A$ consists of a cocycle

(1.9) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70E}}:{\operatorname{CC}_{\bullet }(A)}_{S^{1}}\longrightarrow k[-n]\end{eqnarray}$$

on the cyclic complex so that the induced morphism of $A$ -bimodules

(1.10) $$\begin{eqnarray}A[n]\longrightarrow A^{\ast }\end{eqnarray}$$

is a quasi-isomorphism where $A^{\ast }$ denotes the $k$ -linear dual of $A$ .

The properness hypothesis implies that $A^{\ast }$ is the Morita-theoretic right adjoint of the evaluation functor $A^{\operatorname{op}}\otimes A\rightarrow \operatorname{Ch}(k)$ explaining our terminology. In this terminology, the fact that a chosen fundamental class (1.3) induces a nondegenerate trace pairing (1.4) is now expressed in the statement that a left Calabi–Yau structure on a smooth dg category $A$ induces a right Calabi–Yau structure on its Morita dual $A^{\vee }$ , i.e., the derived dg category of functors from $A$ to $\operatorname{Perf}_{k}$ (pseudo-perfect $A$ -modules in the terminology of [Reference Toën and VaquiéTV07]).

For example, to relate to the initial example of an $n$ -dimensional oriented manifold $X$ , we introduce a dg category $\unicode[STIX]{x2112}(X)$ , called the linearization of $X$ , which can be informally described as follows: the objects of $\unicode[STIX]{x2112}(X)$ are the points of $X$ , and the mapping complex between two points is the chain complex of the space of paths between the points. The Morita dual of $\unicode[STIX]{x2112}(X)$ can then be identified with the full dg subcategory of $\operatorname{Perf}_{\unicode[STIX]{x2112}(X)}$ consisting of $\infty$ -local systems of complexes of vector spaces on $X$ with perfect fibers. From the point of view of noncommutative geometry, as already strongly advocated in [Reference KontsevichKon09], the dg category $\unicode[STIX]{x2112}(X)$ is much better behaved than its Morita dual $\unicode[STIX]{x2112}(X)^{\vee }$ : the fact that $X$ is homotopy equivalent to a finite CW complex implies that $\unicode[STIX]{x2112}(X)$ is of finite type in the sense of [Reference Toën and VaquiéTV07]. Most importantly for us, as shown in loc. cit, its derived moduli stack of pseudo-perfect modules is locally geometric and of finite presentation so that it has a perfect cotangent complex. This is in stark contrast to the Morita dual $\unicode[STIX]{x2112}(X)^{\vee }$ which is (locally) proper but almost never of finite type so that a reasonable moduli stack in the sense of [Reference Toën and VaquiéTV07] does not exist. It is for these reasons that, in the context of most examples appearing in this work, we are led to consider left Calabi–Yau structures as more fundamental than right Calabi–Yau structures.

Now assume that $X$ is an oriented compact manifold but with possibly nonempty boundary $\unicode[STIX]{x2202}X$ . We obtain a corresponding dg functor

$$\begin{eqnarray}\unicode[STIX]{x2112}(\unicode[STIX]{x2202}X)\longrightarrow \unicode[STIX]{x2112}(X)\end{eqnarray}$$

of linearizations. To capture the orientation of $(X,\unicode[STIX]{x2202}X)$ in terms of this functor, we propose the following relative version of Definition 1.5.

Definition 1.11. An $n$ -dimensional relative left Calabi–Yau structure on a functor $f:A\rightarrow \unicode[STIX]{x212C}$ of smooth $k$ -linear dg categories consists of a cycle

(1.12) $$\begin{eqnarray}\widetilde{[\unicode[STIX]{x212C},A]}:k[n]\longrightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}^{S^{1}}\end{eqnarray}$$

in the relative negative cyclic complex, defined as the cone of the induced map $f_{\ast }:{\operatorname{CC}_{\bullet }(A)}^{S^{1}}\rightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C})}^{S^{1}}$ on absolute negative cyclic complexes, such that all vertical morphisms in the induced diagram of $\unicode[STIX]{x212C}$ -bimodules

(1.13)

are quasi-isomorphisms where the morphism $c$ represents the counit of the derived Morita-adjunction

(1.14) $$\begin{eqnarray}Lf_{!}:D(\operatorname{Mod}_{A})\longleftrightarrow D(\operatorname{Mod}_{\unicode[STIX]{x212C}}):Rf^{\ast }.\end{eqnarray}$$

A relative left Calabi–Yau structure on the zero functor $0\rightarrow \unicode[STIX]{x212C}$ can be identified with an absolute left Calabi–Yau structure on $\unicode[STIX]{x212C}$ . Diagram (1.13) admits a particularly nice interpretation if we assume the dg functor $f$ to be spherical in the sense of [Reference Anno and LogvinenkoAL17] (this is satisfied in many of our examples): in this case, the $\unicode[STIX]{x212C}$ -bimodule $\operatorname{cone}(c)$ represents an autoequivalence of $\operatorname{Mod}_{\unicode[STIX]{x212C}}$ known as a spherical twist of the adjunction. Therefore, in this context, (1.13) amounts to an identification of the inverse dualizing bimodule $\unicode[STIX]{x212C}^{!}$ with a shifted spherical twist of the adjunction (1.14). There are also relative variants of right Calabi–Yau structures (these have already been introduced in [Reference ToënToë14, 5.3]) and, generalizing the absolute case, the two notions are related via Morita duality.

A basic operation in cobordism theory is to glue two manifolds along a common boundary component to produce a new manifold: given manifolds $X$ and $X^{\prime }$ with boundary decompositions $Z\amalg Z^{\prime }=\unicode[STIX]{x2202}X$ and $Z^{\prime }\amalg Z^{\prime \prime }=\unicode[STIX]{x2202}X^{\prime }$ , we have a commutative diagram

where the square is a homotopy pushout square of spaces. Linearizing by applying $\unicode[STIX]{x2112}$ yields the commutative diagram

where the square is a homotopy pushout of dg categories. The fact that composition of cobordisms is compatible with orientations admits the following noncommutative generalization which is the main result of this work.

Theorem 1.1. Let

$$\begin{eqnarray}A\amalg A^{\prime }\longrightarrow \unicode[STIX]{x212C}\end{eqnarray}$$

and

$$\begin{eqnarray}A^{\prime }\amalg A^{\prime \prime }\longrightarrow \unicode[STIX]{x212C}^{\prime }\end{eqnarray}$$

be functors of smooth dg categories equipped with relative left Calabi–Yau structures which are compatible on $A^{\prime }$ . Then the functor

$$\begin{eqnarray}A\amalg A^{\prime \prime }\longrightarrow \unicode[STIX]{x212C}\amalg _{A^{\prime }}\unicode[STIX]{x212C}^{\prime }\end{eqnarray}$$

inherits a canonical relative left Calabi–Yau structure.

As an application of this result, we construct relative left Calabi–Yau structures on topological Fukaya categories of punctured framed Riemann surfaces.

We would like to mention that this work has a sequel [Reference Brav and DyckerhoffBD18] in which we relate left Calabi–Yau structures to derived symplectic geometry in the sense of [Reference Pantev, Toën, Vaquié and VezzosiPTVV13]. We announce the following main result of that work.

Theorem 1.2. Let $k$ be a field of characteristic $0$ .

  1. (1) Let $A$ be a $k$ -linear dg category of finite type. Then an $n$ -dimensional left Calabi–Yau structure on $A$ determines a canonical $(2-n)$ -shifted symplectic form on the derived moduli stack $\unicode[STIX]{x2133}_{A}$ of pseudo-perfect modules.

  2. (2) Let $f:A\rightarrow \unicode[STIX]{x212C}$ be a functor of $k$ -linear dg categories of finite type. Assume that $f$ carries an $(n+1)$ -dimensional left Calabi–Yau structure so that the corresponding negative cyclic class on $A$ determines an $n$ -dimensional left Calabi–Yau structure. Then the induced pullback morphism of derived stacks

    $$\begin{eqnarray}f^{\ast }:\unicode[STIX]{x2133}_{\unicode[STIX]{x212C}}\longrightarrow \unicode[STIX]{x2133}_{A}\end{eqnarray}$$
    carries a canonical Lagrangian structure.

Note that this is a variant of a statement announced in [Reference ToënToë14, 5.3]. However, the result stated in [Reference ToënToë14] uses right Calabi–Yau structures. The use of left Calabi–Yau structures in Theorem 1.2 allows for applications to finite type categories which are not necessarily proper. For example, in the context of topological Fukaya categories, Theorems 7.2 and 1.2 imply the following statement.

Theorem 1.3. Let $(S,M)$ be a stable marked surface with framing on $S\setminus M$ , and let $F(S,M)$ denote its topological Fukaya category. Then pullback along the boundary functor induces a morphism of derived stacks

$$\begin{eqnarray}i^{\ast }:\unicode[STIX]{x2133}_{F(S,M)}\longrightarrow \mathop{\prod }_{\unicode[STIX]{x1D70B}_{0}(\unicode[STIX]{x2202}S\setminus M)}\unicode[STIX]{x2133}_{\text{}\underline{k}}\end{eqnarray}$$

where the right-hand side carries a $2$ -shifted symplectic structure and $i^{\ast }$ has a Lagrangian structure. In particular, if $\unicode[STIX]{x2202}S$ is empty, then $\unicode[STIX]{x2133}_{F(S,M)}$ has a $1$ -shifted symplectic structure.

Here it is crucial to use versions of topological Fukaya categories which arise as global sections of cosheaves of dg categories (as opposed to sheaves) since these are of finite type.

In light of Theorem 1.2 one can interpret the theory of absolute (respectively relative) left Calabi–Yau structures as a noncommutative predual of the geometric theory of shifted symplectic (respectively Lagrangian) structures. For example, Theorem 1.1 is a predual of [Reference CalaqueCal15, Theorem 4.4].

We provide an outline of the contents of this work. In §2, we introduce the technical context of this work: derived Morita theory for dg categories. In §§3 and 4 we give a detailed account of the absolute and relative Calabi–Yau structures sketched above. Section 5 provides examples of Calabi–Yau structures in topology, algebraic geometry and representation theory. In §6 we provide a proof of the main result on the composition of Calabi–Yau cospans. The final section, §7, contains the applications to topological Fukaya categories of surfaces.

We conclude with a discussion of relations to the previously existing literature. The concept of a boundary algebra introduced in [Reference SeidelSei12] as well as the notion of a pre-CY structure introduced in [Reference Kontsevich and VlassopoulosKV13] are very close in spirit to the relative Calabi–Yau structures studied here. In fact, as we were informed by the authors, Kontsevich and Vlassopoulos had contemplated a modified definition very close to ours with results similar to the ones in this work. The appearance of left and right Calabi–Yau structures with applications to mirror symmetry is discussed in [Reference Ganatra, Perutz and SheridanGPS15] where the terminology of ‘smooth’ and ‘proper’ Calabi–Yau structures is used instead. Techniques similar to ours to construct Calabi–Yau structures on Fukaya-type categories appear in the recent work [Reference Shende and TakedaST16].

2 Morita theory of differential graded categories

We introduce some basic ingredients of Morita theory of differential graded categories which will form the technical context for this work.

2.1 Modules over differential graded categories

Let $k$ be a field. We denote by $\operatorname{Ch}(k)$ the category of unbounded cochain complexes of vector spaces over $k$ equipped with its usual monoidal structure. A differential graded (dg) category is a category enriched over $\operatorname{Ch}(k)$ equipped with its usual monoidal structure. We refer the reader to [Reference KellyKel82] for the foundations of enriched category theory and to [Reference ToënToë07] for more details on derived Morita theory. Given dg categories $A$ , $\unicode[STIX]{x212C}$ , there is a dg category

$$\begin{eqnarray}A\otimes \unicode[STIX]{x212C}\end{eqnarray}$$

called the tensor product of $A$ and $\unicode[STIX]{x212C}$ . A dg functor $A\rightarrow \unicode[STIX]{x212C}$ of dg categories is defined to be a $\operatorname{Ch}(k)$ -enriched functor. The collection of functors from $A$ to $\unicode[STIX]{x212C}$ organize into a dg category

$$\begin{eqnarray}\text{}\underline{\operatorname{Fun}}(A,\unicode[STIX]{x212C})\end{eqnarray}$$

which is adjoint to the above tensor product.

Given a dg category $A$ , we introduce the dg category

$$\begin{eqnarray}\operatorname{Mod}_{A}=\text{}\underline{\operatorname{Fun}}(A^{\operatorname{op}},\operatorname{Ch}(k))\end{eqnarray}$$

of right modules over $A$ . We will mostly use right modules, but it is notationally convenient to further introduce the dg category

$$\begin{eqnarray}\operatorname{Mod}^{A}=\text{}\underline{\operatorname{Fun}}(A,\operatorname{Ch}(k))\end{eqnarray}$$

of left modules over $A$ , and, given another dg category $\unicode[STIX]{x212C}$ , the dg category

$$\begin{eqnarray}\operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}=\text{}\underline{\operatorname{Fun}}(A\otimes \unicode[STIX]{x212C}^{\operatorname{op}},\operatorname{Ch}(k))\cong \text{}\underline{\operatorname{Fun}}(A,\operatorname{Mod}_{\unicode[STIX]{x212C}})\end{eqnarray}$$

of $A$ - $\unicode[STIX]{x212C}$ -bimodules. There is a canonical dg functor

$$\begin{eqnarray}A\longrightarrow \operatorname{Mod}_{A},\quad a\mapsto A(-,a)\end{eqnarray}$$

given by the $\operatorname{Ch}(k)$ -enriched Yoneda embedding. We set $A^{e}=A^{\operatorname{op}}\otimes A$ , and call the $A$ - $A$ -bimodule

$$\begin{eqnarray}A:A^{\operatorname{op}}\otimes A\longrightarrow \operatorname{Mod}_{k},\quad (a,a^{\prime })\mapsto A(a^{\prime },a)\end{eqnarray}$$

the diagonal bimodule. Note that, via the equivalence $A^{e}\simeq (A^{e})^{\operatorname{op}}$ , we may identify left and right $A^{e}$ -modules. In what follows, we will leave this distinction implicit and refer to either of these modules as $A$ -bimodules.

The category $\operatorname{Mod}_{A}$ admits a natural cofibrantly generated $\operatorname{Ch}(k)$ -model structure in the sense of [Reference HoveyHov99]: it is obtained from the projective model structure on $\operatorname{Ch}(k)$ by defining weak equivalences and fibrations pointwise.

Hochschild homology

Let $A$ be a dg category. We define the Hochschild complex

$$\begin{eqnarray}\operatorname{CC}_{\bullet }(A)=A\otimes _{A^{e}}^{L}A\end{eqnarray}$$

where $A$ denotes the diagonal $A$ -bimodule. If we use the bar resolution of $A$ as a particular choice of cofibrant replacement, then the right-hand side complex becomes the cyclic bar construction. This complex arises as the realization of a cyclic object in $\operatorname{Ch}(k)$ which equips $\operatorname{CC}_{\bullet }(A)$ with an action of the circle $S^{1}$ (cf. [Reference HoyoisHoy15]). This model further exhibits an explicit functoriality: a functor $f:A\rightarrow \unicode[STIX]{x212C}$ induces an $S^{1}$ -equivariant morphism

$$\begin{eqnarray}\operatorname{CC}_{\bullet }(A)\longrightarrow \operatorname{CC}_{\bullet }(\unicode[STIX]{x212C}).\end{eqnarray}$$

In virtue of the circle action, we obtain the cyclic complex

$$\begin{eqnarray}{\operatorname{CC}_{\bullet }(A)}_{S^{1}}=(\operatorname{CC}_{\bullet }(A)[u^{-1}],b+uB)\end{eqnarray}$$

by passing to homotopy orbits and the negative cyclic complex

$$\begin{eqnarray}{\operatorname{CC}_{\bullet }(A)}^{S^{1}}=(\operatorname{CC}_{\bullet }(A)[[u]],b+uB)\end{eqnarray}$$

by passing to homotopy fixed points. As explained in [Reference HoyoisHoy15], the circle action is captured algebraically in terms of the structure of a mixed complex so that the above orbit and fixed point constructions can be computed by the well-known complexes (cf. [Reference KasselKas87, Reference LodayLod13]).

Given a functor $f:A\rightarrow \unicode[STIX]{x212C}$ of dg categories, we define the relative Hochschild complex $\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)$ as the cofiber (or cone) of the morphism $\operatorname{CC}_{\bullet }(A)\rightarrow \operatorname{CC}_{\bullet }(\unicode[STIX]{x212C})$ . Similarly, we obtain the relative cyclic complex ${\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}_{S^{1}}$ and the relative negative cyclic complex ${\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}^{S^{1}}$ .

Derived $\infty$ -categories

It will be convenient to formulate some of the constructions below in terms of $\infty$ -categories. Given a dg category $A$ , let $\operatorname{Mod}_{A}^{\circ }$ denote the full dg subcategory of $\operatorname{Mod}_{A}$ spanned by the cofibrant objects. We call the dg nerve

$$\begin{eqnarray}D(\operatorname{Mod}_{A})=\operatorname{N}_{\operatorname{dg}}(\operatorname{Mod}_{A}^{\circ })\end{eqnarray}$$

the derived $\infty$ -category of $A$ -modules (cf. [Reference LurieLur11b]).

Morita localization

A dg functor $f:A\rightarrow \unicode[STIX]{x212C}$ is called a quasi-equivalence if the following hold:

  1. (1) the functor $H^{0}(A)\rightarrow H^{0}(\unicode[STIX]{x212C})$ is an equivalence of categories;

  2. (2) for every pair $(a,a^{\prime })$ of objects in $A$ , the map

    $$\begin{eqnarray}A(a,a^{\prime })\rightarrow \unicode[STIX]{x212C}(f(a),f(a^{\prime }))\end{eqnarray}$$
    is a quasi-isomorphism of complexes.

Given a dg category $A$ , we define the dg category $\operatorname{Perf}_{A}$ of perfect $A$ -modules as the full dg subcategory of $\operatorname{Mod}_{A}$ spanned by those cofibrant objects which are compact in $\operatorname{Ho}(\operatorname{Mod}_{A})$ . Here, an object $M$ in $\operatorname{Mod}_{A}$ is called compact, if the functor

$$\begin{eqnarray}\text{}\underline{\operatorname{Hom}}_{A}(M,-):\;\operatorname{Mod}_{A}\longrightarrow \operatorname{Mod}_{k}\end{eqnarray}$$

commutes with filtered homotopy colimits. In fact, to show that $M\in \operatorname{Mod}_{A}$ is compact it is enough to check that the functor $\text{}\underline{\operatorname{Hom}}_{A}(M,-)$ preserves arbitrary direct sums. Furthermore, it is known that the compact objects in $\operatorname{Mod}_{A}$ are precisely the homotopy retracts of finite colimits of representable modules. A dg functor $f:A\rightarrow \unicode[STIX]{x212C}$ induces via enriched left Kan extension a functor

$$\begin{eqnarray}f_{!}:\;\operatorname{Perf}_{A}\longrightarrow \operatorname{Perf}_{\unicode[STIX]{x212C}}\!.\end{eqnarray}$$

We say $f$ is a Morita equivalence if the functor $f_{!}$ is a quasi-equivalence. We are usually interested in dg categories up to quasi-equivalence (respectively Morita equivalence) and will implement this by working with the $\infty$ -categories $\operatorname{L}_{\operatorname{eq}}(Cat_{\operatorname{dg}}(k))$ (respectively $\operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}}(k))$ ) obtained by localizing $Cat_{\operatorname{dg}}(k)$ along the respective collection of morphisms.

2.2 Morita theory

Let $A$ , $\unicode[STIX]{x212C}$ be dg categories and let $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ be a cofibrant bimodule. We denote by

$$\begin{eqnarray}-\!\otimes _{A}M:\operatorname{Mod}_{A}\longrightarrow \operatorname{Mod}_{\unicode[STIX]{x212C}}\end{eqnarray}$$

the $\operatorname{Ch}(k)$ -enriched left Kan extension of $M:A\rightarrow \operatorname{Mod}_{\unicode[STIX]{x212C}}$ along the Yoneda embedding $A\rightarrow \operatorname{Mod}_{A}$ . We further introduce the dg functor $\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(M,-):\operatorname{Mod}_{\unicode[STIX]{x212C}}\rightarrow \operatorname{Mod}_{A}$ given by the composite

$$\begin{eqnarray}\operatorname{Mod}_{\unicode[STIX]{x212C}}\overset{M^{\operatorname{op}}\otimes \mathbf{1}}{\longrightarrow }\text{}\underline{\operatorname{Fun}}(A^{\operatorname{op}},\operatorname{Mod}_{\unicode[STIX]{x212C}}^{\operatorname{op}}\otimes \operatorname{Mod}_{\unicode[STIX]{x212C}})\overset{\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(-,-)}{\longrightarrow }\operatorname{Mod}_{A},\end{eqnarray}$$

obtaining a $\operatorname{Ch}(k)$ -enriched Quillen adjunction

$$\begin{eqnarray}-\!\otimes _{A}M:\operatorname{Mod}_{A}\longleftrightarrow \operatorname{Mod}_{\unicode[STIX]{x212C}}:\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(M,-).\end{eqnarray}$$

Concretely, the dg functor $\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(M,-):\operatorname{Mod}_{\unicode[STIX]{x212C}}\rightarrow \operatorname{Mod}_{A}$ takes a module $N\in \operatorname{Mod}_{\unicode[STIX]{x212C}}$ to the $A$ -module $a\mapsto \operatorname{Hom}_{\unicode[STIX]{x212C}}(M(a),N)$ . We call the bimodule $M^{\vee }:\unicode[STIX]{x212C}\rightarrow \operatorname{Mod}_{A}$ given by the restriction of $\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(M,-)$ along the enriched Yoneda embedding $\unicode[STIX]{x212C}\longrightarrow \operatorname{Mod}_{\unicode[STIX]{x212C}}$ the right dual of  $M$ . By the universal property of its enriched left Kan extension

$$\begin{eqnarray}-\!\otimes _{\unicode[STIX]{x212C}}M^{\vee }:\operatorname{Mod}_{\unicode[STIX]{x212C}}\longrightarrow \operatorname{Mod}_{A},\end{eqnarray}$$

we obtain a canonical natural transformation

$$\begin{eqnarray}\unicode[STIX]{x1D702}:-\!\otimes _{\unicode[STIX]{x212C}}M^{\vee }\longrightarrow \text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(M,-).\end{eqnarray}$$

Definition 2.1. The bimodule $M$ is called right dualizable if, for every cofibrant $\unicode[STIX]{x212C}$ -module $N$ , the morphism $\unicode[STIX]{x1D702}(N)$ in $\operatorname{Mod}_{A}$ is a weak equivalence.

Remark 2.2. A bimodule $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ is right dualizable if and only if, for every $a\in A$ , the right $\unicode[STIX]{x212C}$ -module $M(a)$ is perfect.

Remark 2.3. A right dualizable bimodule $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ induces a $D(\operatorname{Ch}(k))$ -enriched adjunction of derived categories

$$\begin{eqnarray}-\!\otimes _{A}^{L}M:D(\operatorname{Mod}_{A})\longleftrightarrow D(\operatorname{Mod}_{\unicode[STIX]{x212C}}):-\!\otimes _{\unicode[STIX]{x212C}}^{L}M^{\vee }\end{eqnarray}$$

with unit and counit induced via Kan extensions from bimodule morphisms

$$\begin{eqnarray}A\longrightarrow M\otimes _{\unicode[STIX]{x212C}}^{L}M^{\vee }\end{eqnarray}$$

and

$$\begin{eqnarray}M^{\vee }\otimes _{A}^{L}M\longrightarrow \unicode[STIX]{x212C},\end{eqnarray}$$

respectively.

Dually, we may consider $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ as a dg functor

$$\begin{eqnarray}M:\unicode[STIX]{x212C}^{\operatorname{op}}\longrightarrow \operatorname{Mod}^{A}.\end{eqnarray}$$

By the above construction, we obtain a $\operatorname{Ch}(k)$ -enriched Quillen adjunction

$$\begin{eqnarray}M\otimes _{\unicode[STIX]{x212C}}-:\operatorname{Mod}^{\unicode[STIX]{x212C}}\longleftrightarrow \operatorname{Mod}^{A}:\text{}\underline{\operatorname{Hom}}^{A}(M,-)\end{eqnarray}$$

and call the bimodule $\text{}^{\vee }M\in \operatorname{Mod}_{A}^{\unicode[STIX]{x212C}}$ given by restricting $\text{}\underline{\operatorname{Hom}}^{A}(M,-)$ along $A^{\operatorname{op}}\rightarrow \operatorname{Mod}^{A}$ the left dual of $M$ . From the universal property of left Kan extension, we obtain a canonical natural transformation

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D709}:\text{}^{\vee }M\otimes _{A}-\longrightarrow \text{}\underline{\operatorname{Hom}}^{A}(M,-).\end{eqnarray}$$

Definition 2.5. The bimodule $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ is called left dualizable if, for every cofibrant $N\in \operatorname{Mod}^{A}$ , the morphism $\unicode[STIX]{x1D709}(N)$ is a weak equivalence in $\operatorname{Mod}^{\unicode[STIX]{x212C}}$ .

Remark 2.6. A bimodule $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ is left dualizable if and only if, for every $b\in \unicode[STIX]{x212C}$ , the left $A$ -module $M(b)$ is perfect.

Remark 2.7. A left dualizable bimodule $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ induces a $D(\operatorname{Ch}(k))$ -enriched adjunction of derived categories

$$\begin{eqnarray}M\otimes _{\unicode[STIX]{x212C}}^{L}-:D(\operatorname{Mod}^{\unicode[STIX]{x212C}})\longleftrightarrow D(\operatorname{Mod}^{A}):\text{}^{\vee }M\otimes _{A}^{L}-\end{eqnarray}$$

with unit and counit induced via Kan extensions from bimodule morphisms

$$\begin{eqnarray}\unicode[STIX]{x212C}\longrightarrow \text{}^{\vee }M\otimes _{A}^{L}M\end{eqnarray}$$

and

respectively.

Proposition 2.1. Let $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ be a cofibrant bimodule.

  1. (1) Assume that $M$ is right dualizable. Then the cofibrant replacement $Q(M^{\vee })\in \operatorname{Mod}_{A}^{\unicode[STIX]{x212C}}$ of the right dual of $M$ is left dualizable and its left dual is canonically equivalent to $M$ .

  2. (2) Assume that $M$ is left dualizable. Then the cofibrant replacement $Q(\text{}^{\vee }M)\in \operatorname{Mod}_{A}^{\unicode[STIX]{x212C}}$ of the left dual of $M$ is right dualizable and its right dual is canonically equivalent to $M$ .

Proof. We give an argument for statement (1). We observe that the unit and counit morphisms for the adjunction

$$\begin{eqnarray}-\!\otimes _{A}^{L}M:D(\operatorname{Mod}_{A})\longleftrightarrow D(\operatorname{Mod}_{\unicode[STIX]{x212C}}):R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}}(M,-)\end{eqnarray}$$

from Remark 2.3 can be interpreted as unit and counit morphisms for the adjunction

$$\begin{eqnarray}M^{\vee }\otimes _{A}^{L}-:D(\operatorname{Mod}^{A})\longleftrightarrow D(\operatorname{Mod}^{\unicode[STIX]{x212C}}):R\text{}\underline{\operatorname{Hom}}^{\unicode[STIX]{x212C}}(M^{\vee },-)\end{eqnarray}$$

from Remark 2.7. The statements now follow from the uniqueness of right adjoints. ◻

Corollary 2.2. Let $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ be a cofibrant right dualizable bimodule. Then there are $D(\operatorname{Ch}(k))$ -enriched adjunctions

$$\begin{eqnarray}-\!\otimes _{A}^{L}M:D(\operatorname{Mod}_{A})\longleftrightarrow D(\operatorname{Mod}_{\unicode[STIX]{x212C}}):-\!\otimes _{\unicode[STIX]{x212C}}^{L}M^{\vee }\end{eqnarray}$$

and

$$\begin{eqnarray}M^{\vee }\otimes _{A}^{L}-:D(\operatorname{Mod}^{A})\longleftrightarrow D(\operatorname{Mod}^{\unicode[STIX]{x212C}}):M\otimes _{\unicode[STIX]{x212C}}^{L}-.\end{eqnarray}$$

Similarly, let $M\in \operatorname{Mod}_{\unicode[STIX]{x212C}}^{A}$ be a left dualizable bimodule. Then there are $D(\operatorname{Ch}(k))$ -enriched adjunctions

$$\begin{eqnarray}M\otimes _{\unicode[STIX]{x212C}}^{L}-:D(\operatorname{Mod}^{\unicode[STIX]{x212C}})\longleftrightarrow D(\operatorname{Mod}^{A}):\text{}^{\vee }M\otimes _{A}^{L}-\end{eqnarray}$$

and

Example 2.8. The dg category $A$ is called (locally) proper if the diagonal bimodule, considered as an object of $\operatorname{Mod}_{k}^{A^{e}}$ , is right dualizable. Concretely, $A$ is proper if $A(a,a^{\prime })\in \operatorname{Perf}_{k}$ for all $(a,a^{\prime })\in A^{e}$ . We denote the right dual of $A$ by $A^{\ast }$ . Assuming $A$ is proper, we have, by Corollary 2.2, adjunctions

$$\begin{eqnarray}-\!\otimes _{A^{e}}^{L}A:D(\operatorname{Mod}_{A^{e}})\longleftrightarrow D(\operatorname{Mod}_{k}):-\!\otimes _{k}^{L}A^{\ast }\end{eqnarray}$$

and

$$\begin{eqnarray}A^{\ast }\otimes _{A^{e}}^{L}-:D(\operatorname{Mod}^{A^{e}})\longleftrightarrow D(\operatorname{Mod}^{k}):A\otimes _{k}^{L}-.\end{eqnarray}$$

In particular, we obtain canonical equivalences in $D(\operatorname{Ch}(k))$

(2.9) $$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{k}(A\otimes _{A^{e}}^{L}A,k)\overset{\simeq }{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A,A^{\ast })\end{eqnarray}$$

and

$$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{k}(A^{\ast }\otimes _{A^{e}}^{L}A,k)\overset{\simeq }{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A,A)\end{eqnarray}$$

giving descriptions of the $k$ -linear dual of Hochschild homology and of Hochschild cohomology, respectively, in terms of $A^{\ast }$ . In this context, we will also refer to $A^{\ast }$ as the dualizing bimodule.

Example 2.10. Let $A$ be a dg category. The dg category $A$ is called smooth if the diagonal bimodule, considered as an object $A\in \operatorname{Mod}_{k}^{A^{e}}$ , is left dualizable. Concretely, $A$ is smooth if the diagonal bimodule $A$ is perfect as a bimodule. We denote the left dual of $A$ by $A^{!}$ . Assuming $A$ is smooth, we have, by Corollary 2.2, adjunctions

$$\begin{eqnarray}A\otimes _{k}^{L}-:D(\operatorname{Mod}^{k})\longleftrightarrow D(\operatorname{Mod}^{A^{e}}):A^{!}\otimes _{A^{e}}^{L}-\end{eqnarray}$$

and

$$\begin{eqnarray}-\!\otimes _{k}^{L}A^{!}:D(\operatorname{Mod}_{k})\longleftrightarrow D(\operatorname{Mod}_{A^{e}}):-\!\otimes _{A^{e}}^{L}A.\end{eqnarray}$$

In particular, we obtain canonical equivalences in $D(\operatorname{Ch}(k))$

$$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A,A)\overset{\simeq }{\longrightarrow }A^{!}\otimes _{A^{e}}^{L}A\end{eqnarray}$$

and

(2.11) $$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A^{!},A)\overset{\simeq }{\longrightarrow }A\otimes _{A^{e}}^{L}A\end{eqnarray}$$

providing descriptions of Hochschild cohomology and homology, respectively, in terms of $A^{!}$ .We will refer to $A^{!}$ as the inverse dualizing bimodule.

Remark 2.12. Note that the map (2.9) and the inverse of the map (2.11) can be directly constructed without the assumption on $A$ to be proper or smooth, respectively. However, the context given by the adjunctions from which these maps arise via Corollary 2.2 relies on $A$ being proper and smooth, respectively. Since this context is important in what follows, we will consider the maps (2.9) (respectively (2.11)) only if $A$ is proper (respectively smooth).

2.3 Duality

Let $A$ be a smooth dg category. An object $p\in A$ is called locally perfect if, for every $a\in A$ , the mapping complex $A(a,p)$ is perfect. Let $P\subset A$ be a full dg subcategory of $A$ spanned by some collection of locally perfect objects. Consider the bimodule

$$\begin{eqnarray}A_{P}:A^{\operatorname{op}}\otimes P\longrightarrow \operatorname{Mod}_{k},\;(a,p)\mapsto A(a,p)\end{eqnarray}$$

as an object of $\operatorname{Mod}_{A}^{P}$ . Since $P$ consists of locally perfect objects, the bimodule $A_{P}$ is right dualizable and we denote its right dual by $A_{P}^{\ast }\in \operatorname{Mod}_{P}^{A}$ so that we have an adjunction

$$\begin{eqnarray}-\!\otimes _{A^{\operatorname{op}}\otimes P}^{L}A_{P}:D(\operatorname{Mod}_{A^{\operatorname{op}}\otimes P})\longleftrightarrow D(\operatorname{Mod}_{k}):-\otimes _{k}^{L}A_{P}^{\ast }.\end{eqnarray}$$

On the other hand, since $A$ is smooth, we have an adjunction

$$\begin{eqnarray}-\!\otimes _{k}^{L}A^{!}:D(\operatorname{Mod}_{k})\longleftrightarrow D(\operatorname{Mod}_{A^{e}}):-\!\otimes _{A^{e}}^{L}A\end{eqnarray}$$

where $A^{!}\in \operatorname{Mod}_{A}^{A}$ is the inverse dualizing bimodule. We denote by $\text{}_{P}A^{!}$ the restriction of $A^{!}:A\rightarrow \operatorname{Mod}_{A}$ along $P\subset A$ .

Proposition 2.3. Let $A$ be a smooth dg category.

  1. (1) The above bimodules $A_{P}^{\ast }$ and $\text{}_{P}A^{!}$ form an adjunction

  2. (2) Assume in addition that $A$ is proper. Then we may set $P=A$ and the adjunction becomes an equivalence

    $$\begin{eqnarray}-\!\otimes _{A}^{L}A^{!}:D(\operatorname{Mod}_{A})\overset{\simeq }{\longleftrightarrow }D(\operatorname{Mod}_{A}):-\otimes _{A}^{L}A^{\ast }.\end{eqnarray}$$

Proof. To show (1), we may describe as a Morita composite of the functors

$$\begin{eqnarray}P\otimes _{k}A^{!}\in \operatorname{Mod}_{P\otimes A^{\operatorname{op}}\otimes A}^{P}\end{eqnarray}$$

and

$$\begin{eqnarray}\text{}_{P}A\otimes _{k}A\in \operatorname{Mod}_{A}^{P\otimes A^{\operatorname{op}}\otimes A}\end{eqnarray}$$

so that

Passing to right adjoints, we obtain that

proving the claim. Statement (2) follows from a similar calculation. ◻

Example 2.13. Let $A$ be a smooth dg category, let $a\in A$ be any object, and let $p\in P$ be a locally perfect object. Then we have an equivalence

(2.14) $$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{A}(A^{!}(-,p),A(-,a))\simeq R\text{}\underline{\operatorname{Hom}}_{P}(P(-,p),A_{P}^{\ast }(-,a))\simeq R\text{}\underline{\operatorname{Hom}}_{k}(A(a,p),k)\end{eqnarray}$$

in $D(\operatorname{Ch}(k))$ . Assume, in addition, that $A$ is proper. Then, restricting to compact objects, we obtain inverse autoequivalences

$$\begin{eqnarray}-\!\otimes _{A}^{L}A^{!}:\operatorname{Perf}_{A}\longleftrightarrow \operatorname{Perf}_{A}:-\!\otimes _{A}^{L}A^{\ast }\end{eqnarray}$$

so that, for any pair of perfect modules $M,N$ , we have

$$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{A}(M,N\otimes _{A}A^{\ast })\simeq R\text{}\underline{\operatorname{Hom}}_{k}(R\text{}\underline{\operatorname{Hom}}_{A}(N,M),k).\end{eqnarray}$$

In this situation, the autoequivalence $A^{\ast }$ is known as the Serre functor so that $A^{!}$ becomes the inverse of the Serre functor. In a suitable geometric context, this functor can be described in terms of a dualizing complex which explains the terminology (inverse) dualizing bimodule.

3 Absolute Calabi–Yau structures

3.1 Right Calabi–Yau structures

Let $A$ be a proper dg category so that, by (2.9), we have an equivalence

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}:R\text{}\underline{\operatorname{Hom}}_{k}(A\otimes _{A^{e}}^{L}A,k)\overset{\simeq }{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A,A^{\ast }).\end{eqnarray}$$

Definition 3.2. An $n$ -dimensional right Calabi–Yau structure on $A$ consists of a map of complexes

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D714}}:{\operatorname{CC}_{\bullet }(A)}_{S^{1}}\longrightarrow k[-n]\end{eqnarray}$$

such that the corresponding morphism of $A$ -bimodules

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\unicode[STIX]{x1D714}):A[n]\longrightarrow A^{\ast }\end{eqnarray}$$

is a weak equivalence. Here $\unicode[STIX]{x1D714}$ denotes the pullback of $\widetilde{\unicode[STIX]{x1D714}}$ along $\operatorname{CC}_{\bullet }(A)\rightarrow {\operatorname{CC}_{\bullet }(A)}_{S^{1}}$ .

Remark 3.3. A right Calabi–Yau structure identifies the diagonal bimodule $A$ up to shift with its right dual $A^{\ast }$ .

3.2 Left Calabi–Yau structures

Let $A$ be a smooth dg category and consider the equivalence

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}:A\otimes _{A^{e}}^{L}A\overset{\simeq }{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A^{!},A)\end{eqnarray}$$

from (2.11).

Definition 3.5. Let $A$ be a smooth dg category. An $n$ -dimensional left Calabi–Yau structure on $A$ consists of a map of complexes

$$\begin{eqnarray}\widetilde{[A]}:k[n]\longrightarrow {\operatorname{CC}_{\bullet }(A)}^{S^{1}}\end{eqnarray}$$

such that the corresponding morphism of $A$ -bimodules

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}([A]):A^{!}\longrightarrow A[-n]\end{eqnarray}$$

is a weak equivalence. Here $[A]$ denotes the postcomposition of $\widetilde{[A]}$ with ${\operatorname{CC}_{\bullet }(A)}^{S^{1}}\rightarrow \operatorname{CC}_{\bullet }(A)$ .

Remark 3.6. A left Calabi–Yau structure identifies the diagonal bimodule $A$ up to shift with its left dual $A^{!}$ .

Given a dg category $A$ and a full dg subcategory $P\subset A$ spanned by locally perfect objects, we obtain a dg functor

$$\begin{eqnarray}A^{\operatorname{op}}\otimes P\longrightarrow \operatorname{Perf}_{k},\;(a,p)\mapsto A(a,p).\end{eqnarray}$$

Applying $\operatorname{CC}_{\bullet }(-)$ , we obtain an adjoint morphism of complexes

$$\begin{eqnarray}\operatorname{CC}_{\bullet }(A)\simeq \operatorname{CC}_{\bullet }(A^{\operatorname{op}})\longrightarrow R\text{}\underline{\operatorname{Hom}}(\operatorname{CC}_{\bullet }(P),\operatorname{CC}_{\bullet }(\operatorname{Perf}_{k}))\simeq R\text{}\underline{\operatorname{Hom}}_{k}(\operatorname{CC}_{\bullet }(P),k)\end{eqnarray}$$

which is compatible with the circle actions (it can be realized as a map of cyclic complexes), so that upon passing to homotopy fixed points, we obtain a map

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6E9}}:{\operatorname{CC}_{\bullet }(A)}^{S^{1}}\longrightarrow R\text{}\underline{\operatorname{Hom}}_{k}({\operatorname{CC}_{\bullet }(P)}_{S^{1}},k)\end{eqnarray}$$

which is part of the following commutative square.

(3.7)

We now provide a relation between left and right Calabi–Yau structures. Various incarnations of this result have been established in [Reference Ganatra, Perutz and SheridanGPS15] and [Reference Keller and Van den BerghKVdB11].

Theorem 3.1. Let $A$ be a smooth dg category equipped with an $n$ -dimensional left Calabi–Yau structure $\widetilde{[A]}$ . Let $P\subset A$ be a full dg subcategory spanned by a set of locally perfect objects. Then the map $\widetilde{\unicode[STIX]{x1D6E9}}(\widetilde{[A]})$ provides an $n$ -dimensional right Calabi–Yau structure on $P$ .

Proof. Using the equivalences (3.4) and (3.1), we may augment (3.7) by the following square.

The map $\unicode[STIX]{x1D6E9}^{\prime }$ admits the following description. Consider the dg functor $D$ given by the composite

$$\begin{eqnarray}\operatorname{Mod}_{A}^{A}\longrightarrow \operatorname{Mod}_{A}^{P}\overset{M\mapsto M^{\vee }}{\longrightarrow }(\operatorname{Mod}_{P}^{A})^{\operatorname{op}}\longrightarrow (\operatorname{Mod}_{P}^{P})^{\operatorname{op}}\end{eqnarray}$$

where the first and last functors are given by restriction along $P\subset A$ . We obtain an induced $D(\operatorname{Ch}(k))$ -enriched functor

$$\begin{eqnarray}LD:D(\operatorname{Mod}_{A^{e}})\longrightarrow D(\operatorname{Mod}_{P^{e}})^{\operatorname{op}}.\end{eqnarray}$$

Explicitly, this functor associates to an $A$ -bimodule $M$ , the $P$ -bimodule given by

$$\begin{eqnarray}(p,p^{\prime })\mapsto R\text{}\underline{\operatorname{Hom}}_{A}(M(-,p^{\prime }),A(-,p)).\end{eqnarray}$$

Therefore, the functor $LD$ maps the diagonal bimodule $A$ to the diagonal bimodule $P$ , and, by (2.14), the inverse dualizing bimodule $A^{!}$ to the dualizing bimodule $P^{\ast }$ . An explicit calculation shows that the map $\unicode[STIX]{x1D6E9}^{\prime }$ is the map induced by $LD$ on mapping complexes. In particular, $\unicode[STIX]{x1D6E9}^{\prime }$ preserves equivalences: the equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}([A]):A^{!}\overset{\simeq }{\longrightarrow }A[-n]\end{eqnarray}$$

maps to an equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{\prime }(\unicode[STIX]{x1D6F7}([A])):P\overset{\simeq }{\longrightarrow }P^{\ast }[-n]\end{eqnarray}$$

showing that $\widetilde{\unicode[STIX]{x1D6E9}}(\widetilde{[A]})$ is indeed a right Calabi–Yau structure.◻

Remark 3.8. Essentially all examples of right Calabi–Yau structures in this work are induced from left Calabi–Yau structures via the construction of Theorem 3.1. Therefore, it seems that smooth (or even finite type) dg categories equipped with left Calabi–Yau structures should be considered as the fundamental objects.

4 Relative Calabi–Yau structures

4.1 Relative right Calabi–Yau structures

Let $f:A\rightarrow \unicode[STIX]{x212C}$ be a dg functor of proper dg categories. We have an induced morphism

$$\begin{eqnarray}{\operatorname{CC}_{\bullet }(f)}_{S^{1}}:{\operatorname{CC}_{\bullet }(A)}_{S^{1}}\longrightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C})}_{S^{1}}\end{eqnarray}$$

defined explicitly in terms of cyclic bar constructions. We abbreviate $R\text{}\underline{\operatorname{Hom}}_{k}(-,k)$ by $(-)^{\ast }$ . Using (2.11), we obtain a coherent diagram

(4.1)

in $D(\operatorname{Ch}(k))$ . We give an explicit description of the map $\unicode[STIX]{x1D6F9}_{f}$ . Consider the $\operatorname{Ch}(k)$ -enriched Quillen adjunction

(4.2) $$\begin{eqnarray}F_{!}:\operatorname{Mod}_{A^{e}}\longleftrightarrow \operatorname{Mod}_{\unicode[STIX]{x212C}^{e}}:F^{\ast }\end{eqnarray}$$

with $F=f^{\operatorname{op}}\otimes f$ . We introduce the morphism

(4.3) $$\begin{eqnarray}u:A\longrightarrow F^{\ast }\unicode[STIX]{x212C}\end{eqnarray}$$

in $D(\operatorname{Mod}_{A^{e}})$ and its derived left adjoint

(4.4) $$\begin{eqnarray}c:LF_{!}A\longrightarrow \unicode[STIX]{x212C}\end{eqnarray}$$

in $D(\operatorname{Mod}_{\unicode[STIX]{x212C}^{e}})$ .

Remark 4.5. The morphisms $u$ and $c$ represent unit and counit, respectively, of the derived adjunction

$$\begin{eqnarray}Lf_{!}:D(\operatorname{Mod}_{A})\longleftrightarrow D(\operatorname{Mod}_{\unicode[STIX]{x212C}}):Rf^{\ast }.\end{eqnarray}$$

Proposition 4.1. The morphism $\unicode[STIX]{x1D6F9}_{f}$ is given by the composite

$$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}^{e}}(\unicode[STIX]{x212C},\unicode[STIX]{x212C}^{\ast })\overset{F^{\ast }}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}(F^{\ast }\unicode[STIX]{x212C},(F^{\ast }\unicode[STIX]{x212C})^{\ast })\overset{u}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A,A^{\ast })\end{eqnarray}$$

where we implicitly use the canonical identification $F^{\ast }(\unicode[STIX]{x212C}^{\ast })\simeq (F^{\ast }\unicode[STIX]{x212C})^{\ast }$ .

Proof. The proposition follows from an explicit calculation using the bar resolution. ◻

Let $\widetilde{\unicode[STIX]{x1D714}}:{\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}_{S^{1}}\rightarrow k[-n+1]$ be a morphism of complexes. We interpret $\unicode[STIX]{x1D714}$ as a coherent diagram

in $D(\operatorname{Ch}(k))$ . By forming the composite with (4.1), we obtain the coherent diagram

from which we extract the datum of a morphism $\unicode[STIX]{x1D709}:\unicode[STIX]{x212C}[n-1]\rightarrow \unicode[STIX]{x212C}^{\ast }$ together with a chosen zero homotopy of the morphism $\unicode[STIX]{x1D6F9}_{f}(\unicode[STIX]{x1D709}):A[n-1]\rightarrow A^{\ast }$ . By Proposition 4.1, this morphism can be identified with the composite

$$\begin{eqnarray}A[n-1]\overset{u}{\longrightarrow }F^{\ast }\unicode[STIX]{x212C}[n-1]\overset{F^{\ast }\unicode[STIX]{x1D709}}{\longrightarrow }(F^{\ast }\unicode[STIX]{x212C})^{\ast }\overset{c}{\longrightarrow }A^{\ast }\end{eqnarray}$$

so that the chosen zero homotopy induces the dashed arrows which make the diagram

(4.6)

in $D(\operatorname{Mod}_{A^{e}})$ coherent.

Definition 4.7. An $n$ -dimensional right Calabi–Yau structure on $f$ consists of a morphism

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D714}_{(\unicode[STIX]{x212C},A)}}:{\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}_{S^{1}}\longrightarrow k[-n+1]\end{eqnarray}$$

such that all vertical morphisms in the corresponding diagram (4.6) are equivalences in $D(\operatorname{Mod}_{A^{e}})$ .

Example 4.8. Let $A$ be a proper dg category and consider the zero functor $f:A\rightarrow 0$ . Then an $n$ -dimensional right Calabi–Yau structure on $f$ translates to a morphism

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D714}_{A}}:{\operatorname{CC}_{\bullet }(A)}_{S^{1}}\longrightarrow k[-n]\end{eqnarray}$$

such that the vertical maps in

are equivalences. This datum, however, is equivalent to an absolute right Calabi–Yau structure on $A$ .

4.2 Relative left Calabi–Yau structures

Let $f:A\rightarrow \unicode[STIX]{x212C}$ be a functor of smooth dg categories. We have an induced morphism

$$\begin{eqnarray}{\operatorname{CC}_{\bullet }(f)}^{S^{1}}:{\operatorname{CC}_{\bullet }(A)}^{S^{1}}\longrightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C})}^{S^{1}}\end{eqnarray}$$

defined explicitly in terms of cyclic bar constructions. Using (2.11), we obtain a coherent diagram

(4.9)

in $D(\operatorname{Ch}(k))$ . Just like for proper dg categories, we give an explicit description of the map $\unicode[STIX]{x1D6F7}_{f}$ which will now involve the counit morphism

$$\begin{eqnarray}c:LF_{!}A\longrightarrow \unicode[STIX]{x212C}\end{eqnarray}$$

in $D(\operatorname{Mod}_{\unicode[STIX]{x212C}^{e}})$ from (4.4).

Lemma 4.2. Let $M\in \operatorname{Perf}_{A^{e}}$ be a perfect bimodule. Then there is a canonical equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}:LF_{!}(M^{!})\simeq (LF_{!}M)^{!}\end{eqnarray}$$

in $D(\operatorname{Mod}_{\unicode[STIX]{x212C}^{e}})$ .

Proof. We may interpret $M$ as a left dualizable module $M\in \operatorname{Mod}_{k}^{A^{e}}$ . Therefore, using (2.4), we have equivalences

Proposition 4.3. The morphism $\unicode[STIX]{x1D6F7}_{f}$ is given by the composite

$$\begin{eqnarray}R\text{}\underline{\operatorname{Hom}}_{A^{e}}(A^{!},A)\overset{LF_{!}}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{A^{e}}((LF_{!}A)^{!},LF_{!}A)\overset{c}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x212C}^{e}}(\unicode[STIX]{x212C}^{!},\unicode[STIX]{x212C})\end{eqnarray}$$

where we implicitly use the identification $\unicode[STIX]{x1D6FF}:LF_{!}(A^{!})\simeq (LF_{!}A)^{!}$ from Lemma 4.2.

Proof. Follows from an explicit calculation using the bar resolution. ◻

Let $\unicode[STIX]{x1D70E}:k[n]\rightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}^{S^{1}}$ be a negative cyclic cycle. We may interpret $\unicode[STIX]{x1D70E}$ as a coherent diagram

in $D(\operatorname{Ch}(k))$ . By forming the composite with (4.9), we obtain the coherent diagram

from which we extract the datum of a morphism $\unicode[STIX]{x1D709}:A^{!}\rightarrow A[-n+1]$ together with a chosen zero homotopy of the morphism $\unicode[STIX]{x1D6F7}_{f}(\unicode[STIX]{x1D709}):\unicode[STIX]{x212C}^{!}\rightarrow \unicode[STIX]{x212C}[-n+1]$ . By Proposition 4.3, this morphism can be identified with the composite

$$\begin{eqnarray}\unicode[STIX]{x212C}^{!}\overset{c^{!}}{\longrightarrow }(LF_{!}A)^{!}\simeq LF_{!}(A^{!})\overset{LF_{!}(\unicode[STIX]{x1D709})}{\longrightarrow }LF_{!}(A[-n+1])\overset{c}{\longrightarrow }\unicode[STIX]{x212C}[-n+1]\end{eqnarray}$$

so that the chosen zero homotopy induces the dashed arrows which make the diagram

(4.10)

in $D(\operatorname{Mod}_{\unicode[STIX]{x212C}^{e}})$ coherent.

Definition 4.11. An $n$ -dimensional left Calabi–Yau structure on $f$ consists of a morphism

$$\begin{eqnarray}\widetilde{[\unicode[STIX]{x212C},A]}:k[n]\longrightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C},A)}^{S^{1}}\end{eqnarray}$$

such that all vertical morphisms in the corresponding diagram (4.10) are equivalences in $D(\operatorname{Mod}_{\unicode[STIX]{x212C}^{e}})$ .

Example 4.12. Let $\unicode[STIX]{x212C}$ be a smooth dg category. Consider the zero functor $f:0\rightarrow \unicode[STIX]{x212C}$ (up to Morita equivalence we may assume that $\unicode[STIX]{x212C}$ has a zero object). Then an $n$ -dimensional relative left Calabi–Yau structure on $f$ translates to a morphism

$$\begin{eqnarray}\widetilde{[\unicode[STIX]{x212C}]}:k[n]\longrightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x212C})}^{S^{1}}\end{eqnarray}$$

such that the vertical maps in

are equivalences. This datum, however, is equivalent to an absolute left Calabi–Yau structure on  $\unicode[STIX]{x212C}$ .

We conclude this section, by noting that there is a relative variant of Theorem 3.1: a relative left Calabi–Yau structure on a functor $f:A\rightarrow \unicode[STIX]{x212C}$ of smooth dg categories induces a right Calabi–Yau structure on the Morita dual functor $f^{\vee }:\unicode[STIX]{x212C}^{\vee }\rightarrow A^{\vee }$ .

5 Examples

5.1 Topology

The material in this section is inspired by various parts of [Reference LurieLur11a] and generalizes results of [Reference Cohen and GanatraCG15].

5.1.1 Poincaré complexes

We say a topological space $Y$ is of finite type if it is homotopy equivalent to a homotopy retract of a finite CW complex. Let $Y$ be of finite type. We define the functor

(5.1) $$\begin{eqnarray}\operatorname{dg}:Cat_{\infty }\overset{\mathfrak{C}}{\longrightarrow }Cat_{\unicode[STIX]{x1D6E5}}\overset{\operatorname{N}}{\longrightarrow }Cat_{\operatorname{dg}}(k)\end{eqnarray}$$

given as the composite of the left Quillen functor $\mathfrak{C}$ from [Reference LurieLur09a] and the functor $\operatorname{N}$ of applying normalized Moore chains to the mapping spaces. We set

$$\begin{eqnarray}\unicode[STIX]{x2112}(Y):=\operatorname{dg}(\operatorname{Sing}(Y)).\end{eqnarray}$$

Remark 5.2. The objects of $\unicode[STIX]{x2112}(Y)$ can be identified with the points of $Y$ , and the mapping complex between objects $y,y^{\prime }\in Y$ is quasi-isomorphic to the complex $C_{\bullet }(P_{y,y^{\prime }}Y)$ of singular chains on the space $P_{y,y^{\prime }}Y$ of paths in $Y$ from $y$ to $y^{\prime }$ . In particular, if $Y$ is path connected, then the choice of any point $y\in Y$ determines a quasi-equivalence of dg categories

$$\begin{eqnarray}\unicode[STIX]{x2112}(Y)\simeq C_{\bullet }(\unicode[STIX]{x1D6FA}_{y}Y)\end{eqnarray}$$

where the right-hand dg algebra of chains on the based loop space is interpreted as a dg category with one object.

Example 5.3. The choice of any point $x$ on the circle $S^{1}$ provides a quasi-equivalence

$$\begin{eqnarray}\operatorname{dg}(\operatorname{Sing}(S^{1}))\simeq k\mathbb{Z}\end{eqnarray}$$

where $k\mathbb{Z}=k[t,t^{-1}]$ denotes the group algebra of the group $\mathbb{Z}=\unicode[STIX]{x1D70B}_{1}(S^{1},x)$ .

Remark 5.4. The functor $\operatorname{dg}$ is weakly equivalent to the left adjoint of the Quillen adjunction

$$\begin{eqnarray}\operatorname{dg}^{\prime }:Cat_{\infty }\longleftrightarrow Cat_{\operatorname{dg}}(k):\operatorname{N}_{\operatorname{dg}}\end{eqnarray}$$

where $\operatorname{N}_{\operatorname{dg}}$ denotes the differential graded nerve of [Reference LurieLur11b]. From an enriched variant of the adjunction (5.1), we obtain an equivalence of $\infty$ -categories

$$\begin{eqnarray}D(\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)})\simeq \operatorname{Fun}_{\infty }(\operatorname{Sing}(Y),\operatorname{N}_{\operatorname{dg}}(\operatorname{Ch}(k)))\end{eqnarray}$$

so that we are naturally led to interpret $\unicode[STIX]{x2112}(Y)$ -modules as $\infty$ -local systems of complexes of $k$ -vector spaces on $Y$ .

Proposition 5.1. The dg category $\unicode[STIX]{x2112}(Y)$ is of finite type. In particular, it is smooth.

Proof. By construction, the functor $\operatorname{dg}$ commutes with homotopy colimits and takes retracts to retracts. The space $Y$ is of finite type and so expressible as a retract of a finite homotopy colimit of the constant diagram with value $\operatorname{pt}$ . Finite type dg categories are stable under retracts and finite homotopy colimits [Reference Toën and VaquiéTV07] so that it suffices to check the statement of the proposition for $Y=\operatorname{pt}$ , where it is apparent.◻

The constant map $\unicode[STIX]{x1D70B}:Y\rightarrow \operatorname{pt}$ induces a dg module

$$\begin{eqnarray}k_{Y}:\unicode[STIX]{x2112}(Y)\overset{\operatorname{dg}(\unicode[STIX]{x1D70B})}{\longrightarrow }k\subset \operatorname{Mod}_{k}\end{eqnarray}$$

which corresponds to the constant local system on $Y$ with value $k$ . We have a corresponding adjunction

$$\begin{eqnarray}-\!\otimes _{\unicode[STIX]{x2112}(Y)}^{L}k_{Y}:D(\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)})\longleftrightarrow D(\operatorname{Mod}_{k}):R\text{}\underline{\operatorname{Hom}}_{k}(k_{Y},-)\end{eqnarray}$$

and

$$\begin{eqnarray}k_{Y}\otimes _{k}^{L}-:D(\operatorname{Mod}^{k})\longleftrightarrow D(\operatorname{Mod}^{\unicode[STIX]{x2112}(Y)}):R\text{}\underline{\operatorname{Hom}}^{\unicode[STIX]{x2112}(Y)}(k_{Y},-).\end{eqnarray}$$

The functors

$$\begin{eqnarray}\displaystyle C_{\bullet }(Y,-) & := & \displaystyle -\!\otimes _{\unicode[STIX]{x2112}(Y)}^{L}k_{Y},\nonumber\\ \displaystyle C^{\bullet }(Y,-) & := & \displaystyle R\text{}\underline{\operatorname{Hom}}^{\unicode[STIX]{x2112}(Y)}(k_{Y},-)\nonumber\end{eqnarray}$$

define homology, respectively cohomology, of $Y$ with $\infty$ -local systems as coefficients. We denote by

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{Y}\in \operatorname{Mod}_{\unicode[STIX]{x2112}(Y)}\end{eqnarray}$$

the left dual of a cofibrant replacement $Q(k_{Y})$ of $k_{Y}\in \operatorname{Mod}_{k}^{\unicode[STIX]{x2112}(Y)}$ .

Proposition 5.2 (Poincaré duality).

The module $Q(k_{Y})$ is left dualizable so that the canonical map

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{Y}\otimes _{\unicode[STIX]{x2112}(Y)}^{L}-\longrightarrow C^{\bullet }(Y,-)\end{eqnarray}$$

is an equivalence.

Proof. It is immediate that $k_{Y}$ is locally perfect and hence right dualizable. But since, by Proposition 5.1, $\unicode[STIX]{x2112}(Y)$ is smooth, every cofibrant locally perfect $\unicode[STIX]{x2112}(Y)$ -module is perfect [Reference Toën and VaquiéTV07]. Therefore, $Q(k_{Y})$ is left dualizable.◻

Remark 5.6. Evaluating (5.5) at $k_{Y}$ , we obtain an equivalence

$$\begin{eqnarray}C_{\bullet }(Y,\unicode[STIX]{x1D701}_{Y})\simeq C^{\bullet }(Y,k_{Y})\end{eqnarray}$$

which can be understood as a version of Poincaré duality.

Remark 5.7. By definition, the functor $\unicode[STIX]{x1D701}_{Y}$ is given as the restriction of the dg functor $C^{\bullet }(Y,-)$ along $\unicode[STIX]{x2112}(Y)^{\operatorname{op}}\rightarrow \operatorname{Mod}^{\unicode[STIX]{x2112}(Y)}$ . In particular, for a point $y\in Y$ , the complex $\unicode[STIX]{x1D701}_{Y}(y)$ is the cohomology of the $\infty$ -local system on $Y$ which assigns to a point $y^{\prime }$ the complex $C_{\bullet }(P_{y,y^{\prime }}Y)$ (cf. [Reference LurieLur11a]).

We define dg functors

$$\begin{eqnarray}\unicode[STIX]{x2112}(Y)\otimes \unicode[STIX]{x2112}(Y)\overset{\unicode[STIX]{x1D6FB}}{\longrightarrow }\operatorname{N}(\mathfrak{C}(\operatorname{Sing}(Y))\times \mathfrak{C}(\operatorname{Sing}(Y)))\overset{\unicode[STIX]{x1D6E5}}{\longleftarrow }\unicode[STIX]{x2112}(Y)\end{eqnarray}$$

where the functor $\unicode[STIX]{x1D6FB}$ is given by applying the Eilenberg–Zilber construction to mapping complexes, and the functor $\unicode[STIX]{x1D6E5}$ is induced by the diagonal map $Y\rightarrow Y\times Y$ . We obtain a dg functor

$$\begin{eqnarray}-\!\otimes _{Y}-:\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)}\otimes \operatorname{Mod}_{\unicode[STIX]{x2112}(Y)}\overset{\unicode[STIX]{x1D6E5}^{\ast }\unicode[STIX]{x1D6FB}_{!}}{\longrightarrow }\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)}\end{eqnarray}$$

which we call the internal tensor product. We may consider the restriction of this dg functor to $\unicode[STIX]{x2112}(Y)\otimes \unicode[STIX]{x2112}(Y)$ as a module $M_{Y}\in \operatorname{Mod}_{\unicode[STIX]{x2112}(Y)^{\operatorname{op}}\otimes \unicode[STIX]{x2112}(Y)}^{\unicode[STIX]{x2112}(Y)}$ and obtain, via enriched Kan extension, another dg functor

$$\begin{eqnarray}-\!\otimes _{\unicode[STIX]{x2112}(Y)}M_{Y}:\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)}\longrightarrow \operatorname{Mod}_{\unicode[STIX]{x2112}(Y)^{e}}\!.\end{eqnarray}$$

Remark 5.8. If the local system $\unicode[STIX]{x1D701}_{Y}$ is invertible with respect to the tensor product $\otimes _{Y}$ on $D(\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)})$ , then we call $\unicode[STIX]{x1D701}_{Y}$ the $k$ -linear Spivak normal fibration. In this case, the space $Y$ is therefore a topological analogue of a Gorenstein variety.

By Corollary 2.2 and Proposition 5.2, we have an adjunction

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{Y}\otimes _{k}^{L}-:D(\operatorname{Mod}^{k})\longleftrightarrow D(\operatorname{Mod}^{\unicode[STIX]{x2112}(Y)}):k_{Y}\otimes _{\unicode[STIX]{x2112}(Y)}^{L}-\end{eqnarray}$$

which provides us with a canonical equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D713}:C_{\bullet }(Y,k_{Y})\overset{\simeq }{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x2112}(Y)}(\unicode[STIX]{x1D701}_{Y},k_{Y}).\end{eqnarray}$$

Here, we slightly abuse notation and also denote the right $\unicode[STIX]{x2112}(Y)$ -module

$$\begin{eqnarray}\unicode[STIX]{x2112}(Y)^{\operatorname{op}}\overset{\operatorname{dg}(\unicode[STIX]{x1D70B})^{\operatorname{op}}}{\longrightarrow }k^{\operatorname{op}}=k\subset \operatorname{Mod}_{k}\end{eqnarray}$$

by $k_{Y}$ .

Definition 5.9. A class

$$\begin{eqnarray}[Y]\in H^{-n}(C_{\bullet }(Y,k_{Y}))\cong H_{n}(Y,k)\end{eqnarray}$$

is called a fundamental class if the morphism

$$\begin{eqnarray}\unicode[STIX]{x1D713}([Y]):\unicode[STIX]{x1D701}_{Y}\longrightarrow k_{Y}[-n]\end{eqnarray}$$

is an equivalence of $\unicode[STIX]{x2112}(Y)$ -modules. A pair $(Y,[Y])$ of a space equipped with a fundamental class is called a Poincaré complex.

Remark 5.10. Let $(Y,[Y])$ be a Poincaré complex. Combining the equivalence $\unicode[STIX]{x1D713}([Y])$ with (5.5) we obtain, for every $i$ , isomorphisms

$$\begin{eqnarray}H^{i}(Y,k)\cong H_{n-i}(Y,k)\end{eqnarray}$$

recovering classical Poincaré duality.

Example 5.11. A closed topological manifold with chosen orientation provides an example of a Poincaré complex.

We will now explain how a fundamental class for $Y$ gives rise to a left Calabi–Yau structure on the dg category $\unicode[STIX]{x2112}(Y)$ .

Proposition 5.3. Let $Y$ be a topological space of finite type. Then the following hold.

  1. (1) We have canonical equivalences

    $$\begin{eqnarray}\displaystyle k_{Y}\otimes _{\unicode[STIX]{x2112}(Y)}^{L}M_{Y}\simeq \unicode[STIX]{x2112}(Y),\quad \unicode[STIX]{x1D701}_{Y}\otimes _{\unicode[STIX]{x2112}(Y)}^{L}M_{Y}\simeq \unicode[STIX]{x2112}(Y)^{!}. & & \displaystyle \nonumber\end{eqnarray}$$
  2. (2) There is an $S^{1}$ -equivariant equivalence

    $$\begin{eqnarray}j:\operatorname{CC}_{\bullet }(\unicode[STIX]{x2112}(Y))\overset{\simeq }{\longrightarrow }C_{\bullet }(LY,k)\end{eqnarray}$$
    where $LY$ denotes the free loop space of $Y$ with circle action given by loop rotation.
  3. (3) The composite

    $$\begin{eqnarray}\unicode[STIX]{x1D6FC}:C_{\bullet }(Y,k)\simeq R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x2112}(Y)}(\unicode[STIX]{x1D701}_{Y},k_{Y})\overset{-\otimes _{\unicode[STIX]{x2112}(Y)}^{L}M_{Y}}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x2112}(Y)^{e}}(\unicode[STIX]{x2112}(Y)^{!},\unicode[STIX]{x2112}(Y))\overset{j}{\simeq }C_{\bullet }(LY,k)\end{eqnarray}$$
    can be identified with the natural map induced by the inclusion of $Y$ as constant loops in  $LY$ .

Proof. Arguing for each connected component, we may assume $X=BG$ where $G$ is a topological group. The first statement then follows by explicit calculation. Statement (2) is shown in [Reference JonesJon87] and (3) follows by a similar argument.◻

Theorem 5.4 [Reference Cohen and GanatraCG15].

Let $(Y,[Y])$ be a Poincaré complex. Then the dg category $\unicode[STIX]{x2112}(Y)$ is equipped with a canonical left Calabi–Yau structure.

Proof. Since the map $\unicode[STIX]{x1D6FC}$ from Proposition 5.3 is $S^{1}$ -invariant, it induces a canonical map

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:C_{\bullet }(Y,k)\longrightarrow {\operatorname{CC}_{\bullet }(\unicode[STIX]{x2112}(Y))}^{S^{1}}.\end{eqnarray}$$

Define $[\unicode[STIX]{x2112}(Y)]=\unicode[STIX]{x1D6FC}([Y])$ . The corresponding map $\unicode[STIX]{x2112}(Y)^{!}\rightarrow \unicode[STIX]{x2112}(Y)[-n]$ is an equivalence since it is the image of the equivalence $\unicode[STIX]{x1D701}_{Y}\rightarrow k_{Y}[-n]$ under the dg functor $-\!\otimes _{\unicode[STIX]{x2112}(Y)}M_{Y}$ .◻

5.1.2 Poincaré pairs

We provide a generalization of Theorem 5.4 in the relative context. Let $X,Y$ be topological spaces of finite type, and let $\unicode[STIX]{x1D711}:X\rightarrow Y$ a continuous map. Applying $\operatorname{dg}(\operatorname{Sing}(-))$ to $\unicode[STIX]{x1D711}$ , we obtain a dg functor

$$\begin{eqnarray}f:\unicode[STIX]{x2112}(X)\longrightarrow \unicode[STIX]{x2112}(Y).\end{eqnarray}$$

Lemma 5.5. There is a canonical equivalence $Lf_{!}(\text{}^{\vee }k_{X})\simeq \text{}^{\vee }(Lf_{!}k_{Y})$ .

The equivalence $k_{X}\rightarrow f^{\ast }k_{Y}$ has an adjoint map

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}:Lf_{!}k_{X}\longrightarrow k_{Y}\end{eqnarray}$$

with left dual

$$\begin{eqnarray}\text{}^{\vee }\unicode[STIX]{x1D6FE}:\unicode[STIX]{x1D701}_{Y}\longrightarrow Lf_{!}\unicode[STIX]{x1D701}_{X}.\end{eqnarray}$$

Lemma 5.6. The composite

$$\begin{eqnarray}\displaystyle C_{\bullet }(X;k) & \simeq & \displaystyle R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x2112}(X)}(\unicode[STIX]{x1D701}_{X},k_{X})\overset{Lf_{!}}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x2112}(Y)}(Lf_{!}\unicode[STIX]{x1D701}_{X},Lf_{!}k_{X})\overset{\unicode[STIX]{x1D6FE}\circ -\circ \text{}^{\vee }\unicode[STIX]{x1D6FE}}{\longrightarrow }R\text{}\underline{\operatorname{Hom}}_{\unicode[STIX]{x2112}(Y)}(\unicode[STIX]{x1D701}_{Y},k_{Y})\nonumber\\ \displaystyle & \simeq & \displaystyle C_{\bullet }(Y;k)\nonumber\end{eqnarray}$$

can be identified with the map $C_{\bullet }(\unicode[STIX]{x1D711};k)$ .

As a consequence of Lemma 5.6, the choice of a class $[Y,X]\in H_{n}(Y,X;k)$ gives a coherent diagram in $D(\operatorname{Mod}_{\unicode[STIX]{x2112}(Y)})$ of the following form.

(5.12)

Definition 5.13. We call $[Y,X]\in H_{n}(Y,X;k)$ a relative fundamental class if all vertical maps in (5.12) are equivalences. A pair $(X\rightarrow Y,[Y,X])$ consisting of a continuous map of finite type topological spaces and a relative fundamental class is called a Poincaré pair.

Example 5.14. A relative fundamental class for $\emptyset \rightarrow Y$ can be identified with an absolute fundamental class for $Y$ .

Example 5.15. An example of a Poincaré pair is given by a compact oriented manifold $Y$ with boundary $X$ .

Theorem 5.7. Let $(X\rightarrow Y,[Y,X])$ be a Poincaré pair. Then the corresponding dg functor

$$\begin{eqnarray}f:\unicode[STIX]{x2112}(X)\longrightarrow \unicode[STIX]{x2112}(Y)\end{eqnarray}$$

of linearizations carries a canonical relative left Calabi–Yau structure.

Proof. This follows from Proposition 5.3 and Lemma 5.6 by applying the dg functor

5.2 Algebraic geometry

In this subsection, we give examples of (relative) left Calabi–Yau structures coming from anticanonical divisors. We assume all schemes are separated and of finite type over a field $k$ , which for simplicity we take to be of characteristic zero, although the assumption on the characteristic seems to be unnecessary.

5.2.1 Background on ind-coherent sheaves

In this subsection, we briefly review the theory of ind-coherent sheaves, following [Reference Gaitsgory and RozenblyumGR17]. Among other things, the theory of ind-coherent sheaves provides a functorially defined dualizing complex, which we can use to give a geometric computation of Hochschild chains for the dg category of coherent sheaves and a geometric computation for the map on Hochschild chains induced by pushforward along a proper morphism. We shall only need the ‘elementary’ parts of the theory, which can be developed using only basic facts about ind-completion of $\infty$ -categories and the notion of dualizable object in a monoidal $\infty$ -category. If the reader is willing to make smoothness assumptions, then ind-coherent sheaves can be identified with (the dg derived category of) quasi-coherent sheaves, simplifying the formalism.

Given a separated scheme $X$ of finite type over a field $k$ , let $\operatorname{Perf}(X)$ denote the dg category of perfect complexes on $X$ and $\operatorname{QCoh}(X)$ the dg unbounded derived category of quasi-coherent sheaves. It is known that $\operatorname{QCoh}(X)$ is compactly generated and that the compact objects in $\operatorname{QCoh}(X)$ are exactly the perfect complexes $\operatorname{Perf}(X)$ (cf. [Reference Thomason and TrobaughTT90]), so that we have identifications

(5.16) $$\begin{eqnarray}\operatorname{QCoh}(X)\simeq \text{Ind}(\operatorname{Perf}(X))\simeq \operatorname{Mod}_{\operatorname{Perf}(X)}\!.\end{eqnarray}$$

For every morphism $f:X\rightarrow Y$ , we have an adjunction

(5.17) $$\begin{eqnarray}f^{\ast }:\operatorname{QCoh}(Y)\leftrightarrow \operatorname{QCoh}(X):f_{\ast }.\end{eqnarray}$$

Note that since $f^{\ast }$ sends perfects to perfects and hence compacts to compacts, the right adjoint $f_{\ast }$ preserves colimits.

Let $\operatorname{Coh}(X)$ denote the full dg subcategory of $\operatorname{QCoh}(X)$ spanned by objects with bounded coherent cohomology sheaves. By definition, ind-coherent sheaves on $X$ are obtained by ind-completion of $\operatorname{Coh}(X)$ :

(5.18) $$\begin{eqnarray}\operatorname{IndCoh}(X):=\text{Ind}(\operatorname{Coh}(X))\simeq \operatorname{Mod}_{\operatorname{Coh}(X)}\!.\end{eqnarray}$$

For smooth $X$ , we have $\operatorname{Coh}(X)=\operatorname{Perf}(X)$ and so $\operatorname{IndCoh}(X)=\operatorname{QCoh}(X)$ . For singular $X$ , we have proper inclusions $\operatorname{Perf}(X)\subset \operatorname{Coh}(X)\subset \operatorname{QCoh}(X)$ . Ind-completion along the first and second inclusion gives an adjunction

(5.19) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{X}:\operatorname{QCoh}(X)\leftrightarrow \operatorname{IndCoh}(X):\unicode[STIX]{x1D6F9}_{X}\end{eqnarray}$$

in which the left adjoint is fully faithful and hence $\unicode[STIX]{x1D6F9}_{X}$ realizes $\operatorname{QCoh}(X)$ as a colocalization of $\operatorname{IndCoh}(X)$ . There is moreover a natural t-structure on $\operatorname{IndCoh}(X)$ for which $\unicode[STIX]{x1D6F9}_{X}$ is t-exact and, for every $n$ , the restricted functor $\unicode[STIX]{x1D6F9}_{X}:\operatorname{IndCoh}(X)^{{\geqslant}n}\rightarrow \operatorname{QCoh}(X)^{{\geqslant}n}$ is an equivalence. Taking the union over all $n$ , we have $\operatorname{IndCoh}(X)^{+}\simeq \operatorname{QCoh}(X)^{+}$ . The categories $\operatorname{IndCoh}(X)$ and $\operatorname{QCoh}(X)$ therefore differ only in their ‘tails’ at $-\infty$ .

Given a morphism $f:X\rightarrow Y$ , we can restrict the functor $f_{\ast }:\operatorname{QCoh}(X)\rightarrow \operatorname{QCoh}(Y)$ to $\operatorname{Coh}(X)$ , obtaining a functor $f_{\ast }:\operatorname{Coh}(X)\rightarrow \operatorname{QCoh}(X)^{+}\simeq \operatorname{IndCoh}(X)^{+}\subset \operatorname{IndCoh}(X)$ . Passing to ind-completion and slightly abusing notation, we obtain a functor $f_{\ast }:\operatorname{IndCoh}(X)\rightarrow \operatorname{IndCoh}(Y)$ . When $f:X\rightarrow Y$ is proper, we have an adjunction

(5.20) $$\begin{eqnarray}f_{\ast }:\operatorname{IndCoh}(X)\leftrightarrow \operatorname{IndCoh}(Y):f^{!}.\end{eqnarray}$$

Note that since $f$ is proper, $f_{\ast }$ sends compact objects to compact objects ( $f_{\ast }\operatorname{Coh}(X)\subseteq \operatorname{Coh}(Y)$ ), hence the right adjoint $f^{!}$ preserves colimits. Note in particular that since we are working with separated schemes, the diagonal morphism $\unicode[STIX]{x1D6E5}:X\rightarrow X\times X$ is a closed immersion, so that the functor $\unicode[STIX]{x1D6E5}^{!}$ is right adjoint to $\unicode[STIX]{x1D6E5}_{\ast }$ .

Like quasi-coherent sheaves, ind-coherent sheaves enjoy good monoidal properties. In particular, there is a natural equivalence

(5.21) $$\begin{eqnarray}\boxtimes :\operatorname{IndCoh}(X)\otimes \operatorname{IndCoh}(Y)\rightarrow \operatorname{IndCoh}(X\times Y)\end{eqnarray}$$

which, for $F\in \operatorname{Coh}(X)$ , $G\in \operatorname{Coh}(Y)$ , is given by the usual formula

$$\begin{eqnarray}F\boxtimes G=\unicode[STIX]{x1D70B}_{X}^{\ast }F\otimes _{{\mathcal{O}}_{X}}^{L}\unicode[STIX]{x1D70B}_{Y}^{\ast }G\end{eqnarray}$$

while (5.21) is obtained by ind-extension (see [Reference Gaitsgory and RozenblyumGR17, II.2.6.3]).

Using the equivalence 5.21, we construct a pairing

(5.22)

where $p:X\rightarrow \text{pt}=\text{Spec}(k)$ is the structure map and we use the equivalence $\operatorname{IndCoh}(\text{pt})\simeq \operatorname{Mod}_{k}$ . One can show that the pairing in 5.22 is nondegenerate in the symmetric monoidal $\infty$ -category of presentable dg categories with colimit preserving dg functors. The copairing is given by the functor

(5.23)

where $p^{!}:\operatorname{Mod}_{k}\rightarrow \operatorname{IndCoh}(X)$ sends $k$ to the dualizing complex $\unicode[STIX]{x1D714}_{X}$ (see [Reference Gaitsgory and RozenblyumGR17, II.2.4.3], which also uses this duality pairing to give an ‘elementary’ construction of the functor $f^{!}$ for an arbitrary morphism $f:X\rightarrow Y$ ).

From the above discussion, we obtain the following computation of Hochschild chains of $\operatorname{Coh}(X)$ .

Lemma 5.8. $\operatorname{CC}_{\bullet }(\operatorname{Coh}(X))\simeq R\unicode[STIX]{x1D6E4}(X,\unicode[STIX]{x1D6E5}^{!}\unicode[STIX]{x1D6E5}_{\ast }\unicode[STIX]{x1D714}_{X})\simeq R\text{Hom}_{X\times X}(\unicode[STIX]{x1D6E5}_{\ast }{\mathcal{O}}_{X},\unicode[STIX]{x1D6E5}_{\ast }\unicode[STIX]{x1D714}_{X})$ .

Proof. In general, the Hochschild chains of a small dg category $A$ can be computed as the composition of the functor $\operatorname{Mod}_{k}\rightarrow \operatorname{Mod}_{A^{e}}$ sending $k$ to the diagonal bimodule $A$ , and the functor $\operatorname{Mod}_{A^{e}}\rightarrow \operatorname{Mod}_{k}$ given as left Kan extension of the Yoneda pairing $A^{e}\rightarrow \operatorname{Mod}_{k}$ . Note that under the equivalence $\operatorname{Mod}_{A^{\text{op}}}\otimes \operatorname{Mod}_{A}\rightarrow \operatorname{Mod}_{A^{e}}$ , the Yoneda pairing exhibits $\operatorname{Mod}_{A^{\text{op}}}$ as the dual of  $\operatorname{Mod}_{A}$ . Letting $A=\operatorname{Coh}(X)$ and using the duality pairing 5.22, we obtain an identification of  $\operatorname{Mod}_{\operatorname{Coh}(X)^{\text{op}}}$ with $\operatorname{Mod}_{\operatorname{Coh}(X)}\simeq \operatorname{IndCoh}(X)$ . Under this identification, the diagonal bimodule corresponds to $\unicode[STIX]{x1D6E5}_{\ast }\unicode[STIX]{x1D714}_{X}$ , and we obtain the isomorphism $\operatorname{CC}_{\bullet }(\operatorname{Coh}(X))\simeq R\unicode[STIX]{x1D6E4}(X,\unicode[STIX]{x1D6E5}^{!}\unicode[STIX]{x1D6E5}_{\ast }\unicode[STIX]{x1D714}_{X})$ . The isomorphism $R\unicode[STIX]{x1D6E4}(X,\unicode[STIX]{x1D6E5}^{!}\unicode[STIX]{x1D6E5}_{\ast }\unicode[STIX]{x1D714}_{X})\simeq R\text{Hom}_{X\times X}(\unicode[STIX]{x1D6E5}_{\ast }{\mathcal{O}}_{X},\unicode[STIX]{x1D6E5}_{\ast }\unicode[STIX]{x1D714}_{X})$ follows from properness of the diagonal morphism, which gives adjointness between $\unicode[STIX]{x1D6E5}_{\ast }$ and $\unicode[STIX]{x1D6E5}^{!}$ .◻

The next lemma describes the functoriality of Lemma 5.8 for proper morphisms, and is simply a translation of Proposition 4.3.

Lemma 5.9. Let $f:X\rightarrow Y$ be a proper morphism, so that we have a morphism $f_{\ast }:\operatorname{Coh}(X)\rightarrow \operatorname{Coh}(Y)$ and an induced morphism $\operatorname{CC}_{\bullet }(f_{\ast }):\operatorname{CC}_{\bullet }(\operatorname{Coh}(X))\rightarrow \operatorname{CC}_{\bullet }(\operatorname{Coh}(Y))$ . Then under the identifications provided by Lemma 5.8, the induced morphism $\operatorname{CC}_{\bullet }(f_{\ast })$ is given as the composition

where the first arrow is given by pushforward along $f\times f$ and the natural isomorphism $(f\times f)_{\ast }\unicode[STIX]{x1D6E5}_{\ast }\simeq \unicode[STIX]{x1D6E5}_{\ast }f_{\ast }$ and the second arrow is given by pre-composition with $\unicode[STIX]{x1D6E5}_{\ast }$ applied to the natural map ${\mathcal{O}}_{Y}\rightarrow f_{\ast }{\mathcal{O}}_{Y}$ and post-composition with $\unicode[STIX]{x1D6E5}_{\ast }$ applied to the natural counit map $f_{\ast }\unicode[STIX]{x1D714}_{X}\simeq f_{\ast }f^{!}\unicode[STIX]{x1D714}_{Y}\rightarrow \unicode[STIX]{x1D714}_{Y}$ .

In some cases arising in classical algebraic geometry and representation theory, we have vanishing of Hochschild homology above a certain degree. The following lemma describes the consequence of this for negative cyclic homology and provides a means of constructing $S^{1}$ -equivariance data for left Calabi–Yau structures.

Lemma 5.10.

  1. (1) Let $A$ be a small dg category and suppose $HH_{i}(A)=0$ for $i>d$ . Then the natural map $HC_{i}^{-}(A)\rightarrow HH_{i}(A)$ is an isomorphism for $i\geqslant d$ .

  2. (2) Let