Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects

In order to aid later calculations, we give a few definitions and make a few observations about dualisable objects and morphisms between them.

We introduce some notation and recall common notions. Let ${\mathcal{C}}$ be a symmetric monoidal category. An object $C\in{\mathcal{C}}$ is dualisable if there is another object ${C}^{\vee}$, together with an evaluation $\operatorname{ev}_{{C}}:{C}^{\vee}\otimes{C}\rightarrow 1_{{\mathcal{C}}}$ and coevaluation $\operatorname{co}_{{C}}:1_{{\mathcal{C}}}\rightarrow{C}\otimes{C}^{\vee}$ satisfying the usual axioms. Given a morphism $f:{C}\rightarrow{D}$ with dualisable source, the adjoint morphism $\varphi:1\rightarrow{C}^{\vee}\otimes{D}$ is given as the composition

$$1\stackrel{{\scriptstyle\operatorname{co}_{{C}}}}{{\longrightarrow}}{C}\otimes{C}^{\vee}\simeq{C}^{\vee}\otimes{C}\stackrel{{\scriptstyle\operatorname{Id}_{{C}^{\vee}}\otimes f}}{{\longrightarrow}}{C}^{\vee}\otimes{D}.$$

(2.1)

Conversely, given a morphism $\varphi:1\rightarrow{C}^{\vee}\otimes{D}$, we obtain the adjoint morphism $f:{C}\rightarrow{D}$ as the composition

$${C}\stackrel{{\scriptstyle\operatorname{Id}_{{C}}\otimes\varphi}}{{\longrightarrow}}{C}\otimes{C}^{\vee}\otimes{D}\simeq{C}^{\vee}\otimes{C}\otimes{D}\stackrel{{\scriptstyle\operatorname{ev}_{{C}}\otimes\operatorname{Id}_{{D}}}}{{\longrightarrow}}{D}.$$

(2.2)

Note that these two constructions are inverse to each other. Given a morphism $f:{C}\rightarrow{D}$ with dualisable source and target, the dual morphism $f^{\vee}:{D}^{\vee}\rightarrow{C}^{\vee}$ is given as the composition

$${D}^{\vee}\stackrel{{\scriptstyle\operatorname{ev}_{{C}}^{\vee}\otimes\operatorname{Id}_{{D}^{\vee}}}}{{\longrightarrow}}{C}^{\vee}\otimes{C}\otimes{D}^{\vee}\stackrel{{\scriptstyle\operatorname{Id}_{{C}^{\vee}}\otimes f\otimes\operatorname{Id}_{{D}^{\vee}}}}{{\longrightarrow}}{C}^{\vee}\otimes{D}\otimes{D}^{\vee}\stackrel{{\scriptstyle\operatorname{Id}_{{C}^{\vee}}\otimes\operatorname{co}_{{D}}^{\vee}}}{{\longrightarrow}}{C}^{\vee}.$$

(2.3)

In this subsection we discuss the formalism in which we deal with dg categories. Mostly we follow Gaitsgory-Rozenblyum [8].

$\operatorname{DGCat}^{2}_{\rm cont}$ denotes the symmetric monoidal $2$-category of presentable dg categories, continuous dg functors, and dg natural transformations. Here continuous means colimit preserving. The underlying $1$-category, with presentable dg categories as objects and continuous dg functors as $1$-morphisms, is denoted $\operatorname{DGCat_{cont}}$. We denote by ${\operatorname{Fun}}$ the internal Hom adjoint to tensor product. ³³3In some sources, ${\operatorname{Fun}}$ is denoted ${\operatorname{Fun}}^{L}$, to emphasise that morphisms preserve colimits. The unit with respect to the tensor product is the dg category $\operatorname{Vect}_{k}$ of dg vector spaces.

Given a dg category $C\in\operatorname{DGCat_{cont}}$, we denote its subcategory of compact objects by $C^{c}$. A dg category $C$ is compactly generated if $C=\operatorname{Ind}(C^{c})$. Note that for any presentable dg category $C$, $C^{c}$ is a small, idempotent complete dg category. The category of such small dg categories is denoted $\operatorname{dgcat}$.

As a matter of convention, objects of $\operatorname{DGCat_{cont}}$ shall be called simply `dg categories', while objects of $\operatorname{dgcat}$ shall be called `small dg categories'. Let us emphasise here that in the prequel to this paper [4], we worked with a model for small dg categories $\operatorname{dgcat}$ in terms of small categories enriched over cochain complexes and Morita equivalences between them. In the present paper, it is both more convenient and also necessary to work with $\operatorname{DGCat_{cont}}$, since we to handle not-necessarily compactly generated dg categories when dealing with quasi-coherent sheaves on prestacks.

The dualisable objects in $\operatorname{DGCat_{cont}}$ ($1$-dualisable objects in $\operatorname{DGCat}^{2}_{\rm cont}$) are simply called dualisable dg categories. Concretely, a dg category ${C}$ is dualisable if there is another dg category ${C}^{\vee}$ and a pairing $\operatorname{ev}_{{C}}:{C}^{\vee}\otimes{C}\rightarrow\operatorname{Vect}_{k}$ and copairing $\operatorname{co}_{{C}}:\operatorname{Vect}_{k}\rightarrow{C}\otimes{C}^{\vee}$ satisfying the usual properties. Note that if ${C}$ is compactly generated, then it is dualisable with dual ${C}^{\vee}=\operatorname{Ind}(({C}^{c})^{op})$. One shows that $\operatorname{ev}_{{C}}$ and $\operatorname{co}_{{C}}$ are dual up to a switch of tensor factors. Furthermore one shows that for a dualisable dg category, we have a natural equivalence ${C}^{\vee}\otimes{D}\simeq{\operatorname{Fun}({C},{D})}$, and that under this equivalence, the composition $\operatorname{Vect}_{k}\stackrel{{\scriptstyle\operatorname{co}_{{C}}}}{{\rightarrow}}{C}\otimes{C}^{\vee}\simeq{C}^{\vee}\otimes{C}\simeq{\operatorname{Fun}({C},{C})}$ sends $k\in\operatorname{Vect}_{k}$ to $\operatorname{Id}_{{C}}$.

Given a continuous dg functor $f:{C}\rightarrow{D}$ between presentable dg categories (that is, a map in $\operatorname{DGCat_{cont}}$), the adjoint functor theorem ensures the existence of a formal right adjoint $f^{r}:{D}\rightarrow{C}$. When the right adjoint $f^{r}$ is itself continuous, we call $f:{C}\leftrightarrow{D}:f^{r}$ a continuous adjunction. When ${C}$ and ${D}$ are dualisable, passing to duals gives a continuous adjunction $(f^{r})^{\vee}:{C}^{\vee}\longleftrightarrow{D}^{\vee}:f^{\vee}$, by Lemma 2.1. One shows that if ${C}$ is compactly generated, then a continuous functor $f:{C}\rightarrow{D}$ has continuous right adjoint if and only $f$ sends compact objects to compact objects.

A dualisable dg category ${C}$ is called proper if the evaluation functor ${C}^{\vee}\otimes{C}\stackrel{{\scriptstyle\operatorname{ev}_{{C}}}}{{\rightarrow}}\operatorname{Vect}_{k}$ has a continuous right adjoint and is called smooth if the evaluation functor has a left adjoint. Equivalently, ${C}$ is smooth if the coevaluation functor $\operatorname{Vect}_{k}\stackrel{{\scriptstyle\operatorname{co}_{{C}}}}{{\rightarrow}}{C}\otimes{C}^{\vee}$ has a continuous right adjoint. Since $\operatorname{Vect}_{k}$ is generated by the compact object $k$, $\operatorname{co}_{{C}}$ has a continuous right adjoint if and only $\operatorname{co}_{{C}}(k)\in{C}\otimes{C}^{\vee}$ is compact if and only if $\operatorname{Id}_{{C}}\in{\operatorname{Fun}({C},{C})}$ is compact. (We note in passing that the $2$-dualisable objects in $\operatorname{DGCat}^{2}_{\rm cont}$ are precisely the dualisable dg categories ${C}$ that are both smooth and proper.)

In this subsection, we review the notion of rigid dg category, following [8], and prove a corepresentability result (Corollary 2.6) for continuous adjunctions between smooth and rigid dg categories. This corepresentability lemma will be important for understanding the tangent complex of the moduli space of objects.

By monoidal/symmetric monoidal dg category, we mean an algebra/commutative algebra object in $\operatorname{DGCat_{cont}}$.

Given a monoidal dg category ${A}$, we denote the tensor product functor by $\operatorname{m}_{{A}}:{A}\otimes{A}\rightarrow{A}$, and the unit functor by $-\otimes 1_{{A}}:\operatorname{Vect}_{k}\rightarrow{A}$. Since ${A}$ is an algebra object in $\operatorname{DGCat_{cont}}$, $\operatorname{m}_{{A}}$ and $1_{{A}}$ are continuous, hence for every object $a\in{A}$, the functors $a\otimes-,-\otimes a:{A}\rightarrow{A}$ are continuous.

By ${A}$-module category we mean a (left) module ${C}$ for ${A}$ internal to $\operatorname{DGCat_{cont}}$. By definition, the action functor $\operatorname{m}_{{C}}:{A}\otimes{C}\rightarrow{C}$ is continuous. In particular, given any object $c\in{C}$, the functor $-\otimes c:{A}\rightarrow{C}$ is continuous. By the adjoint functor theorem, $-\otimes c$ has a (not necessarily continuous) right adjoint $\underline{\operatorname{Hom}}_{{A}}(c,-):{C}\rightarrow{A}$, called `relative Hom'.

We use the notation

$$\underline{\operatorname{End}}_{{A}}(c):=\underline{\operatorname{Hom}}_{{A}}(c,c).$$

$\underline{\operatorname{End}}_{{A}}(c)$ admits a natural structure of algebra in ${A}$. See [16], 4.7.2.

Given an associative algebra ${\mathcal{A}}$ in a monoidal dg category ${A}$ and an ${A}$-module category ${C}$, there is a dg category of ${\mathcal{A}}$-modules in ${C}$, denoted

$${{\mathcal{A}}}\operatorname{-mod}({C})$$

The datum of an object $c\in{{\mathcal{A}}}\operatorname{-mod}({C})$ is equivalent to giving an algebra morphism ${\mathcal{A}}\rightarrow\underline{\operatorname{End}}_{{A}}(c)$.

We shall need the following fact, proved in [8], I.1.8.5.7:

A monoidal dg category ${A}$ is called rigid if the unit $1_{{A}}$ is compact, the monoidal product $\operatorname{m}_{{A}}:{A}\otimes{A}\rightarrow{A}$ has a continuous right adjoint $\operatorname{m}_{{A}}^{r}$, and $\operatorname{m}_{{A}}^{r}$ is a map of ${A}$-bimodules. It is easy to see that $\operatorname{ev}_{{A}}:={\operatorname{Hom}_{{A}}}(1_{{A}},\operatorname{m}_{{A}}(-)):{A}\otimes{A}\rightarrow{A}\rightarrow\operatorname{Vect}_{k}$ induces a self-duality equivalence ${A}\simeq{A}^{\vee}$. When ${A}$ is compactly generated, the condition that $\operatorname{m}_{{A}}^{r}$ be a bimodule functor can replaced with the requirement that an object is compact if and only if it admits a left and right dual. See [8], I.1.9.

If ${C}$ is dualisable, then one can show that there is an equivalence of dg categories ${C}^{\vee}\simeq\operatorname{Fun}_{{A}}{{C}}{{A}}$ and that there is an ${A}$-linear relative evaluation functor $\operatorname{ev}_{{C}/{A}}:{C}^{\vee}\otimes_{{A}}{C}\rightarrow{A}$ exhibiting ${C}^{\vee}$ as the ${A}$-module dual of ${C}$ ([8], I.1.9.5.4). We say that ${C}$ is smooth over ${A}$ if the relative evaluation $\operatorname{ev}_{{C}/{A}}$ has a left adjoint $\operatorname{ev}^{l}_{{C}/{A}}$ and proper over ${A}$ if there is a continuous right adjoint $\operatorname{ev}_{{C}/{A}}^{r}$.

For a rigid dg category ${A}$, the induction-restriction adjunction

$$-\otimes 1_{{A}}:\operatorname{Vect}_{k}\longleftrightarrow{A}:{\operatorname{Hom}_{{A}}}(1_{{A}},-)$$

(2.7)

is continuous. Tensoring 2.7 with a dg category ${C}$, we obtain a continuous induction-restriction functor for ${C}$ and ${C}_{{A}}:={A}\otimes{C}$, which for brevity we denote

$$i:{C}\longleftrightarrow{C}_{{A}}:i^{r}.$$

Concretely, we have $i(c)=1_{{A}}\otimes c$ and $i^{r}(a\otimes c)={\operatorname{Hom}_{{A}}}(1_{{A}},a)\otimes c$.

Next, we specialise to the case of dualisable and smooth sources and rigid target, where standard diagram chases establish the following.

Let ${A}$ be a rigid, compactly generated dg category, ${C}$ a compactly generated ${A}$-module category. An object $c\in{C}$ is called left proper over ${A}$ if $\underline{\operatorname{Hom}}_{{A}}(c,-):{C}\rightarrow{A}$ is continuous with continuous right adjoint, and right proper over ${A}$ if $\underline{\operatorname{Hom}}_{{A}}(-,c)^{\vee}:{C}\rightarrow{A}$ is continuous with continuous right adjoint. ⁴⁴4Here, $\underline{\operatorname{Hom}}_{{A}}(-,c)^{\vee}$ is a slight abuse of notation. Strictly speaking, the formula is correct on compact objects, and is then defined everywhere by left Kan extension.

The functor $\operatorname{Id}_{{C}/{A}}^{!}$ adjoint to $\operatorname{ev}^{l}_{{C}/{A}}(k)\in{C}^{\vee}\otimes_{A}{C}$ is called the (relative) inverse dualising functor, since by the following corollary it behaves like an `inverse Serre functor' relative to ${A}$.

Combining Lemma 2.3 and Proposition 2.4, we have the following corepresentability result, which will be essential in understanding the tangent complex of the moduli space of objects ${\mathcal{M}}_{{C}}$ in a smooth dg category ${C}$.

We end this section with a computation that will be useful later for computing fibres of certain canonical perfect complexes on the moduli space of objects in a dg category.

We review some basic notions in derived algebraic geometry that we shall need later, mostly following [8], Chapters 2-6. For more subtle points, we give precise references.

For now on, we take $k$ be a field of characteristic $0$.

By definition, the category of (derived) affine schemes $\operatorname{Aff}$ is opposite to the category $\operatorname{CAlg}_{k}^{\leq 0}\subset\operatorname{CAlg}_{k}$ of connective commutative algebras in $\operatorname{Vect}_{k}$. ⁵⁵5Since we are working in characteristic $0$, it is possible to model $\operatorname{CAlg}_{k}^{\leq 0}$ in terms of cohomologically non-positive commutative differential graded algebras. See [16], Proposition 7.1.4.11.

An affine scheme $U={{\operatorname{Spec}(R)}}$ is said to be of finite type over the ground field $k$ if $H^{0}(R)$ is finitely generated as a commutative algebra over $k$, $H^{i}(R)$ is finitely generated as a module over $H^{0}(R)$, and $H^{-i}(R)=0$ for $i>>0$. The category of affine schemes of finite type is denoted $\operatorname{{Aff}_{ft}}$.

By definition, the dg-category of quasi-coherent sheaves $\operatorname{QCoh}(U)$ on an affine scheme $U={{\operatorname{Spec}(R)}}$ is the dg category of dg modules over the commutative algebra $R$. Given a map $f:U\rightarrow V$, the pullback functor $f^{*}:\operatorname{QCoh}(V)\rightarrow\operatorname{QCoh}(U)$ is given by induction of modules along the corresponding map of rings. As such, $f^{*}$ is symmetric monoidal. The naturality of pullback is expressed via a functor

$$\operatorname{QCoh}(-)^{*}:\operatorname{Aff}^{\rm op}\rightarrow\operatorname{DGCat_{cont}}$$

Since we are so far considering only affine schemes, $f^{*}$ always has a continuous right adjoint $f_{*}$.

One can show that $\operatorname{QCoh}(U)$ is a rigid symmetric monoidal dg category, and in particular that $\otimes$-dualisable objects coincide with compact objects. In this case, the structure sheaf ${\mathcal{O}}_{U}$, corresponding to the ring $R$, is a compact generator. The compact objects in $\operatorname{QCoh}(U)$ are called perfect complexes, which form a small idempotent complete dg category denoted $\operatorname{Perf}(U)$. They are preserved by pullback. In the present affine case, we therefore have $\operatorname{Ind}(\operatorname{Perf}(U))=\operatorname{QCoh}(U)$.

Given a pullback square of affine schemes

(3.1)

naturality of pullback gives an isomorphism $q^{*}f^{*}\simeq p^{*}g^{*}$, so by adjunction we obtain a base-change map

$$f^{*}g_{*}\rightarrow q_{*}p^{*},$$

(3.2)

which is easily checked to be an isomorphism by considering its action on the generator ${\mathcal{O}}_{V}\in\operatorname{QCoh}(V)$.

For affine schemes $U={{\operatorname{Spec}(R)}}$ of finite type, define the small subcategory $\operatorname{Coh}(U)\subseteq\operatorname{QCoh}(U)$ of coherent sheaves to consist of quasi-coherent sheaves with bounded, finitely generated cohomology: $F\in\operatorname{Coh}(U)$ if $H^{i}(F)$ is finitely generated over $H^{0}(R)$ and $H^{i}(F)=0$ for $|i|>>0$. The dg category of ind-coherent sheaves is defined to be the ind-completion of the category of coherent sheaves:

$$\operatorname{IndCoh}(U):=\operatorname{Ind}(\operatorname{Coh}(U)).$$

The category of ind-coherent sheaves $\operatorname{IndCoh}(U)$ is a module category for quasi-coherent sheaves $\operatorname{QCoh}(U)$, with the action given by ind-completion of the action of $\operatorname{Perf}(U)$ on $\operatorname{Coh}(U)$. For a map of affine schemes of finite type $f:U\rightarrow V$, there is a functor ⁶⁶6For an ‘elementary’ definition of $f^{!}$, see [8], II.5.4.3.

$$f^{!}:\operatorname{IndCoh}(V)\rightarrow\operatorname{IndCoh}(U).$$

More precisely, we have a functor

$$\operatorname{IndCoh}()^{!}:\operatorname{{Aff}_{ft}}^{\rm op}\rightarrow\operatorname{DGCat_{cont}},$$

$\operatorname{IndCoh}(U)$ has a natural symmetric monoidal structure, the product of which is denoted $\otimes^{!}$, and the unit of which is $\omega_{U}:=p^{!}(k)$ for $p:U\rightarrow*$. Using the action of $\operatorname{QCoh}(U)$ on $\operatorname{IndCoh}(U)$, tensoring with $\omega_{U}$ gives a symmetric monoidal functor

$$\Upsilon_{U}={-}\otimes\omega_{U}:\operatorname{QCoh}(U)\rightarrow\operatorname{IndCoh}(U).$$

The functor $\Upsilon$ intertwines $*$-pullback and $!$-pullback: $\Upsilon_{U}f^{*}\simeq f^{!}\Upsilon_{V}$.

More precisely, $\Upsilon$ is a natural transformation

$$\Upsilon:\operatorname{QCoh}(-)^{*}\rightarrow\operatorname{IndCoh}(-)^{!}$$

of functors from $\operatorname{{Aff}_{ft}}^{\rm op}$ to $\operatorname{DGCat_{cont}}$.

There is a self-duality equivalence $\operatorname{IndCoh}(U)\simeq\operatorname{IndCoh}(U)^{\vee}$. The corresponding equivalence between compact objects is denoted

$$\operatorname{\mathbf{D}}_{U}(-)=\operatorname{Coh}(U)^{\rm op}\simeq\operatorname{Coh}(U)$$

One can show that there is an isomorphism of functors $\operatorname{\mathbf{D}}_{U}(-)\simeq\underline{\operatorname{Hom}}_{\operatorname{QCoh}(U)}(-,\omega_{U})$. The functor $\operatorname{\mathbf{D}}_{U}(-)$ can be used to define a contravariant Grothendieck-Serre duality functor

$${\operatorname{QCoh}(U)^{-}}^{op}\rightarrow{\operatorname{Fun^{ex}}(\operatorname{Coh}(U)^{op},\operatorname{Vect}_{k})}\simeq\operatorname{IndCoh}(U)$$

(3.3)

given explicitly by $E\mapsto{\operatorname{Hom}_{\operatorname{QCoh}(U)}}(E,\operatorname{\mathbf{D}}_{U}(-))$ ⁷⁷7Here, $\operatorname{QCoh}(U)^{-}$ denotes quasi-coherent sheaves that are cohomologically bounded above. For more on Grothendieck-Serre duality, see [9], I.1.3.4.. If $E$ is a perfect complex, then for any $F\in\operatorname{Coh}(U)\subset\operatorname{IndCoh}(U)$, we have isomorphisms

$${\operatorname{Hom}_{\operatorname{QCoh}(U)^{-}}}(E,\operatorname{\mathbf{D}}_{U}(F))\simeq{\operatorname{Hom}_{\operatorname{IndCoh}(U)}}(E\otimes F,\omega_{U})\simeq{\operatorname{Hom}_{\operatorname{IndCoh}(U)}}(F,E^{\vee}\otimes\omega_{U})$$

hence the functor 3.3 is given on perfect complexes by

$$E\mapsto\Upsilon(E^{\vee}).$$

(3.4)

In particular, it is symmetric monoidal and fully faithful when restricted to perfect complexes. More generally, one can show that $\operatorname{\mathbf{D}}_{U}(-)$ is fully faithful on bounded above quasi-coherent sheaves having coherent cohomology sheaves.

In this subsection, we fix notation by reviewing some basic constructions concerning prestacks and dg categories of sheaves on prestacks. Our basic reference is [8],[9].

We denote by $\operatorname{PrStk}:={\operatorname{Fun}(\operatorname{CAlg}_{k}^{\leq 0},\operatorname{Spc})}$ the category of prestacks on $\operatorname{Aff}$. Being a topos, $\operatorname{PrStk}$ is cocomplete, Cartesian closed, and colimits commute with pullbacks. We denote the internal/local mapping space adjoint to $X\times-$ by $\underline{\operatorname{Map}}(X,-)$, and the global mapping space by $\operatorname{Map}(X,-)$. Moreover, there is a continuous faithful embedding $\operatorname{Spc}\hookrightarrow\operatorname{PrStk}$ sending a space $K$ to the constant prestack with value $K$.

The embedding $\operatorname{Spc}\hookrightarrow\operatorname{PrStk}$ is symmetric monoidal for the Cartesian monoidal structures, so (abelian) groups in $\operatorname{Spc}$ map to (abelian) groups in $\operatorname{PrStk}$. We shall be especially interested in the circle group $B\mathbf{Z}=S^{1}$.

The free loop space $LX$ carries a natural action of the circle group $S^{1}$, which we call `loop rotation'. Decomposing a circle into two intervals and using the fact that mapping out of a colimit gives a limit, we obtain an isomorphism of the free loop space with the self-intersection of the diagonal:

$$LX\simeq X\times_{X\times X}X$$

In particular, if $X$ is affine, then the free loop space is again affine.

Mostly we shall be interested in prestacks that are laft (locally almost of finite type) and def (`have deformation theory'). Roughly, a prestack $X$ is laft if it is determined by maps $U\rightarrow X$ with $U$ an affine of finite type, and is def if it has a (pro-)cotangent complex ${T}^{*}(X)$ that behaves as expected. See the next section for what we expect of a (pro-)cotangent complex.

Recall from Section 3.1 the functor of quasi-coherent sheaves on affine schemes:

$$\operatorname{QCoh}(-)^{*}:\operatorname{Aff}^{\rm op}\rightarrow\operatorname{DGCat_{cont}}$$

Taking the right Kan extension of $\operatorname{QCoh}(-)^{*}$, we obtain a functorial notion of quasi-coherent sheaves on general prestacks:

$$\operatorname{QCoh}(-)^{*}:\operatorname{PrStk}^{\rm op}\rightarrow\operatorname{DGCat_{cont}}.$$

Since every prestack $X$ is tautologically a colimit over all affines mapping into it, $X=\operatorname*{colim}_{\operatorname{Aff}/X}U$, we have by definition an identification

$$\operatorname{QCoh}(X)=\lim_{(\operatorname{Aff}/X)^{op}}\operatorname{QCoh}(U).$$

For each map of prestacks $f:X\rightarrow Y$, we have by definition a pullback functor $f^{*}:\operatorname{QCoh}(Y)\rightarrow\operatorname{QCoh}(X)$. The adjoint functor theorem provides a right adjoint, denoted $f_{*}$, but in general it can be poorly behaved. However, for `qca' morphisms $f$, $f_{*}$ is continuous and satisfies base change and the projection formula for pullbacks along maps of affines $U\rightarrow Y$ (see Corollary 1.4.5 [7]). A morphism $f:X\rightarrow Y$ is qca if the pullback of $X$ along a map from any affine $U\rightarrow Y$ is a nice Artin $1$-stack with affine stabilisers. This will be obvious in the situations where we need it.

One can similarly define perfect complexes on a prestack by right Kan extension from affines, so that in particular we have an identification

$$\operatorname{Perf}(X)=\lim_{(\operatorname{Aff}/X)^{op}}\operatorname{Perf}(U).$$

For a general prestack $X$, perfect complexes need not be compact as objects in $\operatorname{QCoh}(X)$, but they always identify with the subcategory of $\otimes$-dualisable objects in $\operatorname{QCoh}(X)$. In particular, $\operatorname{QCoh}(X)$ is not always rigid, nor even dualisable in $\operatorname{DGCat_{cont}}$. It shall therefore be convenient for us to formally introduce the category of ind-perfect sheaves $\operatorname{Ind}(\operatorname{Perf}(X))$. Note that by construction $\operatorname{Ind}(\operatorname{Perf}(X))$ is compactly generated and that pullback preserves compact objects, hence for a map of prestacks $f:X\rightarrow Y$, we have a continuous adjunction

$$f^{*}:\operatorname{Ind}(\operatorname{Perf}(Y))\longleftrightarrow\operatorname{Ind}(\operatorname{Perf}(X)):f_{*}$$