2.1.1 Dualizability of dg categories

Recall that an object $X \in {\mathcal {C}}$ is dualizable in a symmetric monoidal category $\mathcal {C}$ if there is another object $X^\vee$ and evaluation/coevaluation maps $\textrm {ev}_X: X^\vee \otimes X \to 1_{\mathcal {C}}$ and $\textrm {coev}_X: 1_{\mathcal {C}} \to X \otimes X^\vee$ , where $1_{\mathcal {C}}$ denotes the monoidal unit, such that the compositions

\begin{align*} X &\xrightarrow {\textrm {coev}_X \otimes {\textrm {id}}_X} X \otimes X^\vee \otimes X \xrightarrow {{\textrm {id}}_X \otimes \textrm {ev}_X} X, \\ X^\vee &\xrightarrow {{\textrm {id}}_{X^\vee } \otimes \textrm {coev}_X} X^\vee \otimes X \otimes X^\vee \xrightarrow {\textrm {ev}_X \otimes {\textrm {id}}_{X^\vee }} X \end{align*}

are equivalent to the identity functors on $X$ and $X^\vee$ . We now recall the classification of dualizable objects and morphisms in categories of dg categories; see, for example, [Reference Hoyois, Scherotzke and SibillaHSS17, Proposition 4.23].

The statement regarding morphisms is equivalent to the statement that admitting a continuous right adjoint is equivalent to preserving compact objects. Dualizability is more complicated in the category $\textrm {dgcat}$ of small dg categories. A small dg category $\mathcal {A}$ is said to be smooth if it is perfect as an $\mathcal {A}$ -bimodule. More precisely, the Yoneda embedding ${\mathcal {A}} \hookrightarrow \textrm {Ind}({\mathcal {A}})$ corresponds to a bimodule ${\mathcal {A}}_\Delta \in \textrm {Ind}({\mathcal {A}}^{\textrm{op}} \otimes _k {\mathcal {A}})$ , and $\mathcal {A}$ is smooth if ${\mathcal {A}}_\Delta$ is compact. The category $\mathcal {A}$ is said to be proper if for any two objects $a,a'$ the mapping object ${\textrm {Hom}}_{\mathcal {A}}(a,a') \in \textrm {Vect}$ is compact, i.e. is in $\textrm {Perf}$ . We will also call the category saturated if it is both smooth and proper. Let us rephrase these definitions of smooth and proper. Consider the category $\textrm {Ind}({\mathcal {A}})$ as an object of $\textrm {DGCat}$ . There are (co)evaluation morphisms

\begin{align*} \textrm {ev}_{\textrm {Ind}({\mathcal {A}})}: & \textrm {Ind}({\mathcal {A}}^{\textrm{op}}) \otimes \textrm {Ind}({\mathcal {A}}) \to \textrm {Vect}, \\ \textrm {coev}_{\textrm {Ind}({\mathcal {A}})}: & \textrm {Vect} \to \textrm {Ind}({\mathcal {A}}) \otimes \textrm {Ind}({\mathcal {A}}^{\textrm{op}}), \end{align*}

satisfying the usual relations. Then $\mathcal {A}$ is smooth if and only $\textrm {coev}_{\textrm {Ind}({\mathcal {A}})}$ preserves compact objects, and proper if and only if $\textrm {ev}_{\textrm {Ind}({\mathcal {A}})}$ preserves compact objects. From this, one concludes the following proposition.

2.1.2 Pseudo-perfect modules

It will important for us to relate smooth and proper dg categories using the following construction. Let $\mathcal {A}$ be any dg category.

Definition 2.3. The category of pseudo-perfect modules over $\mathcal {A}$ is the object of $\textrm {dgcat}$ given by

\begin{equation*} {\mathcal {A}}^{\textrm{pp}} := \textrm {Fun}^{ex}({\mathcal {A}}^{\textrm{op}}, \textrm {Perf}), \end{equation*}

i.e. it is the dg category whose objects are $k$ -linear exact functors ${\mathcal {A}} \to \textrm {Perf}$ .

We have a natural embedding ${\mathcal {A}}^{\textrm{pp}} \hookrightarrow \textrm {Ind}({\mathcal {A}})$ . The category $\mathcal {A}$ itself also embeds into $\textrm {Ind}({\mathcal {A}})$ by the $k$ -linear Yoneda embedding; this allows one to compare these two categories as subcategories of $\textrm {Ind}({\mathcal {A}})$ . The following result just follows directly from the definitions of smooth and proper.

We have then the following easy corollary for smooth dg categories, which follows from restricting the canonical map ${\mathcal {A}}^{\textrm{op}} \otimes {\mathcal {A}}^{\textrm{pp}} \to \textrm {Perf}$ along the inclusion ${\mathcal {A}}^{\textrm{pp}} \subset {\mathcal {A}}$ .

2.2.1 Dualizability of bimodules

Consider the ‘linear dual’ functor ${\textrm {Hom}}_k(-,k)$ on the dg category $\textrm {Vect}$ . Note that this is an anti-automorphism when restricted to the subcategory $\textrm {Perf}$ . For any $\mathcal {M}$ in ${\mathcal {A}}\text{-}\textrm {Mod}\text{-}{\mathcal {B}}$ , we define its linear dual

\begin{equation*} \mathcal {M}^\vee = {\textrm {Hom}}_k(\mathcal {M}, k),\end{equation*}

which is an object of ${\mathcal {B}} \text{-} \textrm {Mod}\text{-}{\mathcal {A}}$ . Explicitly, as a functor $\mathcal {M}^\vee : {\mathcal {B}}\otimes {\mathcal {A}}^{\textrm{op}} \to \textrm {Vect}$ , it is given by

\begin{equation*} (b,a) \mapsto {\textrm {Hom}}_k(\mathcal {M}(a,b), k). \end{equation*}

For any bimodule $\mathcal {M}$ there is an adjunction

\begin{equation*} - \otimes _{{\mathcal {A}}\otimes {\mathcal {B}}^{\textrm{op}}} \mathcal {M}: {\mathcal {B}} \text{-} \textrm {Mod}\text{-}{\mathcal {A}} \rightleftarrows \textrm {Vect} : {\textrm {Hom}}_k(\mathcal {M},-). \end{equation*}

$\mathcal {M}$ is called linear dualizable (or right dualizable) if the natural transformation $- \otimes _k \mathcal {M}^\vee \to {\textrm {Hom}}_k(\mathcal {M},-)$ is an equivalence of functors. Equivalently, $\mathcal {M}$ is linear dualizable if it always evaluates to a perfect $k$ -complex, i.e. $\mathcal {M}(a,b) \in \textrm {Perf}$ for every $(a,b) \in {\mathcal {A}}\otimes {\mathcal {B}}^{\textrm{op}}$ . In that case, there is an isomorphism of bimodules $\mathcal {M} \overset {\simeq }\to (\mathcal {M}^\vee )^\vee$ , so we also get another adjunction

\begin{equation*} \mathcal {M}^\vee \otimes _{{\mathcal {A}}\otimes {\mathcal {B}}^{\textrm{op}}} -: {\mathcal {A}} \text{-} \textrm {Mod}\text{-}{\mathcal {B}} \rightleftarrows \textrm {Vect}: \mathcal {M} \otimes _k - .\end{equation*}

There is also the bimodule dual of $\mathcal {M}$ defined using the internal Hom of bimodules

\begin{equation*} \mathcal {M}^! = {\textrm {Hom}}_{{\mathcal {A}}\otimes {\mathcal {B}}^{\textrm{op}}}(\mathcal {M}, {\mathcal {A}} \otimes _k {\mathcal {B}}^{\textrm{op}}), \end{equation*}

where we regard ${\mathcal {A}} \otimes _k {\mathcal {B}}^{\textrm{op}}$ as a ${\mathcal {A}}^e \otimes {\mathcal {B}}^e$ -module (i.e. with two actions of $\mathcal {A}$ and two of $\mathcal {B}$ ). This gives an object of ${\mathcal {B}} \text{-} \textrm {Mod}\text{-}{\mathcal {A}}$ , which maps a pair of objects $(b,a) \in {\mathcal {B}} \times {\mathcal {A}}^{\textrm{op}}$ to

\begin{equation*} \mathcal {M}^!(b,a) = {\textrm {Hom}}_{{\mathcal {A}} \text{-} \textrm {Mod}\text{-}{\mathcal {B}}}(\mathcal {M}(-,-'), {\mathcal {A}}(a,-)\otimes _k {\mathcal {B}}(-',b)). \end{equation*}

For any bimodule $\mathcal {M}$ there is an adjunction

\begin{equation*} \mathcal {M} \otimes _k -: \textrm {Vect} \rightleftarrows {\mathcal {A}} \text{-} \textrm {Mod}\text{-}{\mathcal {B}}: {\textrm {Hom}}_{{\mathcal {A}} \text{-} \textrm {Mod}\text{-}{\mathcal {B}}}(\mathcal {M}, -). \end{equation*}

$\mathcal {M}$ is called bimodule dualizable (or left dualizable) if the natural transformation $\mathcal {M}^! \otimes _{{\mathcal {A}}\otimes {\mathcal {B}}^{\textrm{op}}} - \to {\textrm {Hom}}_{{\mathcal {A}} \text{-} \textrm {Mod}\text{-}{\mathcal {B}}}(\mathcal {M}, -)$ is an equivalence. Equivalently $\mathcal {M}$ is bimodule dualizable if it is perfect as a bimodule (i.e. is a compact object in the category of bimodules). In that case we get an isomorphism of bimodules $\mathcal {M} \overset {\simeq }\to (\mathcal {M}^!)^!$ , and we also have another adjunction

\begin{equation*} - \otimes _k \mathcal {M}^!: \textrm {Vect} \rightleftarrows {\mathcal {A}} \text{-} \textrm {Mod}\text{-}{\mathcal {B}}: - \otimes _{{\mathcal {A}}\otimes {\mathcal {B}}^{\textrm{op}}} \mathcal {M}. \end{equation*}

2.2.2 Hochschild homology

Given a pair $({\mathcal {A}},\mathcal {M})$ of a dg category $\mathcal {A}$ and an $\mathcal {A}$ -bimodule $\mathcal {M}$ , the Hochschild complex $\textrm {HH}({\mathcal {A}},\mathcal {M})$ of $\mathcal {A}$ with coefficients in $\mathcal {M}$ is defined as a (derived) tensor product

\begin{equation*} \textrm {HH}({\mathcal {A}},\mathcal {M}) = {\mathcal {A}}_\Delta \otimes _{{\mathcal {A}}^e} \mathcal {M},\end{equation*}

of right and left modules over ${\mathcal {A}}^e = {\mathcal {A}}^{\textrm{op}} \otimes {\mathcal {A}}$ . When $\mathcal {M} = {\mathcal {A}}_\Delta$ is the identity (or diagonal) bimodule itself, we will denote

\begin{equation*} \textrm {HH}({\mathcal {A}}) := \textrm {HH}({\mathcal {A}},{\mathcal {A}}_\Delta ) = {\mathcal {A}}_\Delta \otimes _{{\mathcal {A}}^e} {\mathcal {A}}_\Delta .\end{equation*}

We will also use the notation $\textrm {HH}_n({\mathcal {A}}) = H^n(\textrm {HH}({\mathcal {A}}))$ to denote the Hochschild homology groups.

The tensor product above can be computed by taking any appropriate resolution; in particular, one can take the bar resolution [Reference LodayLod13]. Tensoring over ${\mathcal {A}}^e$ with the diagonal bimodule then gives the reduced bar complex $\overline {C}^{\mathrm {bar}}({\mathcal {A}})$ [Reference LodayLod13, Reference YeungYeu22] representing $\textrm {HH}({\mathcal {A}})$ . This complex carries a canonical homotopy $S^1$ action, whose homotopy orbits $\textrm {HH}({\mathcal {A}})_{S^1}$ and fixed points $\textrm {HH}({\mathcal {A}})^{S^1}$ were classically termed the cyclic and negative cyclic complexes [Reference HoyoisHoy18, Reference KasselKas87], with homology groups denoted by

\begin{equation*} \textrm {HC}_n({\mathcal {A}}) = H^*(\textrm {HH}({\mathcal {A}})_{S^1}), \quad \textrm {HC}^-_n({\mathcal {A}}) = H^*(\textrm {HH}({\mathcal {A}})^{S^1}).\end{equation*}

Explicit representatives for these complexes can be given by using the formalism of mixed complexes [Reference LodayLod13]; we refer the reader to [Reference YeungYeu22, § 2.4] for an explicit application of this formalism to the context of dg categories. This construction is functorial in $\mathcal {A}$ : using the formalism of trace functors [Reference Brav and DyckerhoffBD21, § 4], one sees that taking Hochschild complexes gives a functor $\textrm {HH}(-): \textrm {dgcat} \to \textrm {Vect}$ . Moreover, the $S^1$ -actions are also compatible, so we also have functors $\textrm {HH}(-)^{S^1}$ and $\textrm {HH}(-)_{S^1}$ computing the negative cyclic and cyclic complexes.

2.2.3 Calabi–Yau structures [Reference Kontsevich and SoibelmanKS08a, Reference GinzburgGin07]

Recall that a category $\mathcal {A}$ is said to be proper if all the Hom spaces are perfect as $k$ -complexes, which is equivalent to the identity bimodule ${\mathcal {A}}_\Delta$ being linear dualizable. In this case the dual of Hochschild homology can be computed in terms of the linear dual ${\mathcal {A}}^\vee$ (by adjunction):

\begin{equation*} {\textrm {Hom}}_k(\textrm {HH}({\mathcal {A}}),k) = {\textrm {Hom}}_k({\mathcal {A}}_\Delta \otimes _{{\mathcal {A}}^e} {\mathcal {A}}_\Delta , k) \simeq {\textrm {Hom}}_{{\mathcal {A}}^e}({\mathcal {A}}_\Delta ,{\mathcal {A}}^\vee ).\end{equation*}

Recall also that a category $\mathcal {A}$ is said to be smooth if the diagonal bimodule is compact as a module over ${\mathcal {A}}^e$ , or equivalently bimodule dualizable. In this case the Hochschild homology of $\mathcal {A}$ can be calculated in terms of ${\mathcal {A}}^!$ :

\begin{equation*} \textrm {HH}({\mathcal {A}}) = {\textrm {Hom}}_k(k, {\mathcal {A}}_\Delta \otimes ^L_{{\mathcal {A}}^e} {\mathcal {A}}_\Delta ) \simeq {\textrm {Hom}}_{{\mathcal {A}}^e}({\mathcal {A}}^!,{\mathcal {A}}_\Delta ).\end{equation*}

In the case where $\mathcal {A}$ is both smooth and proper (i.e. dualizable as an object of $\textrm {dgcat}$ ), tensoring over $\mathcal {A}$ with the bimodules ${\mathcal {A}}^\vee$ and ${\mathcal {A}}^!$ gives inverse autoequivalences of the category $\textrm {Perf}\text{-}{\mathcal {A}} = (\textrm {Mod}\text{-}{\mathcal {A}})^c$ of compact $\mathcal {A}$ -modules; these are respectively a Serre functor and its inverse for this category [Reference Brav and DyckerhoffBD19].

More generally, fix an arbitrary $\mathcal {A}$ -bimodule $\mathcal {M}$ . If $\mathcal {A}$ is proper there is an isomorphism

\begin{equation*} {\textrm {Hom}}_k(\textrm {HH}({\mathcal {A}},\mathcal {M}),k) \simeq {\textrm {Hom}}_{{\mathcal {A}}^e}(\mathcal {M},{\mathcal {A}}^\vee ), \end{equation*}

and if $\mathcal {A}$ is smooth there is an isomorphism

\begin{equation*} \textrm {HH}({\mathcal {A}},\mathcal {M}) \simeq {\textrm {Hom}}_{{\mathcal {A}}^e}({\mathcal {A}}^!, \mathcal {M}), \end{equation*}

allowing one to describe Hochschild homology with coefficients in an arbitrary bimodule.

Definition 2.6. A $d$ -dimensional proper (or right) Calabi–Yau structure on a proper dg category $\mathcal {A}$ is a morphism in $\textrm {Vect}$

\begin{equation*} \textrm {HH}({\mathcal {A}})_{S^1} \to k[-d], \end{equation*}

so that the induced map ${\mathcal {A}}_\Delta \to {\mathcal {A}}^\vee [-d]$ is an isomorphism of $\mathcal {A}$ -bimodules. A $d$ -dimensional smooth (or left) Calabi–Yau structure on a smooth dg category $\mathcal {A}$ is a morphism in $\textrm {Vect}$ ,

\begin{equation*} k[d] \to \textrm {HH}({\mathcal {A}})^{S^1}, \end{equation*}

so that the induced map ${\mathcal {A}}^![d] \to {\mathcal {A}}_\Delta$ is an isomorphism of $\mathcal {A}$ -bimodules.

There are also relative versions of the definitions above [Reference ToënToë14, Reference Brav and DyckerhoffBD19]. A functor $f:{\mathcal {A}}\to {\mathcal {B}}$ induces a map of Hochschild complexes $\textrm {HH}(f): \textrm {HH}({\mathcal {A}}) \to \textrm {HH}({\mathcal {B}})$ , compatible with the $S^1$ action. We define the relative Hochschild complex

\begin{equation*} \textrm {HH}({\mathcal {B}}|{\mathcal {A}}) := {\textrm {Cone}}(\textrm {HH}({\mathcal {A}}) \to \textrm {HH}({\mathcal {B}})), \end{equation*}

and denote similarly by $\textrm {HH}({\mathcal {B}}|{\mathcal {A}})^{S^1}$ and $\textrm {HH}({\mathcal {B}}|{\mathcal {A}})_{S^1}$ the relative negative cyclic and cyclic complexes.

Definition 2.7 [Reference Brav and DyckerhoffBD19]. A $d$ -dimensional proper (or right) relative Calabi–Yau structure on a functor $f:{\mathcal {A}}\to {\mathcal {B}}$ between proper dg categories is a dual class in cyclic homology

\begin{equation*} \phi \in {\textrm {Hom}}_k(\textrm {HH}({\mathcal {B}}|{\mathcal {A}})_{S^1}, k[-d+1]), \end{equation*}

satisfying a nondegeneracy condition. A $d$ -dimensional smooth (or left) relative Calabi–Yau structure on a functor $f:{\mathcal {A}}\to {\mathcal {B}}$ between smooth dg categories is a negative cyclic class

\begin{equation*} \theta \in {\textrm {Hom}}_k(k[d], \textrm {HH}({\mathcal {B}}|{\mathcal {A}})^{S^1}), \end{equation*}

satisfying a nondegeneracy condition.

We refer to [Reference Brav and DyckerhoffBD19, § 4] for the more detailed description of these definitions.

2.3.1 The sheaf of morphisms

We will use the following well-known fact about sheaves of categories: given a sheaf of categories $\mathcal {F}$ on some space $X$ and two objects $x,y \in \mathcal {F}(X)$ , the morphism spaces between the images of $x$ and $y$ form a sheaf of morphisms, which associates $U \mapsto \mathcal {F}(U)(\rho x, \rho y)$ , where $\rho :\mathcal {F}(X) \to \mathcal {F}(U)$ is the restriction map of $\mathcal {F}$ . In the main text of this paper, we will assume that this fact holds equally for sheaves $\mathcal {F}$ valued in the $\infty$ -category $\textrm {dgcat}$ , giving sheaves of morphisms valued in $\textrm {Vect}$ , and that compositions in $\textrm {Vect}$ between the morphism spaces give compositions maps at the level of sheaves of morphisms. More precisely, we make the following assumption.

We could not find in the literature precise statements of the proposition above for the specific setting that we are working in, namely, for diagrams valued in the $\infty$ -category of small stable $\infty$ -categories tensored over some rigid symmetric monoidal $\mathcal {E}$ (in our case, ${\mathcal {E}} = \textrm {Perf}_k$ ). For completeness, we include in Appendix A a proof of this fact using a different, albeit equivalent, model for these categories.

We extend the notation $\mathcal {F}|^{x,y}_U$ to allow for objects of sections on other opens. Given $x \in \mathcal {F}(U'), y \in \mathcal {F}(U'')$ , we write

\begin{equation*} \mathcal {F}|^{x,y}_U = \left\{\begin{array}{l@{\quad}l} \mathcal {F}|^{\rho x, \rho y}_U & \text {if} \, U \subset U' \cap U'', \\ 0 & \text {otherwise}. \end{array}\right.\end{equation*}

3.1.1 Sheafified trace map

Hochschild homology gives a sort of universal trace on morphism spaces of a category, and the same is true for the sheafified version we defined above. For any $U \in \mathcal {U}(X)$ and $x,y$ in the object set $S_U$ , by including into the $(m,n)=(0,0)$ summand of $\textrm {HH}(\mathcal {F})_{(m,n)}$ , we have a morphism of $\textrm {Vect}$ -valued presheaves $\mathcal {F}|_U^{x,y} \otimes \mathcal {F}|_U^{y,x} \to \textrm {HH}(\mathcal {F})$ .

3.2.1 Relation to proper Calabi–Yau structures

We now discuss the relation between our definition of orientations and previously existing notions of Calabi–Yau structures on proper dg categories.

Proof. The data of $\mathcal {F}$ is given just by the morphism $f$ between two proper dg categories $\mathcal {F}([0,1))$ and $\mathcal {F}((0,1))$ . Note that in this case, exactly as happened for the point, there is no difference between $\textrm {HH}(\mathcal {F})$ and $\mathcal {H}\mathcal {H}(\mathcal {F})$ , since $\textrm {HH}(\mathcal {F})$ is already a sheaf. We now calculate the sections of Verdier dualizing sheaf $\omega _X$ to be

\begin{equation*} \omega _X([0,1)) = 0, \quad \omega _X((0,1)) = k[1]. \end{equation*}

Therefore, the data of a morphism $\textrm {HH}(\mathcal {F})_{S^1} \to \omega _X[-d]$ is equivalent to the data of a map $\textrm {HH}(\mathrm {fib}) \to k[-d]$ , where $\textrm {HH}(\mathrm {fib})$ denotes the fibre of the induced morphism $\textrm {HH}(\mathcal {F}([0,1))) \to \textrm {HH}(\mathcal {F}((0,1)))$ . It remains to check that the nondegeneracy condition coincides with that given in [Reference Brav and DyckerhoffBD19, Definition 4.7]; this is the condition that a certain map of distinguished triangles of $\mathcal {F}(X)$ -bimodules be an isomorphism. But isomorphisms of bimodules can be checked by evaluation at each pair $(x,y)$ of representatives for isomorphism classes of objects of $\mathcal {F}(X)$ ; we can pick these objects to be in our chosen object set $S_U$ . The restriction maps from $X=[0,1)$ to $(0,1)$ combined with the isomorphism of sheaves $\mathcal {F}|_X^{x,y} \to \mathcal {H}\kern -1.2pt \mathit {om}(\mathcal {F}|_X^{y,x},\omega _X[-d])$ gives the desired isomorphism of distinguished triangles.

3.2.2 Pushforward

We now recall and prove the pushforward result stated in Theorem 1.3. Consider $f: X \to Y$ a proper and constructible map from a csc space $(X,\mathcal {F})$ . There is a sheaf of proper dg categories $f_*\mathcal {F}$ making $Y$ a csc space. Choosing as object sets for the sheaf $f_*\mathcal {F}$ spanning sets containing the images of the spanning sets $S_{f^{-1}(U)}$ , we get a map of presheaves $\textrm {HH}(f_*\mathcal {F}) \to f_*\textrm {HH}(\mathcal {F})$ , and by the universal property of sheafification there is a canonical morphism of sheaves $\mathcal {H} \mathcal {H}(f_* \mathcal {F}) \to f_* \mathcal {H} \mathcal {H} (\mathcal {F})$ lifting it, and similarly for the corresponding cyclic homology sheaves. Since $f$ is proper (so $f_* = f_!$ ), we consider the composition

\begin{equation*} \mathcal {H} \mathcal {H}(f_* \mathcal {F})_{S^1} \to f_* \mathcal {H} \mathcal {H} (\mathcal {F})_{S^1} \xrightarrow {\Omega } f_* \omega _X \to \omega _Y, \end{equation*}

which we denote by $f_*\Omega$ .

Proof. For any open $U \in \mathcal {U}(Y)$ and objects $x,y \in S_U$ , consider the morphism

\begin{equation*} \mathcal {F}|_{f^{-1}(U)}^{x,y} \to \mathbb {D}' (\mathcal {F}|_{f^{-1}(U)}^{x,y}) [-d], \end{equation*}

determined by the orientation $\Omega$ ; this is an isomorphism by assumption. We apply the pushforward $f_*$ to obtain a morphism

\begin{equation*} \mathcal {F}|_{f^{-1}(U)}^{x,y} \to f_* \mathbb {D}' (\mathcal {F}|_{f^{-1}(U)}^{x,y}) [-d] \simeq \mathbb {D}' (f_*\mathcal {F}|_{f^{-1}(U)}^{x,y}) [-d], \end{equation*}

where we used the fact that there is a canonical equivalence $f_! \simeq f_*$ since $f$ was proper. By construction, this is equivalent to the morphism determined by the pushforward orientation $f_*\Omega$ .

As stated in the introduction, we can combine Propositions 3.8 and 3.9 in the case where the space is of the form $X = X_0 \cup _{\partial X_0} (\partial X_0 \times [0,1))$ , with $X_0$ compact, with $\mathcal {F}$ stratified with respect to some stratification restricting to a product of a stratification on $\partial X_0$ and the stratification we considered above on $[0,1)$ . Then an orientation on $(X,\mathcal {F})$ gives a relative Calabi–Yau structure on the restriction map $\mathcal {F}(X) \to \mathcal {F}(\partial X_0 \times (0,1))$ , as stated in Corollary 1.5 in the introduction.

We can refine that result and rephrase the nondegeneracy condition entirely in terms of relative Calabi–Yau structures on a cover. Consider the cover $X = \bigcup _\sigma U_\sigma$ , indexed over the strata of $X$ , with $U_\sigma = \textrm {Star}(\sigma )$ . For each $\sigma$ , we then choose a small compact retract $K_\sigma$ of $U_\sigma$ , i.e. a compact subset $K$ of $U$ to which $U$ has a stratification-respecting retraction.

In other words, combining the two propositions above gives a way to globalize relative Calabi–Yau structure, as we explained in the introduction.

4.1.1 K-sheaves

Let $X$ be a locally compact Hausdorff space. We write ${\mathcal {K}}(X)$ for the set of all compact subsets of $X$ ; we view it as a partially ordered set with respect to inclusion. Let $\mathcal {A}$ be a stable $\infty$ -category which admits small limits and colimits.

Analogously to the usual sheaves, we write ${\textrm {Cosh}}_{\mathcal {K}}(X,{\mathcal {A}}) = \textrm {Sh}_{\mathcal {K}}(X,{\mathcal {A}}^{\textrm{op}})^{\textrm{op}}$ . One can go back and forth between ordinary (co)sheaves and $\mathcal {K}$ -(co)sheaves.

Following [Reference VolpeVol23a], we will also make use of an intermediate category between the $\mathcal {K}$ -sheaves and $\mathcal {K}$ -presheaves. We denote by $\textrm {Sh}({\mathcal {K}}(X),{\mathcal {A}})$ the full subcategory of $\textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},{\mathcal {A}})$ spanned by the objects satisfying conditions (1) and (2) of Definition 4.1.

Proposition 4.3 [Reference VolpeVol23a, Lemma 5.13]. The inclusion $\textrm {Sh}_{\mathcal {K}}(X,{\mathcal {A}}) \hookrightarrow \textrm {Sh}({\mathcal {K}}(X),{\mathcal {A}})$ has a right-inverse

\begin{equation*} \varphi : \textrm {Sh}({\mathcal {K}}(X),{\mathcal {A}}) \to \textrm {Sh}_{\mathcal {K}}(X,{\mathcal {A}}), \end{equation*}

which preserves filtered colimits and gives an equivalence $\varphi \mathcal {F}(K) \simeq \mathrm{colim}_{K \Subset K'} \mathcal {F}(K')$ for every $\mathcal {F} \in \textrm {Sh}({\mathcal {K}}(X),{\mathcal {A}})$ and $K \in {\mathcal {K}}(X)$ .

4.1.2 Verdier duality functor

Let us now describe the construction of the Verdier duality functor for cosheaves valued in $\mathcal {A}$ . Let $\mathcal {G}$ be an $\mathcal {A}$ -valued precosheaf on $X$ , that is, an object $\mathcal {G} \in \textrm {Fun}(\mathcal {U}(X)^{\textrm{op}},{\mathcal {A}})$ . Given a closed subset $K \subset X$ , the cosections cosupported in $K$ are by definition $\mathcal {G}_K(X) := \mathrm {cofib}(\mathcal {G}(X \setminus K) \to \mathcal {G}(X))$ . $\mathcal {G}_K$ is functorial in $K$ and $\mathcal {G}$ , as it is obtained from the following construction. Consider the map ${\mathcal {K}}(X)^{\textrm{op}} \to \textrm {Fun}(\Delta ^1,\mathcal {U}(X))$ given by $K \mapsto (X \setminus K \to X)$ , and take the composition

\begin{equation*} {\mathcal {K}}(X)^{\textrm{op}} \times \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}(\Delta ^1,\mathcal {U}(X)) \times \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}(\Delta ^1,{\mathcal {A}}) \overset {\mathrm {fib}}{\to } {\mathcal {A}}. \end{equation*}

We will denote by $D_{X,{\mathcal {A}}}: \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},{\mathcal {A}})$ the adjoint $\infty$ -functor; by construction we have an equivalence $D_{X,{\mathcal {A}}}(\mathcal {G})(K) \simeq \mathcal {G}_K(X)$ for each $K$ .

The composition ${\mathbb {D}}_{X, {\mathcal {A}}} = \psi \circ D_{X, {\mathcal {A}}}$ is called the covariant Verdier duality functor; it gives an equivalence of $\infty$ -categories ${\textrm {Cosh}}(X,{\mathcal {A}}) \simeq \textrm {Sh}(X,{\mathcal {A}})$ . It can be proven that its inverse is given, under the identification ${\textrm {Cosh}}(X,{\mathcal {A}}) \simeq \textrm {Sh}(X,{\mathcal {A}}^{\textrm{op}})^{\textrm{op}}$ , by the same procedure with ${\mathcal {A}}^{\textrm{op}}$ substituted for $\mathcal {A}$ , i.e. by the functor $({\mathbb {D}}_{X, {\mathcal {A}}^{\textrm{op}}})^{\textrm{op}}$ . When $X$ or $\mathcal {A}$ is clear from context, we omit it from the notation.

4.1.3 Relation to classical Verdier duality

We cannot simply take $\mathcal {A}$ to be $\textrm {Perf}$ in the description above, since this category is not closed under all small limits or colimits. However, upon restricting to sheaves that are constructible with respect to a finite stratification $A$ , every (co)limit we take when defining the Verdier dual is equivalent to a finite (co)limit, so the equivalence ${\mathbb {D}}_{X, \textrm {Vect}}$ restricts to an analogous equivalenceFootnote ⁴

\begin{equation*} {\mathbb {D}}_{X,\textrm {Perf}}: \textrm {Sh}^A(X,\textrm {Perf}) \xrightarrow {\sim } {\textrm {Cosh}}^A(X,\textrm {Perf}). \end{equation*}

One can further compose this functor with the linear duality functor ${\textrm {Hom}}_k(-,k)$ ; since linear duality exchanges limits and colimits in $\textrm {Perf}$ there is an equivalence

\begin{equation*} \mathbb {D}' _X: \textrm {Sh}^A(X,\textrm {Perf}) \to \textrm {Sh}^A(X,\textrm {Perf})^{\textrm{op}}, \end{equation*}

corresponding to the classical contravariant notion of Verdier duality for sheaves of (complexes) of vector spaces. Namely, there is an object $\omega _X \in \textrm {Sh}(X,\textrm {Perf})$ , called the dualizing complex of the space $X$ , and a canonical equivalence of functors ${\mathbb {D}}'_X(-) \simeq \mathcal {H}\kern -1.2pt \mathit {om}_X(-, \omega _X)$ .

Since ${\mathbb {D}}'_X$ is an equivalence, we have an equivalence $\varpi _X \simeq \omega _X^\vee$ , identifying the object defined above with the $k$ -linear dual of the dualizing complex. Classical Verdier duality does not extend all the way to $\textrm {Vect}$ -coefficients, but it does extend beyond $\textrm {Perf}$ enough for our purposes. Namely, we consider constructible sheaves valued in the following categories.

Note that the $\pm$ subscript refers to which direction the complexes are allowed to be supported in infinitely many degrees. From the fact that the compact objects in $\textrm {Vect}$ are exactly those that are isomorphic to complexes of finite-dimensional vector spaces bounded in both directions, it follows that the intersection of $\textrm {Vect}^b_-$ and $\textrm {Vect}^b_+$ is identified with $\textrm {Perf} \subseteq \textrm {Vect}$ . Each of these three subcategories is closed under finite (but not arbitrary) limits and colimits, so $\mathbb {D}$ restricts to equivalences $\textrm {Sh}^A(X,{\mathcal {C}}) \xrightarrow {\sim } {\textrm {Cosh}}^A(X,{\mathcal {A}})$ for ${\mathcal {A}} = \textrm {Vect}^b_-, \textrm {Vect}^b_+$ or $\textrm {Perf}$ . In terms of classical Verdier duality, we recover the usual description.

Corollary 4.7. Contravariant Verdier duality gives an equivalence

\begin{equation*} \mathbb {D}' : \textrm {Sh}^A(X, \textrm {Vect}^b_\pm ) \xrightarrow {\sim } (\textrm {Sh}^A(X, \textrm {Vect}^b_\mp ))^{\textrm{op}}. \end{equation*}

4.1.4 The bivariant Verdier dual

We now describe a variant of the construction above which we will later need to define the notion of an orientation on a cosheaf of categories. Recall that the Verdier dual $\mathcal {K}$ -presheaf of some $\mathcal {A}$ -valued precosheaf on $X$ is given by applying the composition

\begin{equation*} \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},\quad\textrm {Fun}(\Delta ^1,{\mathcal {A}})) \to \textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},{\mathcal {A}}), \end{equation*}

where the first functor is pullback under $K \mapsto (X \setminus K \to X)$ and the last functor is induced by the functor $\textrm {Fun}(\Delta ^1,{\mathcal {A}}) \to {\mathcal {A}}$ taking the cofibre. Let us now make a variant of this definition by considering instead the functor

\begin{equation*} \mathcal {U}(X) \times {\mathcal {K}}(X)^{\textrm{op}} \to \textrm {Fun}(\Delta ^1,\mathcal {U}(X)), \end{equation*}

given by $(U,K) \mapsto (U \setminus K \to U)$ . We then get a composition

\begin{equation*} (\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}) \times \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}(\Delta ^1,\mathcal {U}(X)) \times \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}(\Delta ^1,{\mathcal {A}}). \end{equation*}

We compose the adjoint to the map above with the three maps $\textrm {Fun}(\Delta ^1,{\mathcal {A}}) \to {\mathcal {A}}$ given by the source, target and cofibre, to get three maps $\textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},{\mathcal {A}})$ , which we call $(-)_{\mathcal {U}-{\mathcal {K}}},(-)_{\mathcal {U}}$ and $D_{\mathcal {U}-{\mathcal {K}}}(-)$ .

We have canonical natural transformations among the three functors out of $\mathcal {A}$ , inducing, for any $\mathcal {G} \in \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}})$ , an equivalence

\begin{equation*} \mathrm {cofib}((\mathcal {G})_{\mathcal {U}-{\mathcal {K}}} \to (\mathcal {G})_{\mathcal {U}}) \xrightarrow {\sim } D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G}). \end{equation*}

Let us describe these objects: for any $(U,K) \in \mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ , we have equivalences

\begin{align*} (\mathcal {G})_{\mathcal {U}-{\mathcal {K}}}(U,K) &\simeq \mathcal {G}(U \setminus K), \\ (\mathcal {G})_{\mathcal {U}}(U,K) &\simeq \mathcal {G}(U), \\ D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G})(U,K) &\simeq \mathrm {cofib}(\mathcal {G}(U \setminus K) \to \mathcal {G}(U)). \end{align*}

We think of the object $D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G})$ as a more local variant of the Verdier dual $\mathcal {K}$ -presheaf to the precosheaf $\mathcal {G}$ ; the restriction of $D_{\mathcal {U},{\mathcal {K}}}(\mathcal {G})$ to $\{X\} \times {\mathcal {K}}(X)$ is equivalent to $D(\mathcal {G}) = D_{X,{\mathcal {A}}}(\mathcal {G})$ . We are interested in the values of $D_{\mathcal {U},{\mathcal {K}}}(\mathcal {G})$ on pairs where $K \subseteq U$ ; therefore, let us denote by $L$ the sub-poset of $\mathcal {U}(X) \times {\mathcal {K}}(X)^{\textrm{op}}$ where that holds.

Remark 3. For a cosheaf $\mathcal {G}$ , the object $D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G})|_L \in \textrm {Fun}(L,{\mathcal {A}})$ contains just as much information as the usual Verdier duality $\mathcal {K}$ -sheaf $D(\mathcal {G})$ ; for any $(U,K) \in L$ , the map

\begin{equation*} D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G})(U,K) \to D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G})(X,K) = D(\mathcal {G})(K)\end{equation*}

is an equivalence since the square obtained by applying $\mathcal {G}(-)$ to the cover of $X$ given by $\{U,X \setminus K\}$ is a pushout square, by the cosheaf property. On the other hand, for an arbitrary precosheaf $\mathcal {G}$ , the object $D_{\mathcal {U}-{\mathcal {K}}}(\mathcal {G})|_L$ cannot be recovered from $D(\mathcal {G})$ , since its ‘local Verdier duals’ may have more information.

4.2.1 $\mathcal {K}$ -presheaf version

We give a $\mathcal {K}$ -presheaf version of these homs. Consider the inclusion of posets ${\mathcal {K}}(X)^{\textrm{op}} \to \mathcal {U}(X)$ given by $K \mapsto X \setminus K$ ; considering compacts contained in a fixed $K$ , this restricts to an inclusion of posets

\begin{equation*} {\mathcal {K}}(X)^{\textrm{op}}_{\subseteq K} \to \mathcal {U}(X)_{X\setminus K \subseteq }, \end{equation*}

into the poset of opens containing $X \setminus K$ .

Just as we did for the hom sheaves in § 3.1, we extend this notation. Given any $x \in \mathcal {F}(X \setminus K')$ and $y \in \mathcal {F}(X \setminus K'')$ , we define

\begin{equation*} \mathcal {F}|^{x,y}_K= \left\{\begin{array}{l@{\quad}l} \mathcal {F}|^{\iota x, \iota y}_K & \text {if}\, K \subseteq K', K \subseteq K'', \\ 0 & \text {otherwise}. \end{array}\right.\end{equation*}

4.2.2 Hochschild homology of a cosheaf

Let us pick spanning sets of objects $S_U \subseteq \mathrm {Ob}(\mathcal {F}(U))$ for every $U \in \mathcal {U}(X)$ , such that for any $U \subseteq V$ , $S_V$ contains the image of $S_U$ under the corestriction map of $\mathcal {F}$ . This is always possible; one can start by independently picking spanning sets of objects independently for every open, and then adjoining all the images of spanning sets of smaller opens. We define an object $\textrm {HH}(\mathcal {F})_{(m,n)}$ for each $(m,n)$ by the same formula we used for the sheaf case, namely,

\begin{equation*} \textrm {HH}(\mathcal {F})_{(m,n)} = \bigoplus \mathcal {F}|_{V_n}^{y_n,x_0} \otimes \mathcal {F}|_{U_0}^{x_0,x_1}[1] \otimes \cdots \otimes \mathcal {F}|_{U_m}^{x_m,y_0} \otimes \mathcal {F}|_{V_0}^{y_0,y_1}[1] \otimes \cdots \otimes \mathcal {F}|_{V_{n-1}}^{y_{n-1},y_n}[1], \end{equation*}

where the sum is taken over all collections of elements of $\mathcal {U}(X)$ and all objects in $S$ .

By the assumption on the object sets $S_U$ , we have the exact analogue of Proposition 3.2.

$\textrm {HH}(\mathcal {F})$ is generally not a cosheaf, and we denote by $\mathcal {H}\mathcal {H}(\mathcal {F})$ its cosheafification. Analogously to the sheaf case, we denote by $\mathcal {H}\mathcal {H}(\mathcal {F})^{S^1}, \mathcal {H}\mathcal {H}(\mathcal {F})_{S^1}$ the negative cyclic and cyclic homology cosheaves, given by cosheafifying the analogous respective precosheaves.

4.3.1 The Verdier dual diagonal $\mathcal {K}$ -sheaf

In order to state the definition of nondegeneracy condition on $\Omega$ , we will need to check a certain map of sheaves of bimodules; to construct that, we use a similar method to that which we used in § 4.2.2, but using sheaves and $\mathcal {K}$ -sheaves valued in bimodules instead. We define the following $\mathcal {F}(X)^e\text{-}\textrm {Mod}$ -valued $\mathcal {K}$ -presheaves on $X$ :

\begin{align*} \mathrm {D}\Delta _X(\mathcal {F})_{m,n} = \bigoplus \mathrm {cofib}(&\mathcal {F}(X)(-,\iota x_0) \otimes \cdots \otimes \mathcal {F}|_{K_m}^{x_m,y_0} \otimes \cdots \otimes \mathcal {F}(X)(\iota y_n,-) \\ &\quad \to \mathcal {F}(X)(-,\iota x_1) \otimes \cdots \otimes \mathcal {F}(X)(\iota x_m,\iota y_0) \otimes \cdots \otimes \mathcal {F}(X)(\iota y_n,-)), \end{align*}

where the sum is over all compacts and objects in $S$ , and the terms $\mathcal {F}(X)(\ldots )$ denote constant presheaves on $X$ , whose sections over any open are given by the denoted object of $\textrm {Vect}$ . Defining $\mathrm {D}\Delta _X(\mathcal {F}) = \bigoplus _{m,n}(\mathrm {D}\Delta _X(\mathcal {F})_{m,n})$ , together with the differential given by the composition maps, we get a $\mathcal {F}(X)^e\text{-}\textrm {Mod}$ -valued $\mathcal {K}$ -presheaf on $X$ whose value on $K$ is equivalent to ${\textrm {Cone}}(I_! \mathcal {F}(X\setminus K)_\Delta \to \mathcal {F}(X)_\Delta )$ , i.e. the cofibre of the map of bimodules from the pushforward of the diagonal bimodule of $\mathcal {F}(X \setminus K)$ to the diagonal bimodule of the global sections category.

Proof. We need to verify that this object satisfies conditions (1) and (2) of Definition 4.1. Condition (1) holds trivially since the sections over $K=\emptyset$ are given by the cofibre of an isomorphism. Using the description of sections of $\mathrm {D}\Delta _X(\mathcal {F})$ as pushforwards of diagonal bimodules, condition (2) says that for every pair $(K,K')$ of compacts

is a pullback square, or equivalently a pushout square, since the category of $\mathcal {F}(X)$ -bimodules is stable. But since $\mathcal {F}$ is a cosheaf, this is exactly the statement of [Reference Brav and DyckerhoffBD19, Theorem 6.1].

A similar argument also proves the following result, concerning functoriality with respect to $X$ .

Proof. We can construct this map using the explicit description of $\mathrm {D}\Delta _U(\mathcal {F})$ and using the fact that $I_!$ can be expressed by tensoring on both sides over $\mathcal {F}_U$ with $\mathcal {F}(X)_\Delta$ . It remains to check that for any $K \in {\mathcal {K}}(U)$ this map induces an isomorphism, which follows from the fact that the square

is a pushout square because $\mathcal {F}$ is a cosheaf.

We can apply the functor $\varphi$ of Proposition 4.3 to get a $\mathcal {K}$ -sheaf of bimodules.

The same functoriality property of Proposition 4.14 holds for this $\mathcal {K}$ -sheaf.

Proposition 4.16. If $\mathcal {F}$ is a cosheaf, then there is a canonical isomorphism

\begin{equation*} I_! \varphi \mathrm {D}\Delta _U(\mathcal {F}|_U) \to \varphi \mathrm {D}\Delta _X(\mathcal {F})|_U. \end{equation*}

4.3.2 The bimodule dual diagonal $\mathcal {K}$ -sheaf

We now define another $\mathcal {K}$ -sheaf, which will be our representative for the ‘bimodule dual’ of the diagonal cosheaf. Recall that the bimodule dual of a perfect $\mathcal {A}$ -bimodule $\mathcal {M}$ is given by $\mathcal {M}^!={\textrm {Hom}}_{{\mathcal {A}}^e}(\mathcal {M},{\mathcal {A}}^e)$ ; to calculate this derived hom one can take, for example, the bar complex resolution. We now use this strategy, but with sheaves of morphisms instead. Let us define the following object of $\textrm {Fun}(\mathcal {U}(X),\mathcal {F}(X)^e\text{-}\textrm {Mod})$ :

\begin{equation*} \Delta _X(\mathcal {F})_{p,q} := \bigoplus \mathcal {F}(X)(-,\iota x_0) \otimes \cdots \otimes \mathcal {F}|_{U_p}^{x_p,y_0} \otimes \cdots \otimes \mathcal {F}(X)(\iota y_q,-).\end{equation*}

Let us also define its left dual as a $\mathcal {F}(X)$ -bimodule, i.e. the object of $\textrm {Fun}(\mathcal {U}(X)^{\textrm{op}},\mathcal {F}(X)^e\text{-}\textrm {Mod})$ given by

\begin{equation*} \Delta ^!_X(\mathcal {F})_{p,q} := \prod {\textrm {Hom}}_{\mathcal {F}(X)^e}( \mathcal {F}(X)(-,\iota x_0) \otimes \cdots \otimes \mathcal {F}|_{U_p}^{x_p,y_0} \otimes \cdots \otimes \mathcal {F}(X)(\iota y_q,-), \mathcal {F}(X)^e). \end{equation*}

We denote by $\Delta _X(\mathcal {F})$ and $\Delta ^!_X(\mathcal {F})$ the objects given by the direct sum and product over all $p,q$ and endowed with the differential coming from the composition maps. We can apply the functor $\theta$ mentioned in the statement of Proposition 4.2 to pass to a $\mathcal {K}$ -presheaf; for any $K$ there is an equivalence $\theta (\Delta ^!_X(\mathcal {F}))(K) \cong \lim _{K \subseteq U} \Delta ^!_X(\mathcal {F})(U)$ .

Proof. We must verify that this object satisfies conditions (1) and (2) of Definition 4.1. Condition (1) is automatically satisfied. To prove condition (2), we choose small enough open neighbourhoods $U, U'$ of $K,K'$ ; it is then sufficient to prove that the square

is a pullback square for all small enough open neighbourhoods $U, U'$ of $K,K'$ . This holds since the square above is obtained from a pushout square of compact bimodules by applying the functor $(-)^!$ , and this latter is an anti-equivalence when restricted to compact bimodules.

We can apply the functor of Proposition 4.3 to get a $\mathcal {K}$ -sheaf.

We have functoriality of $\theta (\Delta ^!_X(\mathcal {F}))$ and $\varphi \theta (\Delta ^!_X(\mathcal {F}))$ with respect to $X$ .

4.3.3 Nondegeneracy condition

Let us recall the definition from the introduction: the data of an orientation of dimension $n$ on the ccc space $(X,\mathcal {F})$ is a morphism of cosheaves

\begin{equation*} \Omega : \varpi _X[d] \to \mathcal {H}\mathcal {H}(\mathcal {F})^{S^1}, \end{equation*}

where $\omega _X^\vee$ is the covariant Verdier dual of the constant sheaf $\underline {k}$ on $X$ , and $\mathcal {H}\mathcal {H}(\mathcal {F})^{S^1}$ is the cosheafification of the negative cyclic homology cosheaf. In order to give an orientation, the corresponding morphism of cosheaves $\Omega :\varpi _X[d] \to \mathcal {H}\mathcal {H}(\mathcal {F})$ must satisfy a certain nondegeneracy condition, namely, we require that under a certain universal map

\begin{equation*} \eta _X: {\textrm {Hom}}_{{\textrm {Cosh}}(X,\textrm {Vect})}(\varpi _X,\mathcal {H}\mathcal {H}(\mathcal {F})) \to {\textrm {Hom}}_{\textrm {Sh}_{\mathcal {K}}(X,\mathcal {F}(X)^e\text{-}\textrm {Mod})}(\varphi \theta \Delta ^!_X(\mathcal {F}),\varphi \mathrm {D}\Delta _X(\mathcal {F})), \end{equation*}

the morphism of cosheaves $\Omega$ maps to an isomorphism of $\mathcal {K}$ -sheaves $\varphi \theta \Delta ^!_X(\mathcal {F})[d] \simeq \varphi \mathrm {D}\Delta _X(\mathcal {F})$ . The remainder of this subsection is devoted to the construction of the map $\eta _X$ .

We can apply the ‘bivariant Verdier dual’ functor of Definition 4.8 to the Hochschild homology precosheaf to get an object $D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F}))$ of the category $\textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},\textrm {Vect})$ . By definition, for any $(U,K) \in \mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ , there is an isomorphism

\begin{equation*} D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F}))(U,K)\simeq \mathrm {cofib}(\textrm {HH}(\mathcal {F}(U \setminus K)) \to \textrm {HH}(\mathcal {F}(U))). \end{equation*}

We note that, using bar complexes, one can construct maps

\begin{equation*} \textrm {HH}(\mathcal {F}(U\setminus K)) \simeq \mathcal {F}(U\setminus K)_\Delta \otimes _{\mathcal {F}(U\setminus K)} \mathcal {F}(U \setminus K)_\Delta \to I_!\mathcal {F}(U)_\Delta \otimes _{\mathcal {F}(X)^e} I_!(\mathcal {F}(X\setminus K)_\Delta ) \end{equation*}

giving a map $D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F}))(U,K) \to I_!\mathcal {F}(U)_\Delta \otimes _{\mathcal {F}(X)^e} \mathrm {cofib}(I_!(\mathcal {F}(X\setminus K)_\Delta ) \to \mathcal {F}(X)_\Delta )$ . Since $\mathcal {F}$ is a cosheaf of smooth dg categories by assumption, we have an isomorphism between this latter complex and the bimodule hom ${\textrm {Hom}}_{\mathcal {F}(X)^e}(I_!(\mathcal {F}(U)_\Delta )^!,\mathrm {cofib}(I_!(\mathcal {F}(X\setminus K)_\Delta ) \to \mathcal {F}(X)_\Delta ))$ .

We now consider the objects

\begin{align*} \Delta ^!_X(\mathcal {F}) &\in \textrm {Fun}(\mathcal {U}(X)^{\textrm{op}},\mathcal {F}(X)^e\text{-}\textrm {Mod}) \simeq \textrm {Fun}(\mathcal {U}(X),(\mathcal {F}(X)^e\text{-}\textrm {Mod})^{\textrm{op}}), \\ \mathrm {D}\Delta _X(\mathcal {F}) &\in \textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},\mathcal {F}(X)^e\text{-}\textrm {Mod}) \end{align*}

constructed in the preceding subsections. We denote by

\begin{equation*} {\textrm {Hom}}_{\mathcal {F}(X)^e\text{-}\textrm {Mod}}(\Delta ^!_X(\mathcal {F}),\mathrm {D}\Delta _X(\mathcal {F})) \in \textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},\textrm {Vect}), \end{equation*}

the object obtained by combining these two objects with the $\textrm {Vect}$ -enriched internal hom

\begin{equation*} {\textrm {Hom}}_{\mathcal {F}(X)^e\text{-}\textrm {Mod}}: (\mathcal {F}(X)^e\text{-}\textrm {Mod})^{\textrm{op}} \times \mathcal {F}(X)^e\text{-}\textrm {Mod} \to \textrm {Vect}. \end{equation*}

Explicitly, the object ${\textrm {Hom}}(\Delta ^!_X(\mathcal {F}),\mathrm {D}\Delta _X(\mathcal {F}))$ associates to a pair $(U,K)$ the object of $\textrm {Vect}$ given by

\begin{align*} {\textrm {Hom}}_{\mathcal {F}(X)^e\text{-}\textrm {Mod}}&(\Delta ^!_X(\mathcal {F})(U),\mathrm {D}\Delta _X(\mathcal {F})(K)) \simeq {\textrm {Hom}}_{\mathcal {F}(X)^e}(I_!(\mathcal {F}(U)_\Delta )^!,\mathrm {cofib}(I_!(\mathcal {F}(X\setminus K)_\Delta ) \to \mathcal {F}(X)_\Delta )). \end{align*}

Therefore, for each fixed $(U,K)$ , combining the identifications above, we have a map of complexes

\begin{equation*} \xi _{X,(U,K)}: D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F}))(U,K) \to {\textrm {Hom}}(\Delta ^!_X(\mathcal {F}),\mathrm {D}\Delta _X(\mathcal {F}))(U,K). \end{equation*}

The following lemma shows that these maps can be all assembled coherently over $\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ .

Lemma 4.20. There is a natural morphism in the category $\textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},\textrm {Vect})$ ,

\begin{equation*} \xi _X: D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F})) \to {\textrm {Hom}}_{\mathcal {F}(X)^e\text{-}\textrm {Mod}}(\Delta ^!_X(\mathcal {F}),\mathrm {D}\Delta _X(\mathcal {F})), \end{equation*}

whose induced map on any $(U,K)$ is equivalent to the map $\xi _{X,(U,K)}$ .

Proof. By definition, the bivariant Verdier dual $D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F}))$ is equivalent to the cofibre of the map $(\textrm {HH}(\mathcal {F}))_{\mathcal {U}-{\mathcal {K}}} \to (\textrm {HH}(\mathcal {F}))_{\mathcal {U}}$ , so we have an equivalence

\begin{align*} D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F})) \simeq \bigoplus \mathrm {cofib}&\big(\mathcal {F}|_{U\setminus K}^{w_q,x_0} \otimes \cdots \otimes \mathcal {F}|_{U \setminus K}^{x_m,y_0} \otimes \cdots \otimes \mathcal {F}|_{U\setminus K}^{y_n,z_0} \otimes \cdots \otimes \mathcal {F}|_{U \setminus K}^{z_p,w_0} \otimes \cdots \\ &\quad \to \mathcal {F}|_U^{w_q,x_0} \otimes \cdots \otimes \mathcal {F}|_U^{x_m,y_0} \otimes \cdots \otimes \mathcal {F}|_U^{y_n,z_0} \otimes \cdots \otimes \mathcal {F}|_U^{z_p,w_0} \otimes \cdots \big), \end{align*}

where again the sum is taken over all $(m,n,p,q),(U,K)$ and objects in our chosen object sets, and the tensor factors are the appropriate functors out of $\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ , and the whole complex is endowed with the differential coming from composition maps.

We map the object above to

\begin{align*} \bigoplus \mathrm {cofib}&\big(\mathcal {F}(X)(w_q,x_0) \otimes \cdots \otimes \mathcal {F}|_U^{x_m,y_0} \otimes \cdots \otimes \mathcal {F}(X)(y_n,z_0) \otimes \cdots \otimes \mathcal {F}|_{X \setminus K}^{z_p,w_0} \otimes \cdots \\ &\quad \to \mathcal {F}(X)(w_q,x_0) \otimes \cdots \otimes \mathcal {F}|_U^{x_m,y_0} \otimes \cdots \otimes \mathcal {F}(X)(y_n,z_0) \otimes \cdots \otimes \mathcal {F}(X)(z_p,w_0) \otimes \cdots \big), \end{align*}

where the factors in the dots are of the forms $\mathcal {F}|_U^{-,-}[1]$ , $\mathcal {F}|_{X\setminus K}^{-,-}$ or $\mathcal {F}(X)(-,-)$ . Since tensor commutes with cofibre, the object above is isomorphic to

\begin{align*} &\bigoplus (\mathcal {F}(X)(-,x_0) \otimes \cdots \otimes \mathcal {F}|_U^{x_m,y_0} \otimes \cdots \otimes \mathcal {F}(X)(y_n,-)) \\ &\quad \otimes _{\mathcal {F}(X)^e} \mathrm {cofib}\big(\mathcal {F}(X)(-,z_0) \otimes \cdots \otimes \mathcal {F}|_{X \setminus K}^{z_p,w_0} \otimes \cdots \mathcal {F}(X)(w_q,-) \\&\quad \to \mathcal {F}(X)(-,z_0) \otimes \cdots \otimes \mathcal {F}(X)(z_p,w_0) \otimes \cdots \mathcal {F}(X)(w_q,-)\big), \end{align*}

i.e. the two-sided tensor $\Delta _X\mathcal {F} \otimes _{\mathcal {F}(X)^e} \mathrm {D}\Delta _X(\mathcal {F})$ . Evaluating the image of this map against $\Delta ^!_X\mathcal {F}$ gives the desired map.

Recall that if $C$ is an ordinary category, morphisms in presheaf categories can be described by the formalism of ‘ends’: the morphism space between two $C$ -valued sheaves on $X$ is given by

\begin{equation*} {\textrm {Hom}}_{\textrm {Sh}(X,C)}(\mathcal {G}_1,\mathcal {G}_2) = \int _{\mathcal {U}(X)^{\textrm{op}}} {\textrm {Hom}}_C \circ (\mathcal {G}_1 \times \mathcal {G}_2), \end{equation*}

the end of the bifunctor given by composing with $\mathcal {G}_1 \times \mathcal {G}_2$ with

\begin{equation*} {\textrm {Hom}}_C: C^{\textrm{op}} \times C \to \mathrm {Set}. \end{equation*}

In the setting of $\infty$ -categories, the appropriate notion of end can be given as the limit over a certain category of ‘twisted arrows’ [Reference HaugsengHau22]. For any $\infty$ -category $I$ , there is an $\infty$ -category $\mathrm {Tw}^\ell (I)$ with a canonical morphism

\begin{equation*} p: \mathrm {Tw}^\ell (I) \to I^{\textrm{op}} \times I, \end{equation*}

such that the end of an $\infty$ -functor $\mathcal {H}:I^{\textrm{op}} \times I \to {\mathcal {A}}$ is given by $\int _I \mathcal {H} = \lim (F \circ p)$ . If we have a $\textrm {Vect}$ -enriched $\infty$ -category $\mathcal {C}$ , and $\mathcal {G}_1,\mathcal {G}_2 \in \textrm {Sh}(X,{\mathcal {C}})$ , we can then express their hom space as an object of $\textrm {Vect}$ as $\lim (p \circ (\mathcal {G}_1 \otimes \mathcal {G}_2))$ , a limit over $\mathrm {Tw}^\ell (\mathcal {U}(X)^{\textrm{op}})$ .

In order to construct the desired morphism $\eta _X$ , we must relate these ends to our previous constructions. Recall the following two functors: Lurie’s covariant Verdier functor

\begin{equation*} \mathbb {D}: \textrm {Fun}(\mathcal {U}(X)^{\textrm{op}},{\mathcal {A}}) \to \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}); \end{equation*}

and the bivariant Verdier duality

\begin{equation*} D_{\mathcal {U}-{\mathcal {K}}}: \textrm {Fun}(\mathcal {U}(X),{\mathcal {A}}) \to \textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},{\mathcal {A}}). \end{equation*}

Recall that the functor $\mathbb {D}$ , when restricted to sheaves, gives an equivalence of categories, with inverse given again by the functor $\mathbb {D}$ on cosheaves. We will now prove a generalization of this statement involving the functor $D_{\mathcal {U}-{\mathcal {K}}}$ .

Note that if $\mathcal {G} \in \textrm {Sh}(X,{\mathcal {A}})$ , upon fixing $U$ , $D_{\mathcal {U}-{\mathcal {K}}} \circ {\mathbb {D}}_X(\mathcal {G})$ restricted to $U \times {\mathcal {K}}(U)$ is isomorphic to $D_U \circ {\mathbb {D}}_U(\mathcal {G})\simeq \theta \mathcal {G}$ . The following proposition is just about making this statement coherent over all opens. Consider the projection map

\begin{equation*} p: \mathcal {U}(X) \times {\mathcal {K}}(X)^{\textrm{op}} \to {\mathcal {K}}(X)^{\textrm{op}}. \end{equation*}

Composing the corresponding pullback with the functor $\theta$ gives a functor

\begin{equation*} p^* \circ \theta : \textrm {Fun}(\mathcal {U}(X)^{\textrm{op}},{\mathcal {A}}) \to \textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},{\mathcal {A}}). \end{equation*}

We now have two functors with the same source and target, $D_{\mathcal {U}-{\mathcal {K}}} \circ {\mathbb {D}}$ and $p^*\circ \theta$ . We can further restrict the source to sheaves and compose the target with restriction to the locus $L \subset \mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ where $K \subseteq U$ .

Proposition 4.21. There is a natural equivalence

\begin{equation*} (-)|_L \circ D_{\mathcal {U}-{\mathcal {K}}} \circ \mathbb {D} \simeq (-)|_L \circ p^*\circ \theta, \end{equation*}

of functors $\textrm {Sh}(X,{\mathcal {A}}) \to \textrm {Fun}(L,{\mathcal {A}})$ .

Proof. We already know that, for any $\mathcal {G} \in \textrm {Sh}(X,{\mathcal {A}})$ , the sections of $D_{\mathcal {U}-{\mathcal {K}}} \circ \mathbb {D}(\mathcal {G})$ and $p^*\circ \theta (\mathcal {G})$ on $(U,K)$ with $K \subseteq U$ are abstractly isomorphic, so it just remains to construct the correct natural transformation of functors. For conciseness let us write $\mathcal {U} = \mathcal {U}(X)$ and ${\mathcal {K}} = {\mathcal {K}}(X)$ . We define the following maps of simplicial sets:

\begin{equation*} a: \Delta ^1 \times \Delta ^1 \times \mathcal {U}\times {\mathcal {K}}^{\textrm{op}} \to \Delta ^1\times \mathcal {U}, \quad b: \Delta ^1 \times {\mathcal {K}} \to \mathcal {U}^{\textrm{op}}, \end{equation*}

given by

\begin{equation*} a(r,s,U,K) = \begin{cases} (r,U \setminus K), & s=0,\\ (r,U), & s=1, \end{cases} \quad b(r,K') = \begin{cases} X, &r=0,\\ X\setminus K', &r=1. \end{cases}\end{equation*}

Denoting by $\mathcal {M} = \mathcal {U} \cup {\mathcal {K}}$ a subset of the poset of subsets of $X$ , we consider the diagram

\begin{equation*} \Delta ^1 \times \Delta ^1 \times \mathcal {U} \times {\mathcal {K}} \xrightarrow {a} \Delta ^1 \times \mathcal {U} \overset {i}{\hookrightarrow } \Delta ^1 \times \mathcal {M} \overset {j}{\hookleftarrow } \Delta ^1 \times {\mathcal {K}} \xrightarrow {b} \mathcal {U}^{\textrm{op}}. \end{equation*}

We then have a functor $a^* i^* j_! b^*: \textrm {Fun}(\mathcal {U}^{\textrm{op}},{\mathcal {A}}) \to \textrm {Fun}(\Delta ^1\times \Delta ^1\times \mathcal {U}\times {\mathcal {K}}^{\textrm{op}},{\mathcal {A}})$ , which gives an adjoint

\begin{equation*} (a^* i^* j_! b^*)^\sharp : \textrm {Fun}(\mathcal {U}^{\textrm{op}},{\mathcal {A}}) \to \textrm {Fun}(\mathcal {U}\times {\mathcal {K}}^{\textrm{op}},\textrm {Fun}(\Delta ^1\times \Delta ^1,{\mathcal {A}}).\end{equation*}

It follows from the definitions of the functors $D_{\mathcal {U}-{\mathcal {K}}}$ and $\mathbb {D}$ that the functor above, composed with the functor induced by $\mathrm {fib}^2:\textrm {Fun}(\Delta ^1\times \Delta ^1,{\mathcal {A}}))\to {\mathcal {A}}$ , gives $D_{\mathcal {U}-{\mathcal {K}}} \circ {\mathbb {D}}$ . Let us define a simplicial set $Z \subset \Delta ^1\times \Delta ^1\times \mathcal {U} \times {\mathcal {K}}^{\textrm{op}} \times {\mathcal {K}}$ given by

\begin{equation*} Z = \left \{ (r,s,u,K,K')\ \bigg| \begin{array}{ll} K' \subseteq U \setminus K, &s=0 \\ K' \subseteq U, &s=1 \end{array} \right \}. \end{equation*}

We form the square

where $\pi _1(r,s,U,K,K') = (r,s,U,K)$ and $\pi _2(r,s,U,K,K') = (r,K')$ . By the definition of $Z$ , the image of $\pi _2$ on some fixed input is always contained in the image of $a \circ \pi _1$ on the same input,so we have a natural transformation of functors $j\circ \pi _2 \Rightarrow i \circ a \circ \pi _1$ ; using (co)unit maps of the relevant adjunction then gives a natural transformation $\pi _{1!}\pi _2^* a^* \Rightarrow a^* i^* j_! b^*$ . We then form another square

where $\lambda _1(r,s,U,K)=(r,s,K)$ and $\lambda _2(V)=(0,0,V)$ . We now have a natural transformation $\lambda _2\circ b \circ \pi _2 \Rightarrow \lambda _1\circ \pi _1$ ; using adjunctions, we get a natural transformation $\pi _{1!}\pi _2^* a^* \Rightarrow \lambda _1^*\lambda _{2!}$ . This latter functor, upon taking adjoints and composing with $\mathrm {fib}^2$ , gives the functor $p^*\circ \theta$ of the statement we are trying to prove. In summary, we get natural transformations

\begin{equation*} D_{\mathcal {U}-{\mathcal {K}}} \circ \mathbb {D} \Leftarrow \mathrm {fib}^2\circ (\pi _{1!} \circ \pi _2^*)^\sharp \Rightarrow p^* \circ \theta, \end{equation*}

of functors $\textrm {Fun}(\mathcal {U}^{\textrm{op}},{\mathcal {A}}) \to \textrm {Fun}(\mathcal {U}\times {\mathcal {K}}^{\textrm{op}},{\mathcal {A}})$ , so to prove the statement we want it is enough to show that these induce equivalences when applied to a sheaf $\mathcal {G}$ and then restricting the result to the locus $L$ . We can see this by calculating these maps explicitly; for example, picking some pair $K\subseteq U$ , we have that $(\mathrm {fib}^2\circ (\pi _{1!} \circ \pi _2^*)^\sharp \mathcal {G})(U,K)$ is given by taking the double fibre of the square

which, when $\mathcal {F}$ is a sheaf, is equivalent to $\mathrm{colim}_{K\subseteq V} \mathcal {F}(V)$ . Similar calculations of the other two functors show that on each $(U,K)$ the maps are isomorphisms.

We now use the maps we described to produce our desired map. We have a composition

\begin{align*} {\textrm {Hom}}_{{\textrm {Cosh}}(X,\textrm {Vect})}(\varpi _X,\mathcal {H}\mathcal {H}(\mathcal {F})) &\to {\textrm {Hom}}_{{\textrm {Cosh}}(X,\textrm {Vect})}(\varpi _X,\textrm {HH}(\mathcal {F})) \\ &\xrightarrow {D_{\mathcal {U}-{\mathcal {K}}}} {\textrm {Hom}}_{\textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},\textrm {Vect})}(D_{\mathcal {U}-{\mathcal {K}}}\mathbb {D} k_X,D_{\mathcal {U}-{\mathcal {K}}}(\textrm {HH}(\mathcal {F}))) \\ & \xrightarrow {\xi _X} {\textrm {Hom}}_{\textrm {Fun}(\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}},\textrm {Vect})}(D_{\mathcal {U}-{\mathcal {K}}}\mathbb {D} k_X,{\textrm {Hom}}_{\mathcal {F}(X)^e}(\Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F})) \\ & \xrightarrow {(-)|_L} {\textrm {Hom}}_{\textrm {Fun}(L,\textrm {Vect})}((D_{\mathcal {U}-{\mathcal {K}}}\mathbb {D} k_X)|_L,{\textrm {Hom}}_{\mathcal {F}(X)^e}(\Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F})|_L) \\ &\to {\textrm {Hom}}_{\textrm {Fun}(L,\textrm {Vect})}((D_{\mathcal {U}-{\mathcal {K}}}\mathbb {D} k)|_L,{\textrm {Hom}}_{\mathcal {F}(X)^e}(\Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F})|_L) \end{align*}

where $k$ denotes the constant functor with value $k$ and the last map is induced by the canonical map from the constant presheaf to the constant sheaf. By Proposition 4.21, we have that $(D_{\mathcal {U}-{\mathcal {K}}}\mathbb {D} k)|_L$ is the constant functor on $L$ with value $k$ . Applying $\theta$ along the first coordinate, to get functors out of $L_{\mathcal {K}} = \{(J,K)| K \subset J\} \subset {\mathcal {K}} \times {\mathcal {K}}^{\textrm{op}}$ , we get a further map from the last complex above:

\begin{align*} &\to {\textrm {Hom}}_{\textrm {Fun}(L_{\mathcal {K}},\textrm {Vect})}(k,{\textrm {Hom}}_{\mathcal {F}(X)^e}(\theta \Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F})|_L). \end{align*}

That is, an element in the complex above is the data of a coherent choice of maps

\begin{equation*} k \to {\textrm {Hom}}_{\mathcal {F}(X)^e}(\theta \Delta ^!_X\mathcal {F}(J),\mathrm {D}\Delta _X\mathcal {F}(K)), \end{equation*}

for any pair of compacts with $K \subseteq J$ . We now take ends, noting that, by definition, the image of the canonical map from the category of twisted arrows $\mathrm {Tw}^\ell ({\mathcal {K}}(X)^{\textrm{op}}) \to {\mathcal {K}}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ lands inside of the locus $L_{\mathcal {K}}$ . Therefore we can pullback along this map and take limits, giving a map of complexes

\begin{align*} {\textrm {Hom}}_{\textrm {Fun}(L_{\mathcal {K}},\textrm {Vect})}(k,{\textrm {Hom}}_{\mathcal {F}(X)^e}(\theta \Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F})|_L) &\to \int ^{{\mathcal {K}}(X)^{\textrm{op}}} {\textrm {Hom}}_{\mathcal {F}(X)^e}(\theta \Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F}) \\ &= {\textrm {Hom}}_{\textrm {Sh}(X,\mathcal {F}(X)^e\text{-}\textrm {Mod})}(\theta \Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F}). \end{align*}

Definition 4.22. The map $\eta _X$ is given by composing the map

\begin{equation*} {\textrm {Hom}}_{{\textrm {Cosh}}(X,\textrm {Vect})}(\varpi _X,\mathcal {H}\mathcal {H}(\mathcal {F})) \to {\textrm {Hom}}_{\textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},\mathcal {F}(X)^e\text{-}\textrm {Mod})}(\theta \Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F}), \end{equation*}

constructed above with the map

\begin{equation*} {\textrm {Hom}}_{\textrm {Fun}({\mathcal {K}}(X)^{\textrm{op}},\mathcal {F}(X)^e\text{-}\textrm {Mod})}(\theta \Delta ^!_X\mathcal {F},\mathrm {D}\Delta _X\mathcal {F}) \to {\textrm {Hom}}_{\textrm {Sh}_{\mathcal {K}}(X,\textrm {Vect})}(\varphi \theta \Delta ^!_X\mathcal {F},\varphi \mathrm {D}\Delta _X\mathcal {F}), \end{equation*}

induced by the functor $\varphi$ of Proposition 4.3.

This map is universal in $X$ , in the following sense. Recall that for any sheaf $\mathcal {F}$ and $U \in \mathcal {U}(X)$ , we have a natural isomorphism $({\mathbb {D}}_X\mathcal {F})_U \simeq {\mathbb {D}}_U(\mathcal {F}_U)$ , so we have a natural isomorphism $\varpi _U \simeq \varpi _X|_U$ . Therefore, restricting on both sides of the map $\eta _X$ , we get a square of complexes.

Example 12. (The half-open interval). Let $X = [0,1)$ with stratification $\{0\} \cup (0,1)$ . Let us denote $U = (0,1)$ ; the data of a constructible cosheaf $\mathcal {F}$ on $X$ is then given by a corestriction functor $\iota : \mathcal {F}(U) \to \mathcal {F}(X)$ . To describe the relevant $\mathcal {K}$ -sheaves, we note that there are only two types of connected compact subsets; let us pick one representative for each type, for example, compact subsets $K_1 = [0, 3/4]$ and $K_2=[1/4,1/2]$ . Denoting by $\sim$ the equivalence relation generated by stratified retracts of open subsets, we have

\begin{equation*} X \setminus K_1 \sim U, \quad X \setminus K_2 \sim X \sqcup U, \end{equation*}

with the inclusion $X \setminus K_1 \hookrightarrow X \setminus K_2$ equivalent to the inclusion $U \hookrightarrow X \sqcup U$ . Let us describe the Verdier dual Hochschild $\mathcal {K}$ -presheaf. We have equivalences

\begin{align*} \textrm {DHH}_X(\mathcal {F})(K_1) &\cong \mathrm {cofib}(\textrm {HH}(\mathcal {F}(U)) \xrightarrow {\iota } \textrm {HH}(\mathcal {F}(X))), \\ \textrm {DHH}_X(\mathcal {F})(K_2) &\cong \mathrm {cofib}(\textrm {HH}(\mathcal {F}(U)) \oplus \textrm {HH}(\mathcal {F}(X)) \xrightarrow {(\iota ,{\textrm {id}})} \textrm {HH}(\mathcal {F}(X))) \cong \textrm {HH}(\mathcal {F}(U))[1],\end{align*}

with restriction map given by the connecting map. We can also describe the version over $\mathcal {U}(X)\times {\mathcal {K}}(X)^{\textrm{op}}$ . By definition we have $\textrm {DHH}(\mathcal {F})(X,K_i) = \textrm {DHH}_X(\mathcal {F})(K_i)$ , and also know that the map $\textrm {DHH}(\mathcal {F})(U,K) \to \textrm {DHH}(\mathcal {F})(X,K)$ is given by the equivalence

\begin{align*} \mathrm {cofib}&\left (\textrm {HH}(\mathcal {F}(U))\oplus \textrm {HH}(\mathcal {F}(U))\to \textrm {HH}(\mathcal {F}(U))\right ) \\ & \quad \xrightarrow {\sim }\mathrm {cofib}\left (\textrm {HH}(\mathcal {F}(X))\oplus \textrm {HH}(\mathcal {F}(U))\to \textrm {HH}(\mathcal {F}(X))\right ) \cong \textrm {HH}(\mathcal {F}(U))[1].\end{align*}

As for the $\mathcal {K}$ -sheaves of bimodules, we note that every sufficiently small open set containing $K_1$ or $K_2$ , is equivalent, by retracts respecting the stratification, to $X$ or $U$ , respectively. Therefore, we have equivalences

\begin{equation*} \varphi \theta \Delta ^!_X(\mathcal {F})(K_1) \cong \mathcal {F}(X)^!, \quad \varphi \theta \Delta ^!_X(\mathcal {F})(K_2) \cong I_!\mathcal {F}(U)^!.\end{equation*}

Calculating the Verdier dual diagonal bimodule $\mathcal {K}$ -sheaf gives

\begin{align*} \varphi \mathrm {D}\Delta _X(\mathcal {F})(K_1) &\cong \mathrm {cofib}\left (I_!\mathcal {F}(U)_\Delta \to \mathcal {F}(X)_\Delta \right ), \\ \varphi \mathrm {D}\Delta _X(\mathcal {F})(K_2) &\cong \mathrm {cofib}\left (\mathcal {F}(X)_\Delta \oplus I_!\mathcal {F}(U)_\Delta \to \mathcal {F}(X)_\Delta \right ) \cong I_!\mathcal {F}(U)_\Delta [1].\end{align*}

Suppose now that we are given a section $s:\underline {k}_L \to \textrm {DHH}(\mathcal {F})|_L$ . This is just given by an element $s_{(X,K_1)} \in \mathrm {cofib}(\textrm {HH}(\mathcal {F}(U)) \xrightarrow {\iota } \textrm {HH}(\mathcal {F}(X)))$ . We can describe the map of complexes explicitly. For each compact $K$ , we choose small enough open $V$ such that $K \subseteq V$ , and denote $W=V\setminus K$ . For all choices of $x_0,\ldots ,y_0,\ldots ,z_0,\ldots ,w_0,\ldots ,$ we have maps from

\begin{align*} \mathrm {cofib} (\mathcal {F}&(W)(w_q,x_0) \otimes \cdots \otimes \mathcal {F}(W)(x_m,y_0) \otimes \cdots \otimes \mathcal {F}(W)(y_n,z_0) \otimes \cdots \otimes \mathcal {F}(W)(z_p,w_0) \otimes \cdots \\ &\to \mathcal {F}(V)(w_q,x_0) \otimes \cdots \otimes \mathcal {F}(V)(x_m,y_0) \otimes \cdots \otimes \mathcal {F}(V)(y_n,z_0) \otimes \cdots \otimes \mathcal {F}(V)(z_p,w_0) \otimes \cdots ), \end{align*}

to

\begin{align*} \mathrm {cofib}(\mathcal {F}&(X)(w_q,x_0) \otimes \cdots \otimes \mathcal {F}(X_K)(x_m,y_0) \otimes \cdots \otimes \mathcal {F}(X)(y_n,z_0) \otimes \cdots \otimes \mathcal {F}(V)(z_p,w_0) \otimes \cdots \\ &\to\mathcal {F}(X)(w_q,x_0) \otimes \cdots \otimes \mathcal {F}(X)(x_m,y_0) \otimes \cdots \otimes \mathcal {F}(X)(y_n,z_0) \otimes \cdots \otimes \mathcal {F}(V)(z_p,w_0) \otimes \cdots ),\end{align*}

which assemble to a map from $\textrm {DHH}(\mathcal {F})(V,K)_{m,n,p,q}$ by taking the sum over the choices. We can evaluate the image $\phi (s_{(V,K)})$ against any

\begin{equation*} \xi \in \Delta ^!_X(\mathcal {F})(V)^{p,q} = \bigvee {\textrm {Hom}}_{\mathcal {F}(X)^e}(\mathcal {F}(X)(-,z_0) \otimes \cdots \otimes \mathcal {F}(V)(z_p,w_0) \otimes \cdots \otimes \mathcal {F}(X)(w_q,-), \mathcal {F}(X)^e), \end{equation*}

to get an element in

\begin{align*} \bigvee \mathrm {cofib}&\left (\mathcal {F}(X)(-,x_0) \otimes \cdots \otimes \mathcal {F}(X \setminus K)(x_m,y_0) \otimes \cdots \otimes \mathcal {F}(X)(y_n,-) \right . \\ &\quad\left .\to \mathcal {F}(X)(-,x_0) \otimes \cdots \otimes \mathcal {F}(X)(x_m,y_0) \otimes \cdots \otimes\mathcal {F}(X)(y_n,-)\right ) \cong \mathrm {D}\Delta _X(\mathcal {F})_{m,n},\end{align*}

giving the desired map of bar complexes.

We can take $V=X$ for $K_1$ and $V=U$ for $K_2$ ; given a section $s_{(X,K_1)}$ as above, the map induced on sections by the corresponding map of $\mathcal {K}$ -sheaves $\theta \Delta ^!_X(\mathcal {F}) \to \mathrm {D}\Delta _X(\mathcal {F})$ on the pair $(K_1,K_2)$ is the data of a homotopy-commuting square.