is an equivalence, where $i$ denotes the canonical inclusion $i\colon
Z\hookrightarrow X$.

##  6\. Consequences

Having proven 5.17, we now explore some of its consequences. In this section,
we mainly focus on improved functoriality results for the exodromy equivalence
and some structural results for the $\infty$-category of hyperconstructible
hypersheaves.

###  6.1. Structural results for hyperconstructible hypersheaves

Let $(X,P)$ be a conically stratified with locally weakly contractible strata.
Applying 5.19, we see that $\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)$ is a
presentable $\infty$-category. In particular, for every
$\mathcal{E}\in\mathcal{P}\mathrm{r}^{\mathrm{L}}$, the tensor product
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\otimes\mathcal{E}$ is well defined. Our
next task is to compare it with
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.

Since $\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)$ is presentable, it follows that
the inclusion
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\hookrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X)$
is a morphism in $\mathcal{P}\mathrm{r}^{\mathrm{L}}$. Therefore, tensoring
with $\mathcal{E}$ yields a fully faithful functor

(6.1)
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\otimes\mathcal{E}\lhook\joinrel\relbar\joinrel\rightarrow\mathrm{Sh}^{\mathrm{hyp}}(X)\otimes\mathcal{E}\
.$

Let $a\in P$ and let $i_{a}\colon X_{a}\hookrightarrow X$ be the inclusion of
the corresponding stratum. Functoriality of the tensor product of presentable
$\infty$-categories immediately implies the commutativity of the following
diagram:

$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\otimes\mathcal{E}}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X)\otimes\mathcal{E}}$$\displaystyle{\mathrm{LC}^{\mathrm{hyp}}(X_{a})\otimes\mathcal{E}}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X_{a})\otimes\mathcal{E}\
.}$$\scriptstyle{i_{a}^{\ast,\mathrm{hyp}}\otimes\mathrm{id}_{\mathcal{E}}}$$\scriptstyle{i_{a}^{\ast,\mathrm{hyp}}\otimes\mathrm{id}_{\mathcal{E}}}$

Recall from [HPT, Observation 3.11] that the equivalence
$\mathrm{Sh}^{\mathrm{hyp}}(X_{a})\otimes\mathcal{E}\simeq\mathrm{Sh}^{\mathrm{hyp}}(X_{a};\mathcal{E})$
restricts to an equivalence
$\mathrm{LC}^{\mathrm{hyp}}(X_{a})\otimes\mathcal{E}\simeq\mathrm{LC}^{\mathrm{hyp}}(X_{a};\mathcal{E})$.
Thus it follows that under the equivalence
$\mathrm{Sh}^{\mathrm{hyp}}(X)\otimes\mathcal{E}\simeq\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$,
the functor (6.1) factors through
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$, yielding the following
commutative diagram:

(6.2)
$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\otimes\mathcal{E}}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X)\otimes\mathcal{E}}$$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\
.}$

######  Corollary 6.3.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. If the assumptions of 5.17 are satisfied, then
the diagram

$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\otimes\mathcal{E}}$$\displaystyle{\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S}\big{)}\otimes\mathcal{E}}$$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E}\big{)}}$$\scriptstyle{\Phi_{X,P}^{\mathrm{hyp}}\otimes\mathrm{id}_{\mathcal{E}}}$$\scriptstyle{\boxtimes}$$\scriptstyle{\boxtimes}$$\scriptstyle{\Phi_{X,P}^{\mathrm{hyp},\mathcal{E}}}$

is canonically commutative. In particular, the left vertical functor is an
equivalence.

###### Proof.

The first half is a direct consequence of 3.8 and the commutativity of the
diagram (6.2). The second half follows from the first one and the 2-out-of-3
property of the equivalences. ∎

######  Corollary 6.4.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. If the assumptions of 5.17 are satisfied, then
the $\infty$-category $\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$
is $\kappa$-presentable.

###### Proof.

Indeed, $\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S})$ is always
compactly generated, and therefore
$\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S})\otimes\mathcal{E}$
is $\kappa$-presentable by [HA, Lemma 5.3.2.11]. ∎

######  Corollary 6.5 (Categorical Künneth formula).

Let $(X,P)$ and $(Y,Q)$ be conically stratified spaces with locally weakly
contractible strata. Then there is a canonical equivalence

$\mathrm{Cons}_{P\times Q}^{\mathrm{hyp}}(X\times
Y)\simeq\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)\otimes\mathrm{Cons}_{Q}^{\mathrm{hyp}}(Y)\
.$

###### Proof.

This follows from the fact that $\operatorname{Fun}(-,\mathcal{S})$ takes
products of $\infty$-categories to tensor products of presentable
$\infty$-categories, and the fact that $\Pi_{\infty}^{\Sigma}$ commutes with
finite products of stratified spaces. ∎

###  6.2. Exodromic morphisms

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Let
$\mathcal{E}$ be a presentable $\infty$-category. Although the natural
transformations

$\psi_{f}^{\mathrm{hyp}}\colon
f^{\ast,\mathrm{hyp}}\circ\Psi_{X,P}^{\mathrm{hyp}}\longrightarrow\Psi_{Y,Q}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f)^{\ast}\quad\text{and}\quad\phi_{f}^{\mathrm{hyp}}\colon\Phi_{Y,Q}^{\mathrm{hyp}}\circ
f^{\ast,\mathrm{hyp}}\longrightarrow\Pi_{\infty}^{\Sigma}(f)^{\ast}\circ\Phi_{X,P}^{\mathrm{hyp}}$

are always defined, they are not always equivalences.

######  Definition 6.6.

We say that the morphism $f\colon(Y,Q)\to(X,P)$ is:

  * •

_right exodromic_ if the natural transformation $\psi_{f}^{\mathrm{hyp}}$ is
an equivalence;

  * •

_left exodromic_ if the natural transformation $\phi_{f}^{\mathrm{hyp}}$ is an
equivalence when evaluated on hyperconstructible hypersheaves on $(X,P)$;

  * •

_strongly left exodromic_ if the natural transformation
$\phi_{f}^{\mathrm{hyp}}$ is an equivalence;

  * •

(_strongly_) _exodromic_ if it is right and (strongly) left exodromic.

With this terminology and with the help of 5.17, 3.20-(2) implies:

######  Lemma 6.7.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces with
locally weakly contractible strata. Let $\mathcal{E}$ be a presentable
$\infty$-category satisfying the assumptions of 5.17. Then $f$ is left
exodromic if and only if it is right exodromic.

######  Proposition 6.8.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces with
locally weakly contractible strata. Let $\mathcal{E}$ be a presentable
$\infty$-category satisfying the assumptions of 5.17. Then $f$ is left
exodromic.

###### Proof.

From 6.7, it is enough to prove that $\psi_{f}^{\mathrm{hyp}}$ is an
equivalence. For this, it is enough to show that for every $a\in Q$, the
hyper-restriction $i_{a}^{\ast,\mathrm{hyp}}(\psi_{f}^{\mathrm{hyp}})$ is an
equivalence, where $i_{a}:Y_{a}\to Y$ is the inclusion. From 4.9 and 3.26, we
find for every $F:\Pi^{\Sigma}_{\infty}(X,P)\to\mathcal{E}$ a canonical
identification

$i_{a}^{\ast,\mathrm{hyp}}(\psi_{f}^{\mathrm{hyp}}(F))\simeq\psi_{f\circ
i_{a}}^{\mathrm{hyp}}(F)\ .$

Let $b\in P$ be the image of $a$, let $i_{b}:X_{b}\to X$ be the inclusion and
let $g:Y_{a}\to X_{b}$ be the morphism morphism induced by $f$. Then, $f\circ
i_{a}=i_{b}\circ g$ so that a second application of 4.9 and 3.26 gives a
canonical identification

$i_{a}^{\ast,\mathrm{hyp}}(\psi_{f}^{\mathrm{hyp}}(F))\simeq\psi_{g}^{\mathrm{hyp}}(\Pi^{\Sigma}_{\infty}(i_{b})^{\ast}(F))\
.$

Thus, the conclusion follows from 4.8. ∎

######  Remark 6.9.

In particular, if $f\colon(Y,Q)\to(X,P)$ is an open immersion, 6.8 implies
that $\phi_{f}^{\mathrm{hyp}}$ is an equivalence on hyperconstructible
hypersheaves. When $f$ is the inclusion of an open union of strata of $(X,P)$,
Corollaries 3.21 and 5.15 show that more is true: indeed, in this case $f$ is
strongly exodromic.

######  Corollary 6.10.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a presentable $\infty$-category satisfying the
assumptions of 5.17. Let $x\in X$ be a point and let
$F\in\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$. We have:

$\Phi_{X,P}^{\mathrm{hyp}}(F)(x)\simeq\operatorname*{colim}_{x\in U}F(U)\ ,$

where the colimit ranges over the open subsets of $X$ containing $x$.

###### Proof.

Review $x$ as a morphism $x\colon*\to X$. The induced stratification on $\ast$
is the trivial one, and in particular it is conical. Thus, 6.8 implies that
$x$ is exodromic, and therefore that the natural transformation
$\phi_{x}^{\mathrm{hyp}}$ is an equivalence. The conclusion follows since the
colimit in the statement is canonically identified with
$x^{\ast,\mathrm{hyp}}(F)$. ∎

The next lemma asserts that the failure for the formula from 6.10 to hold
exactly measures the defect for a hypersheaf to be hyperconstructible.

######  Corollary 6.11.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a presentable $\infty$-category satisfying the
assumptions of 5.17. Let $F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$. Then
$F$ is hyperconstructible on $(X,P)$ if and only if for every $x\in X$, the
canonical morphism

$\phi_{x}^{\mathrm{hyp}}(F):\Phi_{X,P}^{\mathrm{hyp}}(F)(x)\longrightarrow\operatorname*{colim}_{x\in
U}F(U)\ $

is an equivalence.

###### Proof.

The direct implication follows from 6.10. Assume now that
$\phi_{x}^{\mathrm{hyp}}(F)$ is an equivalence for every $x\in X$. Let
$L_{P}^{\mathrm{hyp}}$ be the left adjoint to the inclusion
$\mathrm{Cons}_{P}(X;\mathcal{E})\hookrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
as obtained in 5.21-$(2)$. To prove 6.11, it is enough to prove that the unit
transformation $F\to L_{P}^{\mathrm{hyp}}(F)$ is an equivalence. Since both
source and target are hypersheaves it is enough to show that for every $x\in
X$, the induced morphism $F_{x}\to L_{P}^{\mathrm{hyp}}(F)_{x}$ is an
equivalence. By assumption, $F_{x}$ identifies with
$\Phi_{X,P}^{\mathrm{hyp}}(F)(x)$. On the other hand, 6.10 combined with
5.21-$(3)$ gives the following chain of equivalences

$L_{P}^{\mathrm{hyp}}(F)_{x}\simeq\Phi_{X,P}^{\mathrm{hyp}}(L_{P}^{\mathrm{hyp}}(F))(x)\simeq\Phi_{X,P}^{\mathrm{hyp}}(F)(x)$

6.11 is thus proved. ∎

######  Corollary 6.12.

Let $(X,P)$ be a conically stratified space and let $(X,Q)$ be a conical
refinement of $P$. Assume that the strata of $(X,P)$ and $(X,Q)$ are locally
weakly contractible. Then the natural map

$f\colon\Pi_{\infty}^{\Sigma}(X,Q)\longrightarrow\Pi_{\infty}^{\Sigma}(X,P)$

is a localization.

###### Proof.

We first reduce to the case where the stratification on $X$ is trivial. Let
$W$ be the collection of $f$-local equivalences in
$\Pi_{\infty}^{\Sigma}(X,Q)$. Since $f$ is essentially surjective, [C, 7.1.7,
7.1.11] shows that it is enough to prove that the functor

$f^{*}\colon\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S})\longrightarrow\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,Q),\mathcal{S})$

is fully faithful and its essential image consists of $W$-local objects. Since
by definition $f=\Pi_{\infty}^{\Sigma}(\mathrm{id}_{X})$, 6.8 shows that
$\psi_{f}^{\mathrm{hyp}}$ makes the diagram

$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S})}$$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,Q),\mathcal{S})}$$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)}$$\displaystyle{\mathrm{Cons}_{Q}^{\mathrm{hyp}}(X)}$$\scriptstyle{f^{*}}$$\scriptstyle{\Psi_{X,P}^{\mathrm{hyp}}}$$\scriptstyle{\Psi_{X,Q}^{\mathrm{hyp}}}$$\scriptstyle{\mathrm{id}_{X}^{\ast,\mathrm{hyp}}}$

commutative, while 5.17 shows that the vertical functors are equivalences.
Considering $\mathrm{Cons}_{Q}^{\mathrm{hyp}}(X)$ and
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)$ as full subcategories of
$\mathrm{Sh}^{\mathrm{hyp}}(X)$, we see that $f^{\ast,\mathrm{hyp}}$ acts as
the identity and it is therefore fully faithful. Thus, the commutativity of
the above diagram implies that the functor $f^{*}$ is fully faithful as well.
Since $f$ takes (by definition) arrows in $W$ to equivalences, we see that
$f^{*}$ factors through the full subcategory of $W$-local objects. Thus, we
are left to check the essential surjectivity. Let
$F\colon\Pi_{\infty}^{\Sigma}(X,Q)\to\mathcal{S}$ be a functor and assume that
it is $W$-local. In virtue of 5.17, it is enough to prove that
$\Psi_{X,Q}^{\mathrm{hyp}}(F)$ belongs to
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X)$. Denote by $\varphi\colon Q\to P$ be the
given morphism at the level of posets. Let $a\in P$ be a point and let
$Q_{a}\coloneqq\varphi^{-1}(a)$. Denote by $j_{a}\colon X_{Q_{a}}\to X$ be the
inclusion of the corresponding union of strata. Applying 4.9 we see that the
transformation

$\psi_{j_{a}}^{\mathrm{hyp}}\colon
j_{a}^{\ast,\mathrm{hyp}}\big{(}\Psi_{X,Q}^{\mathrm{hyp}}(F)\big{)}\longrightarrow\Psi_{X_{Q_{a}},Q_{a}}^{\mathrm{hyp}}\big{(}\Pi_{\infty}^{\Sigma}(j_{a})^{\ast}(F)\big{)}$

is an equivalence. Thus, we can replace $X$ by $X_{a}$. Equivalently, we can
assume from the very beginning that $P$ is the trivial stratification (and
therefore that $X$ is locally weakly contractible).

Equipping $*$ with the trivial stratification $S$, we can see the canonical
map $\Gamma_{X}\colon X\to*$ as a map of stratified spaces $(X,Q)\to(*,S)$.
Then 6.8 implies that $\psi_{\Gamma_{X}}^{\mathrm{hyp}}$ yields a canonical
identification

$\Gamma_{X}^{\ast,\mathrm{hyp}}\simeq\Psi_{X,Q}^{\mathrm{hyp}}\circ\Gamma_{\Pi_{\infty}^{\Sigma}(X,Q)}^{\ast}\
.$

Since $X$ is locally weakly contractible, [HPT, § 2.2] shows that
$\Gamma_{X}^{\ast,\mathrm{hyp}}$ has a left adjoint
$\Gamma_{X,\sharp}^{\mathrm{hyp}}$. The above identification implies

$\Gamma_{X,\sharp}^{\mathrm{hyp}}\simeq\Gamma_{\Pi_{\infty}^{\Sigma}(X,Q),!}\circ\Phi_{X,Q}^{\mathrm{hyp}}\
.$

Using [HPT, Corollary 3.5] we find a canonical identification
$\Pi_{\infty}(X)\simeq\Gamma_{X,\sharp}^{\mathrm{hyp}}(\Gamma_{X}^{\ast,\mathrm{hyp}}(*))$.
Thus, the above identification yields:

$\Pi_{\infty}(X)\simeq\Gamma_{X,\sharp}^{\mathrm{hyp}}\Gamma_{X}^{\ast,\mathrm{hyp}}(*)\simeq\operatorname*{colim}_{\Pi_{\infty}^{\Sigma}(X,Q)}\Gamma_{\Pi_{\infty}^{\Sigma}(X,Q)}^{\ast}(*)\simeq\mathrm{Env}(\Pi_{\infty}^{\Sigma}(X,Q))\
,$

whence the conclusion. ∎

We conclude this subsection with 6.13 showing that exodromy implies enhanced
functorialities at the level of hyperconstructible hypersheaves:

######  Observation 6.13.

Let $f\colon(X,P)\to(Y,Q)$ be an exodromic morphism between conically
stratified spaces with locally weakly contractible strata, and let
$\mathcal{E}$ be a presentable $\infty$-category satisfying the assumption of
5.17. The functor $\Pi_{\infty}^{\Sigma}(f)^{\ast}$ has both a left adjoint
given by the left Kan extension $\Pi_{\infty}^{\Sigma}(f)_{!}$ and a right
adjoint given by the right Kan extension $\Pi_{\infty}^{\Sigma}(f)_{\ast}$.
Thus, it follows that the functor

$f^{\ast,\mathrm{hyp}}\colon\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathrm{Cons}_{Q}^{\mathrm{hyp}}(Y;\mathcal{E})$

also has a left and a right adjoint. We set

$f_{\sharp}^{\mathrm{cons}}\coloneqq\Psi_{X,P}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f)_{!}\circ\Phi_{X,P}^{\mathrm{hyp}}\quad\text{and}\quad
f_{\ast}^{\mathrm{cons}}\coloneqq\Psi_{X,P}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f)_{\ast}\circ\Phi_{X,P}^{\mathrm{hyp}}\
.$

Since the functor
$f^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})$
has a right adjoint given by $f_{\ast}$, and the inclusions of
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ and
$\mathrm{Cons}_{Q}^{\mathrm{hyp}}(Y;\mathcal{E})$ have right adjoints
$\mathrm{R}_{P}^{\mathrm{hyp}}$ and $\mathrm{R}_{Q}^{\mathrm{hyp}}$ (see
5.21), it follows from formal reasons that

(6.14) $f_{\ast}^{\mathrm{cons}}\simeq\mathrm{R}_{P}^{\mathrm{hyp}}\circ
f_{\ast}\ .$

When $f^{\ast,\mathrm{hyp}}$ has itself a left adjoint333This is e.g. the case
if $f$ is the projection $S\times X\to S$ for locally weakly contractible
topological spaces, see [HPT, § 2.2]. $f_{\sharp}^{\mathrm{hyp}}$, it follows
similarly that

$f_{\sharp}^{\mathrm{cons}}\simeq\mathrm{L}_{P}^{\mathrm{hyp}}\circ
f_{\sharp}^{\mathrm{hyp}}\ .$

###  6.3. Change of coefficients revisited

######  Lemma 6.15.

Let $(X,P)$ be a conically stratified space. Let $f:\mathcal{E}\to\mathcal{D}$
be a cocontinuous functor between presentable $\infty$-categories with right
adjoint $g:\mathcal{D}\to\mathcal{E}$. Assume that the conditions of 5.17 are
satisfied. Let $p\in P$ and let $i_{p}:X_{p}\to X$ be the inclusion. Then the
following commutative diagram

$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X_{p};\mathcal{D})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X_{p};\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{D})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\
.}$$\scriptstyle{g\circ-}$$\scriptstyle{i_{p,\ast}}$$\scriptstyle{i_{p,\ast}}$$\scriptstyle{g\circ-}$

is verticaly left adjointable on
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})$. That is, for every
$F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})$, the Beck-
Chevalley transformation

$i_{p}^{\ast,\mathrm{hyp}}\circ g(F)\longrightarrow g\circ
i_{p}^{\ast,\mathrm{hyp}}(F)\ $

is an equivalence.

###### Proof.

The question is local on $X$. Hence, we can suppose that $(X,P)$ is of the
form $Z\times C(Y)$ where $Z$ is a locally weakly contractible topological
space and where $(Y,P_{>p})$ is a stratified space. Let $U$ be an open subset
of $Z$. Then, 5.23 gives

$(g\circ
i_{p}^{\ast,\mathrm{hyp}}(F))(U)=g(i_{p}^{\ast,\mathrm{hyp}}(F)(U))\simeq
g(F(U\times C(Y)))$

On the other hand, we have

$(i_{p}^{\ast,\mathrm{hyp}}\circ
g(F))(U)\simeq\operatorname*{colim}_{\varepsilon\in(0,1)}g(F(U\times
C_{\varepsilon}(Y)))$

We know from 5.22 that for every $\varepsilon\in(0,1)$, the restriction
morphism

$F(U\times C(Y))\longrightarrow F(U\times C_{\varepsilon}(Y))$

is an equivalence. Hence, the above colimit is constant and we get

$(i_{p}^{\ast,\mathrm{hyp}}\circ g(F))(U)\simeq g(F(U\times C(Y)))$

6.15 is thus proved. ∎

The following 6.16 contrasts with 2.29.

######  Corollary 6.16.

Let $(X,P)$ be a conically stratified space. Let $f:\mathcal{E}\to\mathcal{D}$
be a cocontinuous functor between presentable $\infty$-categories with right
adjoint $g:\mathcal{D}\to\mathcal{E}$. Assume that the conditions of 5.17 are
satisfied. Then, the following statements hold :

  1. (1)

For every $F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$, the
functor $f\circ F:\mathrm{Open}(X)^{\mathrm{op}}\to\mathcal{D}$ lies in
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})$.

  2. (2)

For every $F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})$, the
functor $g\circ F:\mathrm{Open}(X)^{\mathrm{op}}\to\mathcal{E}$ lies in
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$. In particular, the
adjunction Eq. 2.28 induces an adjunction

$f\circ-\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\leftrightarrows\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})\colon
g\circ-$

  3. (3)

The Exodromy equivalence induces an equivalence of adjunctions

$\displaystyle{f\circ-:\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D}):g\circ-}$$\displaystyle{f\circ-:\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{D}):g\circ-\
.}$$\scriptstyle{\Phi_{X,P}^{\mathrm{hyp},\mathcal{E}}}$$\scriptstyle{\Phi_{X,P}^{\mathrm{hyp},\mathcal{D}}}$

###### Proof.

The first statement follows from 5.17 and 3.9. The statement $(2)$ follows
from 6.15 and 2.30. The statement $(3)$ follows from 3.9. ∎

###  6.4. Monoidal structures

Assume now that our $\infty$-category of coefficients $\mathcal{E}$ carries a
presentably symmetric monoidal structure. In other words, we fix
$\mathcal{E}\in\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L},\otimes})$.

######  Recollection 6.17.

  1. (1)

For every topological space $X$, the $\infty$-category
$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ inherits a canonical symmetric
monoidal structure, and every morphism $f\colon X\to Y$ induces a symmetric
monoidal functor

$f^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\
.$

In particular, it follows that $\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$ is
stable under tensor product, and consequently that for every stratified space
$(X,P)$, $\operatorname{Cons}_{P}^{\mathrm{hyp}}(X,P)$ acquires a symmetric
monoidal structure.

  2. (2)

Similarly,
$\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E}\big{)}$
inherits a symmetric monoidal structure, where the tensor product is computed
objectwise.

######  Proposition 6.18.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}^{\otimes}$ be
a symmetric monoidal $\infty$-category. Then the functor

$\Phi_{X,P}^{\mathrm{hyp},\mathcal{E}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E}\big{)}\
,$

has a natural lax symmetric monoidal structure. Under the assumptions of 5.17,
$\Phi_{X,P}^{\mathrm{hyp},\mathcal{E}}$ restricts to a symmetric monoidal
functor on the full subcategory
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.

###### Proof.

Thanks to 3.8, we immediately reduce to the case where
$\mathcal{E}=\mathcal{S}$. In this case, the monoidal structure induced on
both $\operatorname{Cons}_{P}^{\mathrm{hyp}}(X)$ and
$\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S}\big{)}$ is
the cartesian one. Then $\Phi_{X,P}^{\mathrm{hyp}}$ has a canonical lax
monoidal structure given by [HA, Proposition 2.4.1.7]. Since
$\Phi_{X,P}^{\mathrm{hyp}}$ is an equivalence, it commutes with products and
hence Corollary 2.4.1.8 in _loc. cit._ guarantees that
$\Phi_{X,P}^{\mathrm{hyp}}$ is strong monoidal. The conclusion follows. ∎

As a consequence of our criterion for constructibility (see 5.22), we obtain:

######  Corollary 6.19.

Let $(X,P)$ be a conically stratified space and let
$\mathcal{F},\mathcal{G}\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.
If the assumptions of 5.17 are satisfied, then the internal hom
$\mathcal{H}\mathrm{om}_{X}(\mathcal{F},\mathcal{G})$, computed inside
$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$, belongs to
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.

###### Proof.

We apply 5.22. It is enough to prove that for every inclusion $i\colon
U\subset V$ for which the induced map
$\Pi_{\infty}^{\Sigma}(i)\colon\Pi_{\infty}^{\Sigma}(U,P)\to\Pi_{\infty}^{\Sigma}(V,P)$
is an equivalence, the restriction map

$\mathcal{H}\mathrm{om}_{X}(\mathcal{F},\mathcal{G})(V)\longrightarrow\mathcal{H}\mathrm{om}_{X}(\mathcal{F},\mathcal{G})(U)$

is an equivalence as well. By definition, this map can be rewritten as

(6.20)
$\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(V;\mathcal{E})}(\mathcal{F}|_{V},\mathcal{G}|_{V})\longrightarrow\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(U;\mathcal{E})}(\mathcal{F}|_{U},\mathcal{G}|_{U})\
.$

5.17 allows to write the target of this morphism

$\operatorname{Hom}_{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(U,P),\mathcal{E})}(\Phi_{U,P}^{\mathrm{hyp}}(\mathcal{F}|_{U}),\Phi_{U,P}^{\mathrm{hyp}}(\mathcal{G}|_{U}))\
,$

and similarly for the source. Moreover, since the natural transformation

$\phi^{\mathrm{hyp}}_{i}\colon\Phi_{U,P}^{\mathrm{hyp}}\circ
i^{\ast,\mathrm{hyp}}\longrightarrow\Pi_{\infty}^{\Sigma}(i)^{\ast}\circ\Phi_{V,P}^{\mathrm{hyp}}$

is an equivalence thanks to 6.8, the morphism (6.20) becomes canonically
identified with the morphism induced by $\Pi_{\infty}^{\Sigma}(i)^{\ast}$

$\operatorname{Hom}_{V}(\Phi_{V,P}^{\mathrm{hyp}}(\mathcal{F}|_{V}),\Phi_{V,P}^{\mathrm{hyp}}(\mathcal{G}|_{V}))\longrightarrow\operatorname{Hom}_{U}(\Pi_{\infty}^{\Sigma}(i)^{\ast}\Phi_{V,P}^{\mathrm{hyp}}(\mathcal{F}|_{V}),\Pi_{\infty}^{\Sigma}(i)^{\ast}\Phi_{V,P}^{\mathrm{hyp}}(\mathcal{G}|_{V}))\
,$

where we wrote $\operatorname{Hom}_{V}$ to denote the $\mathcal{E}$-enriched
hom object computed in
$\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(V,P),\mathcal{E})$, and similarly
for $\operatorname{Hom}_{U}$. Since $\Pi_{\infty}^{\Sigma}(i)^{\ast}$ is an
equivalence, so is the above morphism, whence the conclusion. ∎

###  6.5. Compact generation

Using the extra functorialities obtained at the end of the previous
subsection, we are now able to prove the following result, strengthening 6.4:

######  Theorem 6.21.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a compactly generated presentable
$\infty$-category with a single compact generator. Assume that the poset $P$
is finite and that for every $p\in P$, the set $\pi_{0}(X_{p})$ is finite as
well. Then the $\infty$-category
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ has a single compact
generator.

The proof relies on the following two simple observations:

######  Observation 6.22.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $x\in X$ be a point and let $j_{x}\colon\\{x\\}\to X$ be the
natural inclusion. The stratification on $\\{x\\}$ induced by $P$ is the
trivial one. In particular, 6.13 guarantees that the functor

$j_{x}^{\ast,\mathrm{hyp}}\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathcal{E}$

admits a left adjoint $j_{x,\sharp}^{\mathrm{cons}}$ which can be identified
with the left Kan extension along the induced morphism
$\Pi_{\infty}^{\Sigma}(j_{x})\colon\Pi_{\infty}(\\{x\\})\to\Pi_{\infty}^{\Sigma}(X,P)$.
Since $j_{x}^{\ast,\mathrm{hyp}}$ has also a right adjoint, it commutes with
filtered colimits, and hence $j_{x,\sharp}^{\mathrm{cons}}$ preserves
$\kappa$-compact objects for every regular cardinal $\kappa$.

######  Observation 6.23.

Let $X$ be a locally weakly contractible topological space. Assume that $X$ is
connected and let $x\in X$ be a point. Then the induced morphism
$\Pi_{\infty}(\\{x\\})\to\Pi_{\infty}(X)$ is an epimorphism in $\mathcal{S}$.
It follows that for every presentable $\infty$-category $\mathcal{E}$ the
restriction functor

$\Pi_{\infty}(j_{x})^{\ast}:\operatorname{Fun}\big{(}\Pi_{\infty}(X),\mathcal{E}\big{)}\longrightarrow\operatorname{Fun}\big{(}\Pi_{\infty}(\\{x\\}),\mathcal{E}\big{)}\simeq\mathcal{E}$

is conservative. If furthermore $\mathcal{E}$ satisfies the assumptions of
5.17, 6.8 implies that the functor

$j_{x}^{\ast,\mathrm{hyp}}\colon\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathcal{E}$

is conservative.

###### Proof of 6.21.

Since $\mathcal{E}$ is compactly generated, 5.18 ensures that the
hyperrestriction functors

$\left\\{j_{p}^{\ast,\mathrm{hyp}}\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\to\mathrm{LC}^{\mathrm{hyp}}(X_{p};\mathcal{E})\right\\}_{p\in
P}$

are jointly conservative. Furthermore, 6.13 guarantees that they have left
adjoints $j_{p,\sharp}^{\mathrm{cons}}$. Therefore, their product

$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\prod_{p\in
P}\mathrm{LC}^{\mathrm{hyp}}(X_{p};\mathcal{E})\ ,$

is conservative and admits a left adjoint. Furthermore, it commutes with
(filtered) colimits. Thus, [AGV, Lemma 2.8.3] reduces us to check that the
target has a single compact generator. Since this properties are closed under
finite products, it is enough to consider the case of a single stratum.

When the stratification on $X$ is trivial, our assumption guarantees that the
number of connected components of $X$ is finite. Hence, we are left to treat
the case where $X$ is connected. Let then $x\in X$ be a point and consider the
functor

$j_{x}^{\ast,\mathrm{hyp}}\colon\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathcal{E}\
.$

It follows from 6.23 that this functor is conservative, while 6.22 shows that
it has a left adjoint. Since it also commutes with filtered colimits, using
once more [AGV, Lemma 2.8.3] we see that if $\\{E_{i}\\}_{i\in I}$ is a family
of compact generators for $\mathcal{E}$, then
$\big{\\{}j_{x,\sharp}^{\mathrm{cons}}(E_{i})\big{\\}}_{i\in I}$ is a family
of compact generators for $\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$. In
particular, since $\mathcal{E}$ has a single compact generator, then so does
$\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$. ∎

For future reference, let us record the following more specific statement that
can be deduced from the previous proof:

######  Corollary 6.24.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a compactly generated presentable
$\infty$-category with compact generators $\\{E_{\alpha}\\}_{\alpha\in J}$.
Assume that the poset $P$ is finite and that for every $p\in P$, the set
$\pi_{0}(X_{p})$ is finite as well. Then

$\left\\{j_{x_{i},\sharp}^{\mathrm{cons}}(E_{\alpha})\right\\}_{i\in
I,\alpha\in J}$

is a system of compact generators for
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.

This result is already interesting in the case of the trivial stratification:

######  Example 6.25.

Let $X$ be a connected, locally weakly contractible topological space and let
$x\in X$ be a point. Let $\mathcal{E}$ be a presentable $\infty$-category with
a single compact generator $E\in\mathcal{E}$ (e.g.
$\mathcal{E}=\mathrm{Mod}_{A}$ for some $\mathbb{E}_{\infty}$-algebra $A$, and
$E=A$). Then $j_{x,\sharp}(A)\in\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$ is
a single compact generator.

As a consequence, we can express $\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$
as the $\infty$-category of modules over the endomorphism ring of
$j_{x,\sharp}(A)$. It turns out that it is possible to compute quite
explicitly such endomorphism ring:444We warmly thank Julian V. Holstein and
Alexandru Oancea for discussions on this point.

######  Theorem 6.26.

Let $X$ be a connected and locally weakly contractible topological space. Fix
a point $x\in X$ and consider the $x$-based loop space $\Omega_{x}(X)$. Then
for every $\mathbb{E}_{\infty}$-algebra $A$, the endomorphism ring of
$j_{x,\sharp}(A)$ canonically coincides with the chain algebra
$\mathrm{C}_{\ast}(\Omega_{x}(X);A)$. In particular, there is an equivalence

$\mathrm{LC}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\simeq\mathrm{Mod}_{\mathrm{C}_{\ast}(\Omega_{x}(X);A)}\
.$

######  Remark 6.27.

This theorem recovers [Ha, Theorem 26] and [Hb, Theorem 12]. Besides it
generalizes it in several directions, as $A$ is now allowed to be an
$\mathbb{E}_{\infty}$-algebra, and $X$ is now only required to be locally
weakly contractible rather than admitting a bounded locally finite good
hypercover. In particular, the above theorem holds even without any
paracompactness assumption on $X$.

###### Proof of 6.26.

The second half is a direct consequence of the first half and Lurie-Barr-
Beck’s theorem [HA, Theorem 4.7.3.5]. To compute the endomorphism ring of
$j_{x,\sharp}(A)$, we first observe that there is the following natural
equivalence:

$\operatorname{Hom}_{\mathrm{LC}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}\big{(}j_{x,\sharp}^{\mathrm{hyp}}(A),j_{x,\sharp}^{\mathrm{hyp}}(A)\big{)}\simeq
j_{x}^{\ast,\mathrm{hyp}}j_{x,\sharp}^{\mathrm{hyp}}(A)\ .$

Observe now that the square

$\displaystyle{\Pi_{\infty}(\Omega_{x}(X))}$$\displaystyle{*}$$\displaystyle{*}$$\displaystyle{\Pi_{\infty}(X)\
.}$$\scriptstyle{\Gamma}$$\scriptstyle{\Gamma}$$\scriptstyle{j_{x}}$$\scriptstyle{j_{x}}$

is a pullback in $\mathcal{S}$, where we wrote $\Gamma$ instead of
$\Gamma_{\Omega_{x}(X)}$ for brevity. Unraveling the definitions, this implies
that $\Pi_{\infty}(\Omega_{x}(X))$ coincides with the comma category

$\\{x\\}\times_{\Pi_{\infty}(X)}\Pi_{\infty}(X)_{j_{x}/}\ ,$

so that combining 3.32 with the formula for the left Kan extension, we deduce
that the Beck-Chevalley transformation

$\Gamma_{\sharp}^{\mathrm{hyp}}\circ\Gamma^{\ast,\mathrm{hyp}}\longrightarrow
j_{x}^{\ast,\mathrm{hyp}}\circ j_{x,\sharp}^{\mathrm{hyp}}$

is an equivalence. Thus, the endomorphism ring of $j_{x,\sharp}(A)$ is
canonically identified with

$\Gamma_{\sharp}^{\mathrm{hyp}}\Gamma^{\ast,\mathrm{hyp}}(A)\simeq\Gamma_{\sharp}^{\mathrm{hyp}}\Gamma^{\ast,\mathrm{hyp}}(*)\otimes
A\ .$

By definition,
$\Gamma_{\sharp}^{\mathrm{hyp}}\Gamma^{\ast,\mathrm{hyp}}(*)\in\mathcal{S}$ is
the shape of the $\infty$-topos $\mathrm{Sh}^{\mathrm{hyp}}(\Omega_{x}(X))$.
Applying [HPT, Corollary 3.5], we see that this shape is simply identified
with the homotopy type $\Pi_{\infty}(\Omega_{x}(X))$. Thus, the endomorphism
ring of $j_{x,\sharp}(A)$ is identified with

$\Gamma_{\sharp}^{\mathrm{hyp}}\Gamma^{\ast,\mathrm{hyp}}(A)\simeq\Pi_{\infty}(\Omega_{x}(X))\otimes
A\simeq\mathrm{C}_{\ast}(\Omega_{x}(X);A)\ ,$

where the last equivalence holds by definition of singular chains. ∎

######  Remark 6.28.

It is possible to obtain a similar description in the stratified case, in line
with [HL, §6]. It should be possible to obtain an explicit description of the
endomorphism ring of the single compact generator provided by 6.21 akin to the
one of 6.26, at least in the setting of conically smooth stratified spaces.
Indeed, any such description should see the chain algebras on single strata,
but at the same time, it should also have a contribution from the links of the
stratification. We will come back to this subject in a later work.

###  6.6. Exodromy and stalkwise compactness

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata and let $\mathcal{E}$ be a presentable $\infty$-category. The
construction of $\Phi_{X,P}^{\mathrm{hyp}}$ and $\Psi_{X,P}^{\mathrm{hyp}}$
relies a priori on the existence of limits and colimits of diagrams that are
typically not finite. However, the “regularity” of conical charts paired with
the homotopy-invariance property of hyperconstructible hypersheaves [HPT,
Theorem 0.4] actually implies that all infinite colimits that are involved in
the construction of the exodromy adjunction can be ignored. To give a proper
formulation of this idea, let us first introduce the following notation:

######  Notation 6.29.

Let $(X,P)$ be a stratified space and let $\mathcal{E}$ be a presentable
$\infty$-category. Let $\kappa$ be a regular cardinal. We denote by
$\mathrm{Cons}_{P,\kappa}^{\mathrm{hyp}}(X;\mathcal{E})$ the full subcategory
of $\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ spanned by
$\mathcal{E}$-valued hyperconstructible hypersheaves $F$ whose stalks are
$\kappa$-compact objects of $\mathcal{E}$. When the stratification $P$ is
trivial, we denote this $\infty$-category by
$\mathrm{LC}_{\kappa}^{\mathrm{hyp}}(X;\mathcal{E})$.

######  Warning 6.30.

The $\infty$-category $\mathrm{Cons}_{P,\kappa}^{\mathrm{hyp}}(X;\mathcal{E})$
does not coincide neither with
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E}^{\kappa})$ nor with
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})^{\kappa}$. We offer two
counterexamples:

  1. (1)

Take $X\coloneqq\coprod_{\mathbb{N}}*$ to be an infinite disjoint union of
points equipped with the trivial stratification. Take also
$\mathcal{E}=\mathcal{S}$ and $\kappa=\omega$. Fix $K\in\mathcal{S}^{\kappa}$
and let $F\coloneqq\Gamma_{X}^{\ast}(K)$ be the hyperconstant hypersheaf
associated to $K$. Then
$F\in\mathrm{Cons}_{P,\kappa}^{\mathrm{hyp}}(X;\mathcal{E})$, but
$F(X)\simeq\prod_{\mathbb{N}}K$, which does not belong to
$\mathcal{S}^{\omega}$ unless $K$ is contractible. In particular, $F$ does not
belong to $\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{S}^{\omega})$.

  2. (2)

Take $X=S^{1}$, $\mathcal{E}=\mathrm{Mod}_{\mathbb{C}}$ and $\kappa=\omega$.
Once again, equip $X$ with the trivial stratification. Then, we have canonical
identifications

$\mathrm{LC}^{\mathrm{hyp}}(S^{1};\mathrm{Mod}_{\mathbb{C}})\simeq\operatorname{Fun}(\Pi_{\infty}(S^{1}),\mathrm{Mod}_{\mathbb{C}})\simeq\mathrm{QCoh}(\mathbb{G}_{m,\mathbb{C}})\
.$

Therefore,

$\mathrm{LC}^{\mathrm{hyp}}(S^{1};\mathrm{Mod}_{\mathbb{C}})^{\omega}\simeq\mathrm{Perf}(\mathbb{G}_{m,\mathbb{C}})\
.$

In particular, the structure sheaf of $\mathbb{G}_{m,\mathbb{C}}$ corresponds
to the functor $F\colon S^{1}\to\mathrm{Mod}_{\mathbb{C}}$ selecting
$\mathbb{C}[T,T^{-1}]$ with endomorphism given by multiplication by $T$. Then
6.10 implies that the stalk of $\Psi_{S^{1}}^{\mathrm{hyp}}(F)$ at any point
of $S^{1}$ coincides with $\mathbb{C}[T,T^{-1}]$, which is not compact as an
object in $\mathrm{Mod}_{\mathbb{C}}$.

######  Remark 6.31.

In the second example above, it follows from [BZNP] that
$\mathrm{LC}_{\kappa}^{\mathrm{hyp}}(S^{1};\mathrm{Mod}_{\mathbb{C}})$
corresponds to the full subcategory of
$\mathrm{QCoh}(\mathbb{G}_{m,\mathbb{C}})$ spanned by perfect complexes _with
proper support_.

######  Proposition 6.32.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. Let $\kappa$ be a regular cardinal. If the
assumptions of 5.17 are satisfied, then the exodromy equivalence
$\Phi_{X,P}^{\mathrm{hyp}}\dashv\Psi_{X,P}^{\mathrm{hyp}}$ restricts to an
equivalence

$\Phi_{X,P}^{\mathrm{hyp}}\colon\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E}^{\kappa})\leftrightarrows\mathrm{Cons}_{P,\kappa}^{\mathrm{hyp}}(X;\mathcal{E})\colon\Psi_{X,P}^{\mathrm{hyp}}\
.$

###### Proof.

It is enough to prove that both $\Phi_{X,P}^{\mathrm{hyp}}$ and
$\Psi_{X,P}^{\mathrm{hyp}}$ respect these two full subcategories. Let first
$F\in\mathrm{Cons}_{P,\kappa}^{\mathrm{hyp}}(X;\mathcal{E})$. Then we have to
check that the functor $\Phi_{X,P}^{\mathrm{hyp}}(F)$ takes values in
$\mathcal{E}^{\kappa}$. To see this, let $x\in\Pi_{\infty}^{\Sigma}(X,P)$.
Then 6.10 provides a canonical identification

$\Phi_{X,P}^{\mathrm{hyp}}(F)(x)\simeq F_{x}$

so that $\Phi_{X,P}^{\mathrm{hyp}}(F)$ belongs to
$\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E}^{\kappa})$.

Let now $G\colon\Pi_{\infty}^{\Sigma}(X,P)\to\mathcal{E}^{\kappa}$ be a
functor. Let $x\in X$. Then 6.10 and 5.17 provide the following canonical
identifications:

$\Psi_{X,P}^{\mathrm{hyp}}(G)_{x}\simeq\big{(}\Phi_{X,P}^{\mathrm{hyp}}\Psi_{X,P}^{\mathrm{hyp}}(G)\big{)}(x)\simeq
G(x)\ .$

Thus, the stalks of $\Psi_{X,P}^{\mathrm{hyp}}(G)$ belong to
$\mathcal{E}^{\kappa}$. In other words, $\Psi_{X,P}^{\mathrm{hyp}}(G)$ belongs
to $\mathrm{Cons}_{P,\kappa}^{\mathrm{hyp}}(X;\mathcal{E})$. ∎

###  6.7. Constructibility and pushforward. A general criterion

Let $f\colon(Y,Q)\to(X,P)$ be an exodromic morphism of conically stratified
spaces. Then the natural transformation $\psi_{f}^{\mathrm{hyp}}$ makes the
square

$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(Y,Q),\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})}$$\scriptstyle{\Pi_{\infty}^{\Sigma}(f)^{\ast}}$$\scriptstyle{\Psi_{X,P}^{\mathrm{hyp}}}$$\scriptstyle{\Psi_{Y,Q}^{\mathrm{hyp}}}$$\scriptstyle{f^{\ast,\mathrm{hyp}}}$

commutative. In particular, there is an associated Beck-Chevalley
transformation

$\chi_{f}^{\mathrm{hyp}}\colon\Psi_{X,P}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f)_{\ast}\longrightarrow
f_{\ast}\circ
f^{\ast,\mathrm{hyp}}\circ\Psi_{X,P}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f)_{\ast}\xrightarrow{\psi_{f}^{\mathrm{hyp}}}f_{\ast}\circ\Psi_{Y,Q}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f)^{\ast}\circ\Pi_{\infty}^{\Sigma}(f)_{\ast}\longrightarrow
f_{\ast}\circ\Psi_{Y,Q}^{\mathrm{hyp}}\ .$

The identification (6.14) has the following immediate consequence:

######  Lemma 6.33.

Let $f\colon(Y,Q)\to(X,P)$ be an exodromic morphism of conically stratified
spaces with locally weakly contractible strata. Let $\mathcal{E}$ be a
presentable $\infty$-category and assume that the conditions of 5.17 are
satisfied. Then for every $F\colon\Pi_{\infty}^{\Sigma}(Y,Q)\to\mathcal{E}$,
the following statements are equivalent:

  1. (1)

the hypersheaf
$f_{\ast}\big{(}\Psi_{Y,Q}^{\mathrm{hyp}}(F)\big{)}\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
is hyperconstructible on $(X,P)$;

  2. (2)

the transformation
$\chi_{f}^{\mathrm{hyp}}(F)\colon\Psi_{X,P}^{\mathrm{hyp}}\big{(}\Pi_{\infty}^{\Sigma}(f)_{\ast}(F)\big{)}\to
f_{\ast}\big{(}\Psi_{Y,Q}^{\mathrm{hyp}}(F)\big{)}$ is an equivalence.

The following observation gives a convenient sufficient condition to check
whether pushforward along a morphism $f\colon(Y,Q)\to(X,P)$ preserves
hyperconstructible hypersheaves:

######  Lemma 6.34.

Let $f\colon(Y,Q)\to(X,P)$ be a strongly exodromic morphism of conically
stratified spaces with locally weakly contractible strata. Let $\mathcal{E}$
be a presentable $\infty$-category and assume that the conditions of 5.17 are
satisfied. Then $f_{\ast}$ preserves hyperconstructible hypersheaves.

###### Proof.

By assumption, the natural transformation $\phi_{f}^{\mathrm{hyp}}$ makes the
diagram

$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(Y,Q),\mathcal{E})}$$\scriptstyle{f^{\ast,\mathrm{hyp}}}$$\scriptstyle{\Phi_{X,P}^{\mathrm{hyp}}}$$\scriptstyle{\Phi_{Y,Q}^{\mathrm{hyp}}}$$\scriptstyle{\Pi_{\infty}^{\Sigma}(f)^{\ast}}$

commutative. Passing to right adjoints, we deduce that the natural
transformation $\chi_{f}^{\mathrm{hyp}}$ is an equivalence. Then 6.34 follows
from 4.10. ∎

###  6.8. Constructibility and pushforward along immersions

######  Proposition 6.35.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. Let $S\subseteq P$ be a locally closed subset
of $P$ and let $i_{S}\colon X_{S}\to X$ be the inclusion of the corresponding
union of strata. If the assumption of 5.17 are satisfied, then

$i_{S,\ast}\colon\mathrm{Sh}^{\mathrm{hyp}}(X_{S};\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$

takes hyperconstructible hypersheaves on $(X_{S},S)$ to hyperconstructible
hypersheaves on $(X,P)$.

###### Proof.

Combine Corollaries 5.8 and 5.15 with 6.34. ∎

6.35 has the following two immediate corollaries below which are true for
hyperconstructible hypersheaves since they are already true for hypersheaves.

######  Corollary 6.36.

Let $(X,P)$ be a conically stratified space and let $S\subseteq P$ be a
locally closed subset. Let $\mathcal{E}$ and $\mathcal{D}$ be presentable
$\infty$-categories and let $f\colon\mathcal{E}\to\mathcal{D}$ be a
cocontinuous functor. If the assumption of 5.17 are satisfied, then the
diagram

$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\mathrm{Cons}_{S}^{\mathrm{hyp}}(X_{S};\mathcal{E})}$$\displaystyle{\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})}$$\displaystyle{\mathrm{Cons}_{S}^{\mathrm{hyp}}(X_{S};\mathcal{D})}$$\scriptstyle{i_{S}^{\ast,\mathrm{hyp}}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{i_{S}^{\ast,\mathrm{hyp}}}$

is horizontally right adjointable. Than is, the Beck-Chevalley transformation

$f\circ i_{S,\ast}^{\mathrm{hyp}}\longrightarrow
i_{S,\ast}^{\mathrm{hyp}}\circ f$

is an equivalence.

For the notion of recollement of $\infty$-categories, let us refer to [HA,
A.8.1].

######  Corollary 6.37.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. Let $S\subseteq P$ be a closed subset and let
$i_{S}\colon X_{S}\to X$ be the inclusion. Put $U:=P\setminus S$ and let
$i_{U}\colon X_{U}\to X$ be the inclusion. If the assumptions of 5.17 are
satisfied, then the fully-faithful functors

$i_{S,\ast}\colon\mathrm{Cons}_{S}^{\mathrm{hyp}}(X_{S};\mathcal{E})\longrightarrow\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\longleftarrow\mathrm{Cons}_{U}^{\mathrm{hyp}}(X_{U};\mathcal{E})\colon
i_{U,\ast}$

exhibits $\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ as a recollement of
$\mathrm{Cons}_{S}^{\mathrm{hyp}}(X_{S};\mathcal{E})$ and
$\mathrm{Cons}_{U}^{\mathrm{hyp}}(X_{U};\mathcal{E})$.

###### Proof.

6.37 follows immediately from the fact that the fully-faithful functors

$i_{S,\ast}\colon\mathrm{Sh}^{\mathrm{hyp}}(X_{S};\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\longleftarrow\mathrm{Sh}^{\mathrm{hyp}}(X_{U};\mathcal{E})\colon
i_{U,\ast}$

exhibits $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ as a recollement of
$\mathrm{Sh}^{\mathrm{hyp}}(X_{S};\mathcal{E})$ and
$\mathrm{Sh}^{\mathrm{hyp}}(X_{U};\mathcal{E})$. ∎

As for the hyperrestriction of a hyperconstructible hypersheaf to a stratum
computed in 5.23, one can compute the fibre of $i_{S,\ast}$ at a point $x$
explicitly in terms of a conical chart containing $x$. This is the following

######  Lemma 6.38.

Let $Z$ be a weakly contractible locally weakly contractible topological space
and let $(Y,Q)$ be a stratified space such that $\mathrm{Exit}(Y,Q)$ is an
$\infty$-category and $(X,P)\coloneqq(Z\times C(Y),Q^{\vartriangleleft})$ is
conically stratified. Let $\mathcal{E}$ be a presentable $\infty$-category.
Let $F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ and let $x\in
Z$. If the assumptions of 5.17 are satisfied, there is a canonical
identification

$(i_{Q,\ast}F)_{x}\simeq\lim_{\mathrm{Exit}(Y,Q)}\Phi_{X,P}^{\mathrm{hyp}}(F)|_{\mathrm{Exit}(Y,Q)}$

where in the right-hand side, the restriction is performed along the
equivalence $\mathrm{Exit}(Z\times\mathbb{R}_{>0}\times
Y,Q)\to\mathrm{Exit}(Y,Q)$ induced by the canonical projection.

###### Proof.

Write $\mathcal{B}_{x}$ for the collection of weakly contractible open
neighborhoods of $x$ inside $Z$. Then

$i_{Q,\ast}(F)_{x}\simeq\operatorname*{colim}_{U\in\mathcal{B}_{x}}\operatorname*{colim}_{\varepsilon\in(0,1)}F(U\times(0,\varepsilon)\times
Y)\ .$

From 5.17 and Eq. 3.13, we deduce

$i_{Q,\ast}(F)_{x}\simeq\operatorname*{colim}_{U\in\mathcal{B}_{x}}\operatorname*{colim}_{\varepsilon\in(0,1)}\lim_{\mathrm{Exit}(U\times(0,\epsilon)\times
Y,Q)}\Phi_{X,P}^{\mathrm{hyp}}(F)\ .$

Since the $U\in\mathcal{B}_{x}$ are weakly contractible, we have further

$\mathrm{Exit}(U\times(0,\epsilon)\times
Y,Q)\simeq\mathrm{Exit}(U,\ast)\times\mathrm{Exit}((0,\epsilon),\ast)\times\mathrm{Exit}(Y,Q)\simeq\mathrm{Exit}(Y,Q)\
.$

Hence, the above colimits are constant and 6.38 thus follows. ∎

In view of the construction of moduli carried out in Section 7, we need to
understand the interaction of push-forward with proper hyperconstructible
hypersheaves. To this end, additional finiteness assumptions on $(X,P)$ will
be needed. They will be justified by the following general

######  Lemma 6.39.

Let $\mathcal{C}\in\mathrm{Cat}_{\infty}^{\omega}$ be a compact object in
$\mathrm{Cat}_{\infty}$ and let $\mathcal{E}$ be a presentable
$\infty$-category. Assume that filtered colimits are left exact in
$\mathcal{E}$. Then the functor

$\Gamma_{\mathcal{C},\ast}\colon\operatorname{Fun}(\mathcal{C},\mathcal{E})\longrightarrow\mathcal{E}$

commutes with filtered colimits. When $\mathcal{E}$ is stable, both the left
adjoint $\Gamma_{\mathcal{C},!}$ and the right adjoint
$\Gamma_{\mathcal{C},\ast}$ to $\Gamma_{\mathcal{C}}^{\ast}$ restrict to
functors

$\Gamma_{\mathcal{C},!},\
\Gamma_{\mathcal{C},\ast}\colon\operatorname{Fun}(\mathcal{C},\mathcal{E}^{\omega})\longrightarrow\mathcal{E}^{\omega}\
.$

###### Proof.

The functor $\Gamma_{\mathcal{C}}^{\ast}$ takes $\mathcal{E}^{\omega}$ in
$\operatorname{Fun}(\mathcal{C},\mathcal{E}^{\omega})$ by definition. It is
therefore enough to prove that $\Gamma_{\mathcal{C},\ast}$ commutes with
filtered colimits and that in the stable case it restricts it takes
$\operatorname{Fun}(\mathcal{C},\mathcal{E}^{\omega})$ to
$\mathcal{E}^{\omega}$. Let $\mathcal{T}_{0}(\mathcal{E})$ be the full
subcategory of $\mathrm{Cat}_{\infty}$ spanned by those $\infty$-categories
$\mathcal{C}$ for which the functor $\Gamma_{\mathcal{C},\ast}$ commutes with
filtered colimits. Let $\mathcal{T}(\mathcal{E})$ be the full subcategory of
$\mathrm{Cat}_{\infty}$ spanned by those $\infty$-categories $\mathcal{C}$ for
which the functors $\Gamma_{\mathcal{C},!}$ and $\Gamma_{\mathcal{C},\ast}$
take $\operatorname{Fun}(\mathcal{C},\mathcal{E}^{\omega})$ to
$\mathcal{E}^{\omega}$. Inspection reveals that $\mathcal{T}(\mathcal{E})$ is
closed under finite colimits, retractions and contain all finite
$1$-categories (that is, $1$-categories having a finite number of objects and
of morphisms). Since $\mathrm{Cat}_{\infty}$ is compactly generated by the
$1$-categories $\Delta^{n}$, the conclusion follows. ∎

######  Definition 6.40.

We say that a stratified space $(X,P)$ is _categorically compact_ if
$\mathrm{Exit}(X,P)$ is a compact object in $\mathrm{Cat}_{\infty}$.

######  Definition 6.41.

Let $(X,P)$ be a conically stratified space. We say that $(X,P)$ is _locally
categorically compact_ if for every $p\in P$ and every $x\in X_{p}$, there
exists a conical chart of the form $Z\times C(Y)$ containing $x$ such that
$(Y,P_{>p})$ is categorically compact and $Z$ is weakly contractible and
locally weakly contractible.

######  Proposition 6.42.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that $(X,P)$ is locally categorically compact and that $P$ is
finite. Let $S\subseteq P$ be a locally closed subset and let $\mathcal{E}$ be
a stable presentable $\infty$-category. Then the functor

$i_{S,\ast}\colon\mathrm{Cons}_{S}^{\mathrm{hyp}}(X_{S};\mathcal{E})\longrightarrow\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$

commutes with filtered colimits and it restricts to a functor

$i_{S,\ast}\colon\mathrm{Cons}_{S,\omega}^{\mathrm{hyp}}(X_{S};\mathcal{E})\longrightarrow\mathrm{Cons}_{P,\omega}^{\mathrm{hyp}}(X;\mathcal{E})\
.$

###### Proof.

When $S$ is closed inside $P$, the functor $i_{S,\ast}$ coincides with the
extension by zero and 6.42 is trivial in that case. It is then enough to
consider the case where $S$ is open inside $P$. We proceed by induction on
$\mathrm{depth}(P)$. When $\mathrm{depth}(P)=0$, the stratification is trivial
and therefore we have either $S=\emptyset$ or $S=P$. In both cases, the
statement is obvious.

Assume now that $\mathrm{depth}(X)>0$. Since $P$ is finite, the subset
$M\subseteq P$ of its minimal elements is finite as well. Writing
$M=\\{p_{1},\ldots,p_{n}\\}$, we see that $\\{X_{\geqslant
p_{i}}\\}_{i=1,\ldots,n}$ is an open cover of $X$. Since both the compacity of
stalks and the formation of filtered colimits are local statements on $X$, we
can assume that $P$ has a minimum $p$. Using the inductive hypothesis, we
further reduce to the case where $S=P\smallsetminus M$.

Let now $F\in\mathrm{Cons}_{S,\omega}^{\mathrm{hyp}}(X_{S};\mathcal{E})$. We
have to prove that the stalks of $i_{S,\ast}(F)$ belong to
$\mathcal{E}^{\omega}$. Since $i_{S,\ast}$ is fully faithful, we only have to
prove this statement for the stalks at a point $x\in X_{p}$. Choose a conical
chart of the form $Z\times C(Y)$, where $(Y,S)$ is a categorically compact
stratified space and where $Z$ is weakly contractible and locally weakly
contractible. From 6.38, we have a canonical identification

$(i_{S,\ast}F)_{x}\simeq\lim_{\mathrm{Exit}(Y,S)}\Phi_{X,P}^{\mathrm{hyp}}(F)|_{\mathrm{Exit}(Y,S)}$

Since $\mathrm{Exit}(Y,S)$ is a compact object of $\mathrm{Cat}_{\infty}$ and
since the functor $\Phi_{X,P}^{\mathrm{hyp}}(F)$ takes values in
$\mathcal{E}^{\omega}$, 6.42 thus follows from 6.39. The same method also
guarantees that $i_{S,\ast}$ commutes with filtered colimits. ∎

6.37 immediately gives the following

######  Corollary 6.43.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. Let $S\subseteq P$ be a closed subset and let
$i_{S}\colon X_{S}\to X$ be the inclusion. Put $U:=P\setminus S$ and let
$i_{U}\colon X_{U}\to X$ be the inclusion. If the assumptions of 5.17 are
satisfied, then the fully-faithful functors

$i_{S,\ast}\colon\mathrm{Cons}_{S,\omega}^{\mathrm{hyp}}(X_{S};\mathcal{E})\longrightarrow\mathrm{Cons}_{P,\omega}^{\mathrm{hyp}}(X;\mathcal{E})\longleftarrow\mathrm{Cons}_{U,\omega}^{\mathrm{hyp}}(X_{U};\mathcal{E})\colon
i_{U,\ast}$

exhibits $\mathrm{Cons}_{P,\omega}^{\mathrm{hyp}}(X;\mathcal{E})$ as a
recollement of $\mathrm{Cons}_{S,\omega}^{\mathrm{hyp}}(X_{S};\mathcal{E})$
and $\mathrm{Cons}_{U,\omega}^{\mathrm{hyp}}(X_{U};\mathcal{E})$.

6.36 and 6.42 immediately gives the following

######  Corollary 6.44.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that $(X,P)$ is locally categorically compact and that $P$ is
finite. Let $S\subseteq P$ be a locally closed subset and let $\mathcal{E}$
and $\mathcal{D}$ stable presentable $\infty$-categories. Let
$f\colon\mathcal{E}\to\mathcal{D}$ be a cocontinuous functor preserving
compact objects. Then, the diagram

$\displaystyle{\mathrm{Cons}_{P,\omega}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\mathrm{Cons}_{S,\omega}^{\mathrm{hyp}}(X_{S};\mathcal{E})}$$\displaystyle{\mathrm{Cons}_{P,\omega}^{\mathrm{hyp}}(X;\mathcal{D})}$$\displaystyle{\mathrm{Cons}_{S,\omega}^{\mathrm{hyp}}(X_{S};\mathcal{D})}$$\scriptstyle{i_{S}^{\ast,\mathrm{hyp}}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{i_{S}^{\ast,\mathrm{hyp}}}$

is horizontally right adjointable. Than is, the Beck-Chevalley transformation

$f\circ i_{S,\ast}^{\mathrm{hyp}}\longrightarrow
i_{S,\ast}^{\mathrm{hyp}}\circ f$

is an equivalence.

###  6.9. Constructibility, pushforward and weakly stratified bundles

We now present a second result concerning proper pushforward. Lurie proved
proper non abelian base change for sheaves on locally compact Hausdorff
spaces. See [HTT, Corollary 7.3.1.18]. Note however that it is not clear that
proper base change holds for hypersheaves. We are going to see that under some
additional assumptions on the stratifications involved, proper base change
holds for _hyperconstructible_ hypersheaves.

######  Observation 6.45.

Let $f\colon Y\to X$ be a proper morphism between topological spaces.
Following [HTT, Definition 7.3.1.14], this means for us that $f$ is
universally closed. Let $C\subseteq X$ be a locally closed subset of $X$.
Write $\mathcal{B}_{C}$ for the collection of open neighborhoods of $C$ inside
$X$. Then $\\{f^{-1}(U)\\}_{U\in\mathcal{B}_{C}}$ is a fundamental system of
open neighborhoods for the inverse image $f^{-1}(C)$. Indeed, since $f$ is
universally closed we can localize on $X$ and therefore assume that $C$ is
closed. Let now $V$ be an open neighborhood of $f^{-1}(C)$ inside $Y$. Since
$f$ is closed, $f(Y\smallsetminus V)$ is a closed subset of $X$. Furthermore
$f(Y\smallsetminus V)\cap C=\emptyset$, so $U\coloneqq X\smallsetminus
f(Y\smallsetminus V)$ is an open neighborhood of $C$ inside $X$. We now
observe that if $y\in f^{-1}(U)$, then $f(y)\notin f(Y\smallsetminus V)$,
which in turn implies that $y\in V$. Therefore, $f^{-1}(U)\subseteq V$.

Before proving the sought after proper base change for hyperconstructible
hypersheaves, we need a strengthening of the notion of excellency introduced
in 2.18.

######  Definition 6.46.

Let $(X,P)$ be a conically stratified space and let $S\subseteq P$ be a
subset. We say that $(X,P)$ is _hereditary excellent at $S$_ if for every open
subset $U\subseteq X$, the stratified space $(U,P)$ is excellent at $S$.

######  Proposition 6.47.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces with
locally weakly contractible strata and let $S\subseteq P$ be a locally closed
subset. Let $\varphi\colon Q\to P$ be the underlying morphism of posets and
set $R\coloneqq\varphi^{-1}(S)$. Consider the induced commutative square

$\displaystyle{(Y_{R},R)}$$\displaystyle{(Y,Q)}$$\displaystyle{(X_{S},S)}$$\displaystyle{(X,P)\
.}$$\scriptstyle{j}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{i}$

Assume that:

  1. (1)

the underlying morphism $f\colon Y\to X$ is proper;

  2. (2)

$(Y,Q)$ is hereditary excellent at $R$.

Then for every presentable $\infty$-category $\mathcal{E}$ satisfying the
assumptions of 5.17 and every
$F\in\mathrm{Cons}_{Q}^{\mathrm{hyp}}(Y;\mathcal{E})$ the canonical map

$i^{\ast,\mathrm{hyp}}(f_{\ast}(F))\longrightarrow
g_{\ast}(j^{\ast,\mathrm{hyp}}(F))$

is an equivalence.

###### Proof.

The statement is local on $X$. We can therefore suppose that $S$ is a closed
downwards subset of $P$. Since source and target are hypersheaves, it is
enough to prove that for every open subset $U$ of $X_{S}$ the induced morphism

$i^{-1}(f_{\ast}(F))(U)\longrightarrow g_{\ast}(j^{\ast,\mathrm{hyp}}(F))(U)$

is an equivalence. Let $\mathcal{B}_{U}$ for the collection of open
neighborhoods of $U$ inside $X$. Since $f$ is proper and $U$ is locally closed
inside $X$, 6.45 shows that $\\{f^{-1}(V)\\}_{V\in\mathcal{B}_{U}}$ is a
fundamental system of open neighborhoods of $g^{-1}(U)=f^{-1}(U)$ inside $Y$.
Since $(Y,Q)$ is hereditary excellent at $R$, the collection
$\mathcal{B}_{g^{-1}(U)}^{\mathrm{exc}}$ of excellent open neighborhoods of
$g^{-1}(U)$ inside $Y$ is a fundamental system of open neighborhoods for
$g^{-1}(U)$. Thus, 5.17 and 3.3 provide the following chain of natural
equivalences:

$i^{-1}(f_{\ast}(F))(U)\simeq\operatorname*{colim}_{V\in\mathcal{B}_{U}}F(f^{-1}(V))\simeq\operatorname*{colim}_{W\in\mathcal{B}_{g^{-1}(U)}^{\mathrm{exc}}}F(W)\simeq\operatorname*{colim}_{W\in\mathcal{B}_{g^{-1}(U)}^{\mathrm{exc}}}\lim_{\Pi_{\infty}^{\Sigma}(W,Q)}\Phi_{Y,Q}^{\mathrm{hyp}}(F)\
.$

Since each $W\in\mathcal{B}_{g^{-1}(U)}^{\mathrm{exc}}$ is excellent at $R$,
the functor $\Pi_{\infty}^{\Sigma}(g^{-1}(U),R)\to\Pi_{\infty}^{\Sigma}(W,Q)$
is final. Therefore the colimit on the right is constant, and we deduce

$i^{-1}(f_{\ast}(F))(U)\simeq\lim_{\Pi_{\infty}^{\Sigma}(g^{-1}(U),R)}\Phi_{Y,Q}^{\mathrm{hyp}}(F)|_{\Pi_{\infty}^{\Sigma}(Y_{R},R)}$

Since $j:Y_{R}\to Y$ is exodromic, we deduce

$i^{-1}(f_{\ast}(F))(U)\simeq\lim_{\Pi_{\infty}^{\Sigma}(g^{-1}(U),R)}\Phi_{Y_{R},R}^{\mathrm{hyp}}(j^{\ast,\mathrm{hyp}}(F))\simeq
j^{\ast,\mathrm{hyp}}(F)(g^{-1}(U))\simeq
g_{\ast}(j^{\ast,\mathrm{hyp}}(F))(U)\ .$

The conclusion follows. ∎

######  Remark 6.48.

In the above proof, properness is only used in the form of 6.45. It would
therefore be enough to ask that for every $x\in X$ the collection
$\\{f^{-1}(V)\\}_{V\in\mathcal{B}_{x}}$ form a fundamental system of open
neighborhood for the fiber $f^{-1}(x)$ inside $Y$ (where $\mathcal{B}_{x}$
denotes the collection of open neighborhoods of $x$ inside $X$).

######  Definition 6.49.

We say that a morphism between conically stratified spaces
$f\colon(Y,Q)\to(X,P)$ is a _weak stratified bundle_ if for every $p\in P$
every point $x\in X_{p}$ admits an open neighborhood $U$ in $X_{p}$ such that
there exists a conically stratified space $(W,R)$ and an isomorphism of
stratified spaces

$(f^{-1}(U),Q)\xrightarrow{\sim}(U\times W,R)$

over $U$.

######  Proposition 6.50.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces with
locally weakly contractible strata. Let $\mathcal{E}$ be a presentable
$\infty$-category. Assume that:

  1. (1)

the underlying morphism $f\colon Y\to X$ is proper;

  2. (2)

$f$ is a weak stratified bundle whose fibres have locally weakly contractible
strata;

  3. (3)

$(Y,Q)$ is hereditary excellent at every locally closed subset of $Q$;

  4. (4)

the assumptions of 5.17 are satisfied.

Then
$f_{\ast}\colon\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
preserves hyperconstructible hypersheaves and for every
$F\in\mathrm{Cons}_{P}^{\mathrm{hyp}}(Y;\mathcal{E})$ and every $x\in X$ there
is a canonical equivalence

$f_{\ast}(F)_{x}\simeq\Gamma(Y_{x},j_{x}^{\ast,\mathrm{hyp}}(F))\ ,$

where $j_{x}\colon Y_{x}\hookrightarrow Y$ denotes the inclusion of the fiber.

###### Proof.

Using 6.47, we immediately reduce to the case where $P$ is trivial. Since both
statements are local on the target and since $f$ is a weak stratified bundle,
we can assume that $(Y,Q)\simeq(X\times W,R)$ for some conically stratified
space $(W,R)$ and that $f$ coincides with the canonical projection to $X$. Let
$F\in\mathrm{Cons}_{P}^{\mathrm{hyp}}(Y;\mathcal{E})$. We first show that
$f_{\ast}(F)$ is locally hyperconstant on $X$. To do this, [HPT, Proposition
3.1] ensures that it is enough to prove that for every inclusion $U\subseteq
V$ of weakly contractible open subsets in $X$, the restriction map

$f_{\ast}(F)(V)\longrightarrow f_{\ast}(F)(U)$

is an equivalence. Since $F$ is hyperconstructible, 3.21 and 5.17 allow to
rewrite this map as

$\lim_{\Pi_{\infty}^{\Sigma}(V\times W,R)}\Phi_{X\times
W,R}^{\mathrm{hyp}}(F)\longrightarrow\lim_{\Pi_{\infty}^{\Sigma}(U\times
W,R)}\Phi_{X\times W,R}^{\mathrm{hyp}}(F)\ .$

Since $\Pi_{\infty}^{\Sigma}$ commutes with finite products, we have

$\Pi_{\infty}^{\Sigma}(V\times
W,R)\simeq\Pi_{\infty}(V)\times\Pi_{\infty}^{\Sigma}(W,R),\qquad\Pi_{\infty}^{\Sigma}(U\times
W,R)\simeq\Pi_{\infty}(U)\times\Pi_{\infty}^{\Sigma}(W,R)\ .$

Since the map $\Pi_{\infty}(U)\to\Pi_{\infty}(V)$ is an equivalence, the
conclusion follows.
Let now $x\in X$. To compute $f_{\ast}(F)_{x}$, we can further suppose that
$X$ is weakly contractible. In that case, the local hyperconstancy of
$f_{\ast}(F)$ combined with [HPT, Proposition 3.1] ensures that the canonical
map

$f_{\ast}(F)(X)\longrightarrow f_{\ast}(F)_{x}$

is an equivalence. On the other hand, 3.21 combined with the fact that the
morphism $Y_{x}=\\{x\\}\times W\to Y$ is exodromic gives a chain of
equivalences

$\displaystyle f_{\ast}(F)(X)$
$\displaystyle\simeq\lim_{\Pi_{\infty}^{\Sigma}(X\times W,R)}\Phi_{X\times
W,R}^{\mathrm{hyp}}(F)\simeq\lim_{\Pi_{\infty}^{\Sigma}(\\{x\\}\times
W,R)}\Phi_{X\times W,R}^{\mathrm{hyp}}(F)$
$\displaystyle\simeq\lim_{\Pi_{\infty}^{\Sigma}(Y_{x},R)}\Phi_{Y_{x},R}^{\mathrm{hyp}}(j_{x}^{\ast,\mathrm{hyp}}(F))\simeq\Gamma(Y_{x},j_{x}^{\ast,\mathrm{hyp}}(F))$

The proof of 6.50 is thus complete. ∎

##  7\. Moduli spaces

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata and let $R$ be an excellent commutative ring. Consider the derived
prestack

$\mathbf{Cons}_{P}(X)\colon\mathrm{dAff}_{R}^{\mathrm{op}}\longrightarrow\mathcal{S}$

defined by sending a derived affine scheme $\operatorname{Spec}(A)$ over $R$
to the maximal $\infty$-groupoid contained in

$\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})\simeq\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Perf}(A)\big{)}\
,$

where the equivalence is the one provided by 6.32. Observe that the right hand
side commutes with limits in $\mathrm{Perf}(A)$. In particular we obtain:

######  Lemma 7.1.

The derived prestack $\mathbf{Cons}_{P}(X)$ satisfies faithfully flat
hyperdescent. In particular, it is a derived stack.

###### Proof.

Combine [SAG, Corollary D.6.3.3 and Proposition 2.8.4.2-(10)]. ∎

Our goal is to prove that when $(X,P)$ satisfies certain mild finiteness
conditions, $\mathbf{Cons}_{P}(X)$ is locally geometric.

###  7.1. Finiteness conditions on stratified homotopy types

######  Definition 7.2.

For stratified space $(X,P)$, we say that :

  1. (1)

$(X,P)$ is _categorically compact_ if $\mathrm{Exit}(X,P)$ is a compact object
in $\mathrm{Cat}_{\infty}$ ;

If furthermore $(X,P)$ is conically stratified, we say that:

  1. (2)

$(X,P)$ is _locally categorically compact_ if for every $p\in P$ and every
$x\in X_{p}$, there exists a conical chart of the form $Z\times C(Y)$
containing $x$ such that $(Y,P_{>p})$ is categorically compact and $Z$ is
weakly contractible and locally weakly contractible ;

  2. (3)

$(X,P)$ is _of finite stratified type_ if the poset $P$ is finite and for
every $p\in P$, the homotopy type $\Pi_{\infty}(X_{p})$ of the associated
stratum is a compact object in $\mathcal{S}$.

When the stratification is trivial, there is an abundance of examples:

######  Example 7.3.

  1. (1)

Assume that $X$ is Stein complex (smooth) manifold, equipped with a trivial
stratification. Then classical results of Remmert [R], Bishop [B] and
Narasimhan [N] show that $X$ can be realized as a closed subvariety of
$\mathbb{C}^{n}$ for $n\gg 0$. In particular, Andreotti-Frankel’s theorem [AF]
shows that $X$ has the homotopy type of a finite CW complex, and in particular
$\Pi_{\infty}(X)$ is compact.

  2. (2)

Assume now that $X$ is a smooth algebraic variety. If $X$ is affine, then its
analytification $X^{\mathrm{an}}$ is smooth and Stein and therefore
$\Pi_{\infty}(X^{\mathrm{an}})$ is compact by the previous point. In general,
$X$ admits a finite cover by affine smooth open subvarieties, whose
intersections are again smooth and affine. This immediately implies that
$\Pi_{\infty}(X^{\mathrm{an}})$ can be realized as a finite colimit of compact
objects in $\mathcal{S}$, and henceforth that $\Pi_{\infty}(X^{\mathrm{an}})$
is compact itself.

  3. (3)

Assume that $X$ is a _compact_ topological manifold. Then the work of Kirby
and Siebenmann [KS, Theorem III] implies that $X$ has the homotopy type of a
_finite_ CW complex (although $X$ might not be triangulable itself). In
particular, $\Pi_{\infty}(X)$ is a compact object in $\mathcal{S}$.

Another big class of examples comes from the theory of conically smooth
stratified spaces.

######  Recollection 7.4.

We refer the reader to [AFTb, § 3.2] for the notion of conically smooth
stratified space. Although their definition is long and involved, it is an
extremely convenient setting to work in, as it enjoys the following
properties:

  1. (1)

every stratum $X_{p}$ is a smooth manifold;

  2. (2)

for every $p\in P$, every point $x\in X_{p}$ has a fundamental system of open
neighborhoods of the form $U\times C(Z)$, where $U$ is a weakly contractible
(and in fact homeomorphic to $\mathbb{R}^{i}$ for some $i\geqslant 0$) open
neighborhood of $x$ in $X_{p}$ and $Z$ is a _compact_ conically smooth
stratified space over $P_{>p}$;

  3. (3)

if the stratification on $(X,P)$ is Whitney, then it is conically smooth, as
recently shown by Nocera and Volpe [NV, Theorem 2.7].

In other words, properties (1) and (2) enable inductive arguments on the depth
of the stratification, while property (3) guarantees that every reasonable
example considered in practice falls in this class.

Also, in [AFTb, Definition 8.3.6] the authors introduced the notion of
_finitary_ conically smooth stratified space, that intuitively corresponds to
those conically smooth stratified spaces that admit a finite handlebody
decomposition, such as the one provided by Morse theory. Observe that [AFTb,
Theorem 8.3.10-(2)] implies that every conically smooth stratified space
$(X,P)$ whose underlying topological space is compact is automatically
finitary.

Having recalled these facts concerning conically smooth stratified spaces, we
can state the following recent theorem of Volpe:

######  Theorem 7.5 (Volpe, [V, Proposition 2.12 & Corollary 2.13]).

Let $(X,P)$ be a finitary conically smooth stratified space. Then
$\Pi_{\infty}^{\Sigma}(X,P)$ is categorically equivalent to a simplicial set
having only a finite number of non-degenerate simplexes. Hence, $(X,P)$ is
categorically compact.

######  Corollary 7.6.

Let $(X,P)$ be a conically smooth stratified space. Then $(X,P)$ is locally
categorically compact.

###### Proof.

Since $(X,P)$ is a finitary conically smooth stratified space, for every $p\in
P$ and every $x\in X_{p}$ there exists an open neighborhood of $x$ of the form
$U\times C(Z)$, where $U$ is weakly contractible locally weakly contractible
and $(Z,P_{>p})$ is a _compact_ conically smooth stratified space. In
particular, [AFTb, Theorem 8.3.10-(2)] implies that $(Z,P_{>p})$ is finitary.
Then 7.5 implies that $(Z,P_{>p})$ is categorically compact. ∎

When $(X,P)$ is categorically compact, it is particularly easy to prove that
$\mathbf{Cons}_{P}(X)$ is a locally geometric derived stack:

######  Proposition 7.7.

Let $(X,P)$ be a categorically compact conically stratified space. Then the
derived stack $\mathbf{Cons}_{P}(X)$ is locally geometric, and the tangent
complex at a point $x\colon\operatorname{Spec}(A)\to\mathbf{Cons}_{P}(X)$
classifying a constructible sheaf
$F\in\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})$ is
given by

$x^{\ast}\mathbb{T}_{\mathbf{Cons}_{P}(X)}\simeq\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}(F,F)[1]\
,$

where the right hand side denotes the $\mathrm{Mod}_{A}$-enriched
$\operatorname{Hom}$ of $\mathbf{Cons}_{P}(X;\mathrm{Mod}_{A})$.

###### Proof.

By assumption, $\Pi_{\infty}^{\Sigma}(X,P)$ is a compact object in
$\mathrm{Cat}_{\infty}$. Since
$(-)^{\mathrm{op}}\colon\mathrm{Cat}_{\infty}\to\mathrm{Cat}_{\infty}$ is a
self-equivalence, we deduce that $\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}$ is
compact as well. Moreover, every self map in
$\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}$ is automatically an equivalence,
which in turn implies that $\Pi_{\infty}^{\Sigma}(X,P)$ is idempotent
complete. In particular,

$\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}\simeq\mathrm{PSh}\big{(}\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}\big{)}^{\omega}\simeq\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S}\big{)}^{\omega}\
.$

Now, recall from [HTT, Proposition 5.5.7.11 & Corollary 4.4.5.21] that
$\mathrm{Cat}_{\infty}^{\mathsf{rex},\mathsf{idem}}$ is closed under filtered
colimits in $\mathrm{Cat}_{\infty}$. It follows that
$\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}$ is compact in
$\mathrm{Cat}_{\infty}^{\mathsf{rex},\mathsf{idem}}\simeq\mathcal{P}\mathrm{r}^{\mathrm{L},\omega}$.
Thus,

$\mathcal{C}\coloneqq\mathrm{PSh}\big{(}\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}\big{)}\otimes\mathrm{Mod}_{R}\simeq\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}},\mathrm{Mod}_{R})\
.$

is automatically a compact object in
$\mathcal{P}\mathrm{r}^{\mathrm{L},\omega}_{R}$, and hence it is smooth and
proper in the sense of [SAG, §11]. In particular, its moduli of objects
$\mathcal{M}_{\mathcal{C}}$ in the sense of [TV] is well defined. Unraveling
the definitions, we see that the functor of points of
$\mathcal{M}_{\mathcal{C}}$ sends $A\in\mathrm{CAlg}_{R}$ to

$\displaystyle\operatorname{Fun}_{R}\big{(}(\mathcal{C}^{\omega})^{\mathrm{op}},\mathrm{Perf}(A)\big{)}$
$\displaystyle\simeq\operatorname{Fun}_{R}\big{(}(\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}\otimes\mathrm{Perf}(R))^{\mathrm{op}},\mathrm{Perf}(A)\big{)}$
$\displaystyle\simeq\operatorname{Fun}_{R}\big{(}\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}\otimes\mathrm{Perf}(R),\mathrm{Perf}(A)^{\mathrm{op}}\big{)}^{\mathrm{op}}$
$\displaystyle\simeq\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}},\mathrm{Perf}(A)^{\mathrm{op}}\big{)}^{\mathrm{op}}$
$\displaystyle\simeq\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Perf}(A)\big{)}$
$\displaystyle\simeq\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Perf}(A))\
.$

Therefore, the conclusion follows from [TV, Theorem 3.6 & Corollary 3.17]. ∎

We can now state the main theorem of this section:

######  Theorem 7.8.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that:

  1. (1)

$(X,P)$ is of finite stratified type;

  2. (2)

$(X,P)$ is locally categorically compact.

Then $\mathbf{Cons}_{P}(X)$ is a locally geometric derived stack, locally of
finite presentation.

######  Remark 7.9.

  1. (1)

We do not know whether the assumptions of 7.8 imply that $(X,P)$ is
categorically compact. By contrast with 7.7, we are going to apply Lurie’s
representability theorem to prove 7.8.

  2. (2)

7.8 covers _a priori_ a larger class of stratified spaces $(X,P)$ for which
$\mathbf{Cons}_{P}(X)$ is a locally geometric derived stack than the one
provided combining 7.5 and 7.7. Indeed, we do not know whether every conically
smooth stratified space $(X,P)$ which is also of finite stratified type is
automatically finitary. Nevertheless, we formulate the following conjecture,
that would extend Volpe’s theorem:

######  Conjecture 7.10.

Let $(X,P)$ be a conically smooth stratified space. If $(X,P)$ is of finite
stratified type, then $(X,P)$ is also categorically compact.

We do not know how to prove this conjecture at the moment. However we obtain a
very close result below (see 7.11).

###  7.2. Local finite presentation

Recall that if $\\{A_{\alpha}\\}_{\alpha\in I}$ is a filtered diagram in
$\mathrm{CAlg}_{R}$ with colimit $A$, the canonical map

$\operatorname*{colim}_{\alpha\in
I}\mathrm{Perf}(A_{\alpha})\longrightarrow\mathrm{Perf}(A)$

is an equivalence in $\mathrm{Cat}_{\infty}$ as shown in [SAG, Corollary
4.5.1.8]. Therefore, when $\Pi_{\infty}^{\Sigma}(X,P)$ is compact in
$\mathrm{Cat}_{\infty}$, the functor $\mathbf{Cons}_{P}(X)$ is automatically
locally of finite type. More generally, we have the following

######  Lemma 7.11.

Let $(X,P)$ be a conically stratified space satisfying the assumptions of 7.8.
Then the functor

$\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),(-)^{\omega}\big{)}\colon\mathcal{P}\mathrm{r}^{\mathrm{L},\omega}_{\mathrm{st}}\longrightarrow\mathrm{Cat}_{\infty}$

commutes with filtered colimits. Hence $\mathbf{Cons}_{P}(X)$ is locally of
finite presentation.

###### Proof.

We proceed by induction on the cardinality $n$ of $P$. When $n=1$, the
stratification is trivial and therefore
$\Pi_{\infty}^{\Sigma}(X,P)=\Pi_{\infty}(X)$, which is compact in
$\mathrm{Cat}_{\infty}$ by assumption. Hence, this case follows. If $n>1$,
take $p\in P$ a minimal element and consider $S\coloneqq
P\smallsetminus\\{p\\}$. Then both $P_{\leqslant p}$ and $S$ have cardinality
strictly less than $n$. Consider a filtered diagram
$I\to\mathcal{P}\mathrm{r}^{\mathrm{L},\omega}_{\mathrm{st}}$, noted
$\alpha\mapsto\mathcal{E}_{\alpha}$ and let $\mathcal{E}$ be its colimit
computed in $\mathcal{P}\mathrm{r}^{\mathrm{L},\omega}_{\mathrm{st}}$.
Consider the following diagram:

$\displaystyle{\operatorname*{colim}_{\alpha}\operatorname{Cons}^{\mathrm{hyp}}_{S,\omega}(X_{S};\mathcal{E}_{\alpha})}$$\displaystyle{\operatorname*{colim}_{\alpha}\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathcal{E}_{\alpha})}$$\displaystyle{\operatorname*{colim}_{\alpha}\operatorname{Cons}^{\mathrm{hyp}}_{p,\omega}(X_{p};\mathcal{E}_{\alpha})}$$\displaystyle{\operatorname{Cons}^{\mathrm{hyp}}_{S,\omega}(X_{S};\mathcal{E})}$$\displaystyle{\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathcal{E})}$$\displaystyle{\operatorname{Cons}^{\mathrm{hyp}}_{p,\omega}(X_{p};\mathcal{E})\
.}$$\scriptstyle{i_{S,\ast}}$$\scriptstyle{i_{p,\ast}}$$\scriptstyle{i_{S,\ast}}$$\scriptstyle{i_{p,\ast}}$

From 6.44, each square is commutative. The external vertical arrows are
equivalences by inductive hypothesis. The horizontal functors exhibit the
middle terms as recollements of the external ones as a result of 6.43.
Therefore, the middle vertical arrow is an equivalence as a result of [HA,
Proposition A.8.14]. We then conclude using the fact that the interior
groupoïd functor commutes with filtered colimits. ∎

###  7.3. Cohesiveness, nilcompleteness and integrability

For a review of definitions, let us refer to [SAG, 17.3.7]

######  Proposition 7.12.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Then the derived stack $\mathbf{Cons}_{P}(X)$ is infinitesimally
cohesive, nilcomplete and satisfies integrability.

###### Proof.

These statements follow from the analogous ones for $\mathrm{Perf}$, the
description of the category
$\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})$ as
$\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Perf}(A)\big{)}$
and the commutation of the interior groupoid functor with limits.
Specifically, infinitesimally cohesiveness follows from [SAG, Theorem 16.2.0.1
and Proposition 16.2.3.1]. Nilcompleteness follows from [SAG, Propositions
2.7.3.2 and 19.2.1.5]. Integrability follows from [SAG, Remark 8.5.0.5 and
Theorem 8.5.0.3]. ∎

###  7.4. Cotangent complex

The next important step to verify the assumptions of Artin-Lurie’s
representability theorem is to check the existence of the cotangent complex
for $\mathbf{Cons}_{P}(X)$.

######  Notation 7.13.

In proving the existence of the cotangent complex of $\mathbf{Cons}_{P}(X)$,
we will crucially use the fact that for every $A\in\mathrm{CAlg}$, the
$\infty$-category $\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$
is canonically tensor-enriched over $\mathrm{Mod}_{A}$. We will denote by

$\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}(-,-)\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})^{\mathrm{op}}\times\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\longrightarrow\mathrm{Mod}_{A}$

the enriched $\operatorname{Hom}$. Since
$\otimes:\mathrm{Mod}_{A}\times\mathrm{Mod}_{A}\to\mathrm{Mod}_{A}$ is
cocontinuous in each variable, 3.9 ensures the existence of a tensor product

$(-)\otimes_{A}(-)\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\otimes\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\longrightarrow\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$

defined by applying the tensor product objectwise. If
$F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$ and
$M\in\mathrm{Mod}_{A}$, we write $M\otimes F$ for
$\Gamma^{\ast,\mathrm{hyp}}(M)\otimes F$.

######  Remark 7.14.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $f\colon A\to B$ be a morphism in $\mathrm{CAlg}$. From 6.16, the
adjunction

$f^{\ast}\colon\mathrm{Mod}_{A}\leftrightarrows\mathrm{Mod}_{B}\colon
f_{\ast}$

induces an adjunction

$f^{\ast}\circ-\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\leftrightarrows\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{B})\colon
f_{\ast}\circ-$

To simplifiy notations, for
$F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$ we write
$f^{\ast}(F)$ instead of $f^{\ast}\circ F$ and similarly with $f_{\ast}$. From
6.16 again, the above adjunction can be described as the adjunction

$f^{\ast}\circ-\colon\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Mod}_{A}\big{)}\leftrightarrows\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Mod}_{B}\big{)}\colon
f_{\ast}\circ-\ .$

In particular, for every
$F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$ and every
$N\in\mathrm{Mod}_{B}$, the canonical map

$f_{\ast}(N)\otimes_{A}F\longrightarrow f_{\ast}(N\otimes_{B}f^{\ast}(F))$

is an equivalence.

######  Proposition 7.15.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that $P$ is finite and that the strata of $(X,P)$ have finitely
many connected components. Then the derived stack $\mathbf{Cons}_{P}(X)$
admits a global cotangent complex.

###### Proof.

Let $\operatorname{Spec}(A)$ be a derived affine scheme over $R$. Let

$x\colon\operatorname{Spec}(A)\longrightarrow\mathbf{Cons}_{P}(X)$

be a morphism and let
$F\in\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})\simeq\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Perf}(A)\big{)}$
be the corresponding $A$-linear family of hyperconstructible hypersheaves. Let

$\Omega_{x}\mathbf{Cons}_{P}(X)\coloneqq\operatorname{Spec}(A)\times_{\mathbf{Cons}_{P}(X)}\operatorname{Spec}(A)$

be the loop stack at $x$. We view $\Omega_{x}\mathbf{Cons}_{P}(X)$ as a stack
over $A\times A$. Consider the diagonal morphism

$\delta_{x}\colon\operatorname{Spec}(A)\longrightarrow\Omega_{x}\mathbf{Cons}_{P}(X)$

If $d_{0}\in\mathrm{Der}_{\mathbf{Cons}_{P}(X)/R}(A;-)$ denotes the
$0$-derivation, there are canonical equivalences of functors over
$\mathrm{Mod}_{A}$ :

$\mathrm{Der}_{\mathbf{Cons}_{P}(X)/R}(A;-[-1])\simeq\Omega_{d_{0}}\mathrm{Der}_{\mathbf{Cons}_{P}(X)/R}(A;-)\simeq\mathrm{Der}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A}(A;-)$

In particular, $\mathbf{Cons}_{P}(X)$ admits a cotangent complex
$x^{\ast}\mathbb{L}_{\mathbf{Cons}_{P}(X)/R}$ at $x$ if and only if the
morphism
$\Omega_{x}\mathbf{Cons}_{P}(X)\to\operatorname{Spec}A\times\operatorname{Spec}A$
admits a cotangent complex
$\delta_{x}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times A}$ at
$\delta_{x}$. In this case, one has the relation

$x^{\ast}\mathbb{L}_{\mathbf{Cons}_{P}(X)/R}\simeq\delta_{x}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A}[-1]\ .$

We are thus left to check that the functor

$\mathrm{Der}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A}(A,-)\colon\mathrm{Mod}_{A}\longrightarrow\mathcal{S}$

is corepresentable. Unravelling the definitions, we find for every
$M\in\mathrm{Mod}_{A}$ a natural equivalence

$\mathrm{Der}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A}(A,M)\simeq\operatorname{Map}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}\big{(}F,M\otimes_{A}F\big{)}\
.$

Using 5.17, 6.10 and [GHN, Proposition 5.1], we can rewrite the above mapping
space as

$\int_{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}\operatorname{Map}_{\mathrm{Mod}_{A}}(F_{x},M\otimes_{A}F_{y})\
,$

where the integral denotes the limit over the functor

$\mathrm{Tw}(\Pi_{\infty}^{\Sigma}(X,P))\to\Pi_{\infty}^{\Sigma}(X,P)^{\mathrm{op}}\times\Pi_{\infty}^{\Sigma}(X,P)\to\mathcal{S}$

defined as

$x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)\longrightarrow\operatorname{Map}_{\mathrm{Mod}_{A}}(F_{x},M\otimes_{A}F_{y})$

Since the stalks of $F$ are dualizable, we can further simplify the above
expression as

$\displaystyle\int_{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}\operatorname{Map}_{\mathrm{Mod}_{A}}(F_{x},M\otimes_{A}F_{y})$
$\displaystyle\simeq\int_{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}\operatorname{Map}_{\mathrm{Mod}_{A}}(F_{x}\otimes
F_{y}^{\vee},M)$
$\displaystyle\simeq\operatorname{Map}_{\mathrm{Mod}_{A}}\left(\int^{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}F_{x}\otimes F_{y}^{\vee},M\right)\ .$

Hence, $\mathrm{Der}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times A}(A,-)$ is
corepresentable and we have

$\delta_{x}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A}\simeq\int^{x\to y\in\Pi_{\infty}^{\Sigma}(X,P)}F_{x}\otimes F_{y}^{\vee}\
.$

We have to check that the above $A$-module is eventually connective. For every
$x,y\in\Pi_{\infty}^{\Sigma}(X,P)$, the $A$-module $F_{x}\otimes F_{y}^{\vee}$
is perfect. In particular, there exists an integer $n(x,y)$ such that

$F_{x}\otimes F_{y}^{\vee}\in(\mathrm{Mod}_{A})_{\geqslant n(x,y)}\ .$

On the other hand, $P$ is finite and each stratum of $(X,P)$ has finitely many
connected components. Since $F$ is hyperconstructible on $(X,P)$, we deduce
that

$N\coloneqq\max_{x,y\in\Pi_{\infty}^{\Sigma}(X,P)}n(x,y)$

is a well defined integer. It follows that the functor

$\mathrm{Tw}(\Pi_{\infty}^{\Sigma}(X,P))^{\mathrm{op}}\longrightarrow\mathrm{Mod}_{A}$

given by $(x\rightarrow y)\mapsto F_{x}\otimes F_{y}^{\vee}$ takes values in
$(\mathrm{Mod}_{A})_{\geqslant N}$. Since $(\mathrm{Mod}_{A})_{\geqslant N}$
is closed under arbitrary colimits inside $\mathrm{Mod}_{A}$, we finally
deduce that

$\delta_{x}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A}\in(\mathrm{Mod}_{A})_{\geqslant N}\ .$

We are left to check the cotangent complex is global. In the above situation,
let $f\colon\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ be morphism and
let $y\coloneqq x\circ f$ for the induced morphism classifying the $B$-linear
family of hyperconstructible hypersheaves $f^{\ast}(F)$. Then for every
$N\in\mathrm{Mod}_{B}$, we have:

$\displaystyle\operatorname{Hom}_{\mathrm{Mod}_{B}}(\delta_{y}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A},N)$
$\displaystyle\simeq\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{B})}(f^{\ast}(F),N\otimes
f^{\ast}(F))$
$\displaystyle\simeq\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}(F,f_{\ast}(N\otimes
f^{\ast}(F)))$
$\displaystyle\simeq\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}(F,f_{\ast}(N)\otimes
F)$ by 7.14
$\displaystyle\simeq\operatorname{Hom}_{\mathrm{Mod}_{A}}(\delta_{x}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A},f_{\ast}(N))$
$\displaystyle\simeq\operatorname{Hom}_{\mathrm{Mod}_{B}}(f^{\ast}\delta_{x}^{\ast}\mathbb{L}_{\Omega_{x}\mathbf{Cons}_{P}(X)/A\times
A},N).$

Thus, the conclusion follows from the Yoneda lemma. ∎

###  7.5. Representability results

Before giving the proof of 7.8, let us fix the following couple of notation:

######  Notation 7.16.

  1. (1)

For integers $a\leqslant b$, we let $\mathbf{Perf}_{[a,b]}$ be the substack of
$\mathbf{Perf}$ parametrizing perfect complexes of tor-amplitude contained in
$[a,b]$. Notice that [SAG, Proposition 6.1.4.5] implies that this is an open
substack of $\mathbf{Perf}$.

  2. (2)

Given an $\infty$-category $\mathcal{C}$, we write
$\mathbf{Perf}^{\mathcal{C}}$ (resp. $\mathbf{Perf}^{\mathcal{C}}_{[a,b]}$)
for the derived stack sending $S\in\mathrm{dAff}$ to the maximal
$\infty$-groupoid contained in
$\operatorname{Fun}(\mathcal{C},\mathrm{Perf}(S))$ (resp.
$\operatorname{Fun}(\mathcal{C},\mathrm{Perf}(S)_{[a,b]})$).

######  Lemma 7.17.

Let $\mathcal{C}$ be an $\infty$-category with only a finite number of
equivalence classes of objects. Let $a\leqslant b$ be integers. Then the
induced morphism of derived stacks

$\mathbf{Perf}^{\mathcal{C}}_{[a,b]}\longrightarrow\mathbf{Perf}^{\mathcal{C}}$

is representable by an open immersion.

###### Proof.

Choose a finite set $S$ of objects of $\mathcal{C}$ representing all
equivalence classes of objects in $\mathcal{C}$. Then the diagram of derived
stacks

$\displaystyle{\mathbf{Perf}^{\mathcal{C}}_{[a,b]}}$$\displaystyle{\prod_{c\in
S}\mathbf{Perf}_{[a,b]}}$$\displaystyle{\mathbf{Perf}^{\mathcal{C}}}$$\displaystyle{\prod_{c\in
S}\mathbf{Perf}\ .}$

is cartesian. Since $S$ is finite, the right vertical arrow is an open
immersion, as already observed in 7.16-(1). Hence, so is the left vertical
arrow, whence the conclusion. ∎

######  Remark 7.18.

If $\mathcal{C}$ is the homotopy type of a topological space $X$ in 7.17, we
will note $\mathbf{LC}_{[a,b]}(X)$ instead of
$\mathbf{Perf}^{\Pi_{\infty}(X)}_{[a,b]}$.

We are now ready to prove the main result of this section:

###### Proof of 7.8.

Let $a\leqslant b$ be integers and define $\mathbf{Cons}_{P}(X)_{[a,b]}$ as
$\mathbf{Perf}_{[a,b]}^{\Pi_{\infty}^{\Sigma}(X,P)}$. 7.17 shows that the
canonical map

$\mathbf{Cons}_{P}(X)_{[a,b]}\to\mathbf{Cons}_{P}(X)$

is representable by an open immersion. We now apply Artin-Lurie’s
representability theorem to each $\mathbf{Cons}_{P}(X)_{[a,b]}$:

  1. (1)

_Truncatedness_ : if $A$ is discrete, the mapping spaces in
$\mathrm{Perf}(A)_{[a,b]}$ are $(b-a)$-truncated, and therefore
$\mathrm{Perf}(A)_{[a,b]}$ is a $(b-a+1)$-truncated $\infty$-category. Thus,
the same goes for
$\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Perf}(A)_{[a,b]}\big{)}$,
whence the conclusion.

  2. (2)

_Locally of finite presentation_ : this is the content of 7.11.

  3. (3)

_Integrability_ , _Nilcompleteness_ and _infinitesimal cohesiveness_ : this
has been verified in 7.12.

  4. (4)

_Existence of cotangent complex_ : this follows from 7.15.

∎

######  Corollary 7.19.

Under the assumptions of 7.8, let
$x\colon\operatorname{Spec}(A)\to\mathbf{Cons}_{P}(X)$ be a morphism and let
$F\in\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathrm{Perf}(A))$ be the
corresponding $A$-linear family of hyperconstructible hypersheaves. Then

$x^{\ast}\mathbb{T}_{\mathbf{Cons}_{P}(X)}\simeq\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}(F,F)[1]\
.$

###### Proof.

Since $\mathbf{Cons}_{P}(X)$ is a derived geometric stack of finite
presentation, [SAG, Proposition 17.4.2.3-(1)] shows that its cotangent complex
at $x$ is perfect. Therefore,

$x^{\ast}\mathbb{T}_{\mathbf{Cons}_{P}(X)}\simeq\big{(}x^{\ast}\mathbb{L}_{\mathbf{Cons}_{P}(X)}\big{)}^{\vee}\
.$

Using the formula obtained in 7.15, we now obtain:

$\displaystyle x^{\ast}\mathbb{T}_{\mathbf{Cons}_{P}(X)}$
$\displaystyle\simeq\operatorname{Hom}_{\mathrm{Mod}_{A}}\left(\int^{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}F_{x}\otimes F_{y}^{\vee}[-1],A\right)$
$\displaystyle\simeq\int_{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}F_{x}^{\vee}\otimes F_{y}[1]$
$\displaystyle\simeq\int_{x\to
y\in\Pi_{\infty}^{\Sigma}(X,P)}\operatorname{Hom}_{\mathrm{Mod}_{A}}(F_{x},F_{y})[1]$
$\displaystyle\simeq\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})}(F,F)[1]\
,$

where the last equivalence is due to [GHN, Proposition 5.1]. ∎

###  7.6. Perverse sheaves

As corollary of 7.8, we can obtain a general representability result for the
stack of perverse sheaves. To state the result, let us begin by fixing some
notation.

######  Notation 7.20.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a presentable $\infty$-category. For every $p\in
P$, we can factor the inclusion $j_{p}\colon X_{p}\to X$ as

$\displaystyle{X_{p}}$$\displaystyle{X_{\geqslant p}}$$\displaystyle{X\
,}$$\scriptstyle{i_{p}}$$\scriptstyle{j_{\geqslant p}}$

where $i_{p}$ is a closed immersion and $j_{\geqslant p}$ is an open
immersion. Then

$i_{p,\ast}\colon\mathrm{Sh}^{\mathrm{hyp}}(X_{p};\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X_{\geqslant
p};\mathcal{E})$

commutes with colimits so in particular admits a right adjoint
$i_{p}^{!,\mathrm{hyp}}$. We set

$j_{p}^{!,\mathrm{hyp}}\coloneqq i_{p}^{!,\mathrm{hyp}}\circ j_{\geqslant
p}^{\ast,\mathrm{hyp}}\
\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X_{p};\mathcal{E})\
.$

######  Lemma 7.21.

Let $(X,P)$ be conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a presentable stable $\infty$-category. For every
$p\in P$, the functor $j_{p}^{!,\mathrm{hyp}}$ restricts to a functor

$j_{p}^{!,\mathrm{hyp}}\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathrm{LC}^{\mathrm{hyp}}(X_{p};\mathcal{E})\
.$

If furthermore $(X,P)$ is locally categorically compact and $P$ is finite,
then $j_{p}^{!,\mathrm{hyp}}$ is cocontinuous.

###### Proof.

We can assume without loss of generality that $X=X_{\geqslant p}$. In this
case, we simply write $j\colon X_{p}\hookrightarrow X$ for the natural
inclusion, and $i\colon X_{>p}\hookrightarrow X$ for the inclusion of the open
complementary. For every $F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$, we
have the following fiber sequence:

(7.22) $j_{\ast}j^{!,\mathrm{hyp}}(F)\longrightarrow F\longrightarrow
i_{\ast}i^{\ast,\mathrm{hyp}}(F)\ ,$

computed in $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$. Assume now that $F$
is $P$-hyperconstructible. Then the same goes for $i^{\ast,\mathrm{hyp}}(F)$,
and 6.35 implies that $i_{\ast}i^{\ast,\mathrm{hyp}}(F)$ is hyperconstructible
on $(X,P)$ as well. Thus, 5.20 implies that $j_{\ast}j^{!,\mathrm{hyp}}(F)$
belongs to $\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$. In
particular, $j^{!,\mathrm{hyp}}(F)\simeq
j^{\ast,\mathrm{hyp}}j_{\ast}j^{!,\mathrm{hyp}}(F)$ belongs to
$\mathrm{LC}^{\mathrm{hyp}}(X_{p};\mathcal{E})$.

We are left to show that $j_{p}^{!,\mathrm{hyp}}$ commutes with filtered
colimits of $P$-hyperconstructible sheaves when $(X,P)$ is locally
categorically compact. Under this assumption, 6.42 guarantees that the functor

$i_{\ast}\colon\operatorname{Cons}^{\mathrm{hyp}}_{P_{>p}}(X_{>p};\mathcal{E})\longrightarrow\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$

commutes with filtered colimits. Since $j_{\ast}$ commutes with colimits and
is fully faithful, the conclusion follows from observing that the right and
the middle terms of (7.22) commute with filtered colimits. ∎

Changing coefficients commutes with the exceptional inverse image:

######  Lemma 7.23.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $f\colon\mathcal{E}\to\mathcal{E}^{\prime}$ be an exact functor
between presentable $\infty$-categories. Assume that the conditions of 5.17
are satisfied. Then for every $p\in P$, the commutative diagram

$\displaystyle{\mathrm{LC}^{\mathrm{hyp}}(X_{p};\mathcal{E})}$$\displaystyle{\mathrm{LC}^{\mathrm{hyp}}(X_{p};\mathcal{E}^{\prime})}$$\displaystyle{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E}^{\prime})}$$\scriptstyle{f}$$\scriptstyle{j_{p,*}}$$\scriptstyle{j_{p,*}}$$\scriptstyle{f}$

is vertically right adjointable. That is, the Beck-Chevalley transformation

$\gamma\colon f\circ j_{p}^{!,\mathrm{hyp}}\longrightarrow
j_{p}^{!,\mathrm{hyp}}\circ f$

is an equivalence.

###### Proof.

We can without loss of generality assume that $X=X_{\geqslant p}$. In this
case, we simply write $j\colon X_{p}\hookrightarrow X$ for the canonical
closed immersion and $i\colon X_{>p}\hookrightarrow X$ for the inclusion of
the open complementary. Since $j_{\ast}$ is fully faithful, it is enough to
prove that $j_{\ast}(\gamma)$ is an equivalence. Fix
$F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ and consider the
following commutative diagram whose upper and bottom rows are fiber sequences:

$\displaystyle{fj_{\ast}j^{!,\mathrm{hyp}}(F)}$$\displaystyle{f(F)}$$\displaystyle{fi_{\ast}i^{\ast,\mathrm{hyp}}(F)}$$\displaystyle{j_{\ast}fj^{!,\mathrm{hyp}}(F)}$$\displaystyle{f(F)}$$\displaystyle{i_{\ast}fi^{\ast,\mathrm{hyp}}(F)}$$\displaystyle{j_{*}j^{!,\mathrm{hyp}}f(F)}$$\displaystyle{f(F)}$$\displaystyle{i_{\ast}i^{\ast,\mathrm{hyp}}f(F)\
.}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{j_{\ast}(\gamma)}$$\scriptstyle{\delta}$

The morphisms $\alpha$ and $\beta$ are tautologically equivalences. Hence the
middle row is a fiber sequence as well. Finally, $\delta$ is an equivalence as
recalled in Section 2.4. It follows that $j_{\ast}(\gamma)$ is an equivalence
as well, whence the conclusion. ∎

######  Definition 7.24.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathcal{E}$ be a presentable stable $\infty$-category equipped
with a $t$-structure $\tau=(\mathcal{E}_{\geqslant 0},\mathcal{E}_{\leqslant
0})$. Let

$\mathfrak{p}\colon P\longrightarrow\mathbb{Z}$

be a function. We define

${}^{\mathfrak{p}}\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})_{\geqslant
0}\coloneqq\big{\\{}F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\mid\forall
p\in P\ ,\>\pi_{i}(j_{p}^{\ast,\mathrm{hyp}}(F))=0\text{ for every
}i<\mathfrak{p}(p)\big{\\}}\ ,$

and

${}^{\mathfrak{p}}\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})_{\leqslant
0}\coloneqq\big{\\{}F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\mid\forall
p\in P\ ,\>\pi_{i}(j_{p}^{!,\mathrm{hyp}}(F))=0\text{ for every
}i>\mathfrak{p}(p)\big{\\}}$

where the homotopy groups are computed with respect to the naturally induced
$t$-structure on $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$.

It is a standard fact that the above definition gives rise to a $t$-structure
${}^{\mathfrak{p}}\tau$ on the stable $\infty$-category
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$. We write
${}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ for the
heart of this $t$-structure. Following [DPSb], we give the following
definition:

######  Definition 7.25.

Let $S=\operatorname{Spec}(A)\in\mathrm{dAff}_{R}$ be a derived affine. Let
$(X,P)$ be a conically stratified space with locally weakly contractible
strata. Let $\mathfrak{p}\colon P\to\mathbb{Z}$ be a function. We say that an
$A$-linear family of hyperconstructible hypersheaves
$F\in\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})$ is
_${}^{\mathfrak{p}}\tau$ -flat relative to $S$_ if for every
$M\in\mathrm{Mod}_{A}^{\heartsuit}$, one has

$M\otimes_{A}F\in{}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\
.$

######  Warning 7.26.

Recall that $A$ lies in $\mathrm{Mod}_{A}^{\heartsuit}$ if and only if the
canonical morphism $A\to\pi_{0}(A)$ is an equivalence, that is when $A$ is
underived. In that case, a ${}^{\mathfrak{p}}\tau$-flat object automatically
lies in
${}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$. The
converse holds when $A$ is a field but is false otherwise.

######  Construction 7.27.

Let $i\colon Z\hookrightarrow X$ be a closed immersion of topological spaces.
Since the functor
$i^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})\to\mathrm{Sh}^{\mathrm{hyp}}(Z;\mathrm{Mod}_{A})$
is strong monoidal, its right adjoint $i_{\ast}$ is automatically lax
monoidal. In particular, for every $M\in\mathrm{Mod}_{A}$ and
$F\in\mathrm{Sh}^{\mathrm{hyp}}(Z;\mathrm{Mod}_{A})$ there is an induced
morphism

$M\otimes_{A}i_{\ast}(F)\longrightarrow i_{\ast}(M\otimes_{A}F)\ .$

Observe that this is an isomorphism for $M=A$. Since $i$ is a closed
immersion, $i_{\ast}$ commutes with colimits and therefore the above morphism
is an equivalence. In turn, this induces a natural Beck-Chevalley
transformation

$\alpha_{M,F}^{!,\mathrm{hyp}}\colon
M\otimes_{A}i^{!,\mathrm{hyp}}(F)\longrightarrow
i^{!,\mathrm{hyp}}(M\otimes_{A}F)\ .$

######  Lemma 7.28.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that $(X,P)$ is locally categorically compact and that $P$ is
finite. Let $S=\operatorname{Spec}(A)$ be a derived affine. Then for every
$M\in\mathrm{Mod}_{A}$, every
$F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$ and every
$p\in P$, the natural transformation

$\alpha_{M,F}^{!}\colon M\otimes_{A}j_{p}^{!,\mathrm{hyp}}(F)\to
j_{p}^{!,\mathrm{hyp}}(M\otimes_{A}F)$

is an equivalence.

###### Proof.

7.21 guarantees that $j_{p}^{!,\mathrm{hyp}}$ commutes with colimits. Since
the transformation $\alpha_{M,F}^{!}$ is an equivalence for $M=A$, it follows
that it is an equivalence for every $M\in\mathrm{Mod}_{A}$. ∎

The following elementary lemma is an immediate consequence of the Tor spectral
sequence [HA, 7.2.1.19].

######  Lemma 7.29.

Let $S=\operatorname{Spec}(A)$ be a derived affine. Let $F\in\mathrm{Mod}_{A}$
and assume that $\pi_{i}(F)\simeq 0$ for $i\ll 0$. Let $k\in\mathbb{Z}$. Then
the following are equivalent :

  1. (1)

$F$ is $k$-connective, that is it belongs to $(\mathrm{Mod}_{A})_{\geqslant
k}$;

  2. (2)

for every $M\in\mathrm{Mod}_{A}^{\heartsuit}$, $M\otimes_{A}F$ is
$k$-connective;

  3. (3)

$\pi_{0}(A)\otimes_{A}F$ is $k$-connective.

The following proposition provides a useful reformulation of
${}^{\mathfrak{p}}\tau$-flatness.

######  Proposition 7.30.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that $(X,P)$ is locally categorically compact and that $P$ is
finite. Let $S=\operatorname{Spec}(A)$ be a derived affine. Let
$F\in\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})$. Then
the following statements are equivalent:

  1. (1)

$F$ is ${}^{\mathfrak{p}}\tau$-flat relative to $S$;

  2. (2)

$F$ belongs to
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})_{\geqslant 0}$ and
for every $p\in P$, the stalks of $j_{p}^{!,\mathrm{hyp}}(F)$ have tor-
amplitude contained in $(-\infty,\mathfrak{p}(p)]$.

  3. (3)

for every $p\in P$, the stalks of $j_{p}^{\ast,\mathrm{hyp}}(F)$ are
$\mathfrak{p}(p)$-connectives and the stalks of $j_{p}^{!,\mathrm{hyp}}(F)$
have tor-amplitude contained in $(-\infty,\mathfrak{p}(p)]$.

###### Proof.

Let $p\in P$. From 7.29, the stalks of $j_{p}^{\ast,\mathrm{hyp}}(F)$ are
$\mathfrak{p}(p)$-connective if and only if for every
$M\in\mathrm{Mod}_{A}^{\heartsuit}$, the stalks of

$M\otimes_{A}j_{p}^{\ast,\mathrm{hyp}}(F)\simeq
j_{p}^{\ast,\mathrm{hyp}}(M\otimes_{A}F)$

are $\mathfrak{p}(p)$-connective, where the equivalence follows from the fact
that $j_{p}^{\ast,\mathrm{hyp}}$ is strongly monoidal.

On the other hand, by definition of tor-amplitude, we see that the stalks of
$j_{p}^{!,\mathrm{hyp}}(F)$ have tor-amplitude contained in
$(-\infty,\mathfrak{p}(p)]$ if and only if for every
$M\in\mathrm{Mod}_{A}^{\heartsuit}$ the stalks of

$M\otimes_{A}j_{p}^{!,\mathrm{hyp}}(F)\simeq
j_{p}^{!,\mathrm{hyp}}(M\otimes_{A}F)$

are concentrated in homological degrees $(-\infty,\mathfrak{p}(p)]$, where the
equivalence follows from 7.28. This finishes the proof of 7.30. ∎

######  Remark 7.31.

Observe that condition $(3)$ in 7.30 is stable under base change. Hence, the
${}^{\mathfrak{p}}\tau$-flatness condition defines a substack
${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$ of $\mathbf{Cons}_{P}(X)$ referred to
as the derived stack of perverse sheaves on $(X,P)$.

######  Theorem 7.32.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Assume that:

  1. (1)

$(X,P)$ is of finite stratified type;

  2. (2)

$(X,P)$ is locally categorically compact.

Then,

${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)\longrightarrow\mathbf{Cons}_{P}(X)$

is representable by an open immersion. In particular, $\mathbf{Perv}_{P}(X)$
is locally geometric locally of finite presentation.

###### Proof.

7.30 ensures that the following diagram of derived stacks

$\displaystyle{{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)}$$\displaystyle{\prod_{p\in
P}\mathbf{LC}_{\geqslant\mathfrak{p}(p)}(X_{p})\times\mathbf{LC}_{(-\infty,\mathfrak{p}(p)]}(X_{p})}$$\displaystyle{\mathbf{Cons}_{P}(X)}$$\displaystyle{\prod_{p\in
P}\mathbf{LC}(X_{p})\times\mathbf{LC}(X_{p})}$$\scriptstyle{(j_{p}^{\ast,\mathrm{hyp}},j_{p}^{!,\mathrm{hyp}})_{p\in
P}}$$\scriptstyle{(j_{p}^{\ast,\mathrm{hyp}},j_{p}^{!,\mathrm{hyp}})_{p\in
P}}$

is cartesian. Here $\mathbf{LC}_{\geqslant\mathfrak{p}(p)}(X_{p})$ (resp.
$\mathbf{LC}_{(-\infty,\mathfrak{p}(p)]}(X_{p})$) denotes the substack of
$\mathbf{LC}(X_{p})$ parametrizing locally hyperconstant hypersheaves whose
stalks are perfect and $\mathfrak{p}(p)$-connective (resp. have tor-amplitude
within $(-\infty,\mathfrak{p}(p)]$). The first part of 7.32 thus follows from
7.16-(1) and 7.17. We then deduce the representability of
${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$ from 7.8. ∎

###  7.7. Cohomological Hall algebras

As an application, let us sketch the construction of a cohomological Hall
algebra associated to perverse sheaves on a smooth projective curve over
$\mathbb{C}$. From now on, we will assume that $X$ is a smooth curve over
$\mathbb{C}$ equipped with $n$ marked points $\\{p_{1},p_{2},\ldots,p_{n}\\}$.
We see $X$ as a stratified space over the poset $P=\\{0<1\\}$ equipped with
the middle perversity function, that is

$\mathfrak{p}(0)=0\ ,\qquad\mathfrak{p}(1)=-1\ .$

######  Lemma 7.33.

Let $k$ be a field. Let $(X,P)$ as above. Let
$F,G\in{}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})$.
Then

$\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})}(F,G)\in\mathrm{Mod}_{k}$

is concentrated in homological degrees $[-2,0]$.

###### Proof.

Let us denote by
$\mathbb{D}\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})\to\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})$
the Verdier duality functor. Then, there is a canonical equivalence

$\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})}(F,G)\simeq
R\Gamma(X,\mathbb{D}(F\otimes\mathbb{D}G))\ .$

By Poincaré-Verdier duality, showing that the above complex is concentrated in
homological degrees $[-2,0]$ amounts to show that
$R\Gamma_{c}(X,F\otimes\mathbb{D}G)$ is concentrated in homological degrees
$[0,2]$. Since $\mathbb{D}G$ is perverse, we are thus left to show that if
$F,G\in{}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})$,
then $\Gamma_{c}(X,F\otimes G)$ is concentrated in homological degrees
$[0,2]$. By definition $F\otimes G$ is concentrated in homological degrees
$[0,2]$ and both $\pi_{1}(F\otimes G)$ and $\pi_{0}(F\otimes G)$ are
punctually supported. Since $X$ is a curve over $\mathbb{C}$, the only
potential non zero terms of the spectral sequence

$E^{2}_{p,q}\colon\pi_{p}\Gamma_{c}(X,\pi_{q}(F\otimes
G))\Longrightarrow\pi_{p+q}\Gamma_{c}(X,F\otimes G)$

are thus

$E^{2}_{0,0},\ E^{2}_{0,1},\ E^{2}_{0,2},\ E^{2}_{-1,2},\ E^{2}_{-2,2}\ .$

This implies the sought-after range. ∎

Following [PS, §4.1], there is a canonical simplicial object

$\mathcal{S}_{\bullet}{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)\colon\mathbf{\Delta}^{\mathrm{op}}\longrightarrow\mathrm{dSt}_{R}\
,$

which satisfies the $2$-Segal condition. Here
$\mathcal{S}_{0}{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)\simeq\operatorname{Spec}(R)$
is the final object in $\mathrm{dSt}_{R}$, while

$\mathcal{S}_{1}{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)\simeq{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$

is the derived stack of $\mathfrak{p}$-perverse sheaves we just constructed.
Moreover,

${}^{\mathfrak{p}}\mathbf{Perv}_{P}^{\mathrm{ext}}(X)\coloneqq\mathcal{S}_{2}{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$

can be explicitly described as the derived stack sending a derived affine
$S=\operatorname{Spec}(A)$ to the $\infty$-groupoid consisting of fiber
sequences in
$\operatorname{Cons}^{\mathrm{hyp}}_{P,\omega}(X;\mathrm{Mod}_{A})$ of the
form

$F_{1}\longrightarrow F_{2}\longrightarrow F_{3}\ ,$

where we further ask that $F_{1}$, $F_{2}$ and $F_{3}$ are
${}^{\mathfrak{p}}\tau$-flat relative to $S$. The simplicial structure of
$\mathcal{S}_{\bullet}{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$ induces a
canonical correspondence

$\displaystyle{{}^{\mathfrak{p}}\mathbf{Perv}_{P}^{\mathrm{ext}}(X)}$$\displaystyle{{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)\times{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)}$$\displaystyle{{}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)\
,}$$\scriptstyle{p}$$\scriptstyle{q}$

where $p$ sends a fiber sequence as above to its extremes $(F_{3},F_{1})$ and
$q$ sends it to its middle term $F_{2}$. We have:

######  Lemma 7.34.

  1. (1)

The map $q$ is representable by proper algebraic spaces.

  2. (2)

The map $p$ is derived lci.

###### Proof.

We start by point (1). Notice that this question is insensitive to the derived
structure, so we will pass to the truncations. Direct inspection reveals that
$q$ is $0$-truncated; since source and target are Artin stacks, it follows
that $q$ is representable by algebraic spaces. Observe now that if $X$ is a
smooth (possibly open) $\mathbb{C}$-analytic curve equipped with a trivial
stratification, then the statement is known (see e.g. the proof of [PS,
Theorem 4.9]). Since the collection of representable maps by proper algebraic
spaces is closed under fiber products, we can reduce ourselves to the case
where $X$ is an $\mathbb{C}$-analytic disk $\mathrm{D}$, equipped with the
stratification over $P=\\{0<1\\}$ whose closed stratum consists exactly with
the origin of $\mathrm{D}$. Observe that
${}^{\mathfrak{p}}\mathbf{Perv}_{P}(\mathrm{D})$ can be also described in
terms of representations of the following quiver $Q$:

01

More specifically, the stable $\infty$-category $\mathsf{Rep}(Q)$ of
$\mathbb{C}$-linear $Q$-representations, but equip it with the $t$-structure
$\tau$ defined by declaring that an object $F\in\mathsf{Rep}(Q)$ is connective
if and only if

$F(0)\in(\mathrm{Mod}_{k})_{\geqslant 0}\qquad\text{and}\qquad
F(1)\in(\mathrm{Mod}_{k})_{\geqslant 2}\ .$

Then [DPSb] defines a derived moduli stack $\mathbf{Rep}(Q,\tau)$
parametrizing $\tau$-flat families of finite dimensional $Q$-representations.
Explicitly, $\mathbf{Rep}(Q,\tau)$ sends a derived affine
$S=\operatorname{Spec}(A)$ to the $\infty$-groupoid of $A$-linear
$Q$-representations $F$ such that both $F(0)$ and $F(1)[-2]$ are projective
$A$-modules of finite rank. Inspection reveals that $\mathbf{Rep}(Q,\tau)$ and
${}^{\mathfrak{p}}\mathbf{Perv}_{P}(\mathrm{D})$ are equivalent. We now use
the valuative criterion of properness [Stacks, Tag 0A40] to verify that $q$ is
proper. Let therefore $A$ be a discrete valuation ring with fraction field $K$
and consider the lifting problem

$\displaystyle{\operatorname{Spec}(K)}$$\displaystyle{{}^{\mathfrak{p}}\mathbf{Perv}_{P}^{\mathrm{ext}}(\mathrm{D})}$$\displaystyle{\operatorname{Spec}(A)}$$\displaystyle{{}^{\mathfrak{p}}\mathbf{Perv}_{P}(\mathrm{D})\
.}$$\scriptstyle{t}$$\scriptstyle{q}$$\scriptstyle{s}$

let $F$ be the $A$-linear family of perverse sheaves classified by $s$, and
let

$F^{\prime}\longrightarrow F_{K}\longrightarrow F^{\prime\prime}$

be the extension of $K$-linear families of perverse sheaves classified by $t$,
where $F_{K}\coloneqq K\otimes_{A}F$. Write furthermore

$M\coloneqq F(0)\qquad\text{and}\qquad N\coloneqq F(1)\ .$

Notice that we can review this extension as an extension of perverse sheaves
with coefficients in $\mathrm{Mod}_{A}$ (whose stalks will no longer be
perfect). Define $\widetilde{F}^{\prime\prime}$ to be the image of $F\to
F_{K}\to F^{\prime\prime}$, computed in the abelian category
${}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{A})$. By
construction,
$\widetilde{F}^{\prime\prime}=(\widetilde{M}^{\prime\prime},\widetilde{N}^{\prime\prime},\widetilde{\alpha}^{\prime\prime},\widetilde{\beta}^{\prime\prime})$
is a subobject of
$F^{\prime\prime}=(M^{\prime\prime},N^{\prime\prime},\alpha^{\prime\prime},\beta^{\prime\prime})$.
This means that $\widetilde{M}^{\prime\prime}$ and
$\widetilde{N}^{\prime\prime}$ are subobjects of $M^{\prime\prime}$ and
$N^{\prime\prime}$, and hence they are torsion-free. Since $A$ is a valuation
ring, they are automatically free [Stacks, Tag 0539]. Besides,
$M\to\widetilde{M}^{\prime\prime}$ and $N\to\widetilde{N}^{\prime\prime}$ are
surjective, and hence $\widetilde{M}^{\prime\prime}$ and
$\widetilde{N}^{\prime\prime}$ have finite rank. Thus,
$\widetilde{F}^{\prime}\coloneqq\ker(F\to\widetilde{F}^{\prime\prime})$ is
flat and of finite rank as well. This provides a solution of the previous
lifting problem. The uniqueness follows at once from the uniqueness of the
epi-mono factorization for the morphism $F\to F_{K}\to F^{\prime\prime}$.
Thus, $q$ is proper.

We now prove point (2). The computation of the relative cotangent complex for
the map $p$ done in [PS, Proposition 3.6], reduces us to check that for every
field $k$ and every pair $(F_{3},F_{1})$ of $k$-linear families of perverse
sheaves, the object

$\operatorname{Hom}_{\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathrm{Mod}_{k})}(F_{3},F_{1})\in\mathrm{Mod}_{k}$

is concentrated in homological degrees $[-2,0]$. This follows from 7.33. ∎

At this point, following the discussion in [PS, §4.2], we are led to:

######  Corollary 7.35.

Let $X$ be a smooth projective curve over $\mathbb{C}$ equipped with an
algebraic stratification over $P=\\{0<1\\}$. Let $\mathfrak{p}$ be the middle
perversity function. Then
$\mathrm{Coh}^{\mathrm{b}}({}^{\mathfrak{p}}\mathbf{Perv}_{P}(X))$ carries a
canonical $\mathbb{E}_{1}$-monoidal structure of Hall type. Similarly, the
Borel-Moore homology and $G$-theory of ${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$
carry Hall multiplications.

######  Remark 7.36.

These (categorical, cohomological, $K$-theoretical) Hall algebras generalize
the Betti Hall algebras of [PS, M] (a.k.a. the Hall algebras of the character
variety), which in turn can be seen as a generalization of Schiffmann-Vasserot
CoHA of the commuting variety [SV]. Exploiting the microlocal nature of
perverse sheaves, it is natural to expect that these “perverse CoHAs” admit a
presentation as a Jacobi algebra, generalizing the one of [M]. Moreover,
following [DPSa] it is natural to wonder whether there exists a torsion pair
on ${}^{\mathfrak{p}}\mathrm{Perv}_{P}^{\mathrm{hyp}}(X)$ giving rise to a
“perverse geometric Yangian”. We will come back to these questions in a future
work.

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