# Topological exodromy with coefficients

Mauro PORTA  Mauro PORTA, Institut de Recherche Mathématique Avancée, 7 Rue
René Descartes, 67000 Strasbourg, France<EMAIL_ADDRESS>and  Jean-
Baptiste Teyssier  Jean-Baptiste Teyssier, Institut de Mathématiques de
Jussieu, 4 place Jussieu, 75005 Paris, France  jean-baptiste.teyssier@imj-
prg.fr

###### Abstract.

We improve the exodromy equivalence of MacPherson, Treumann and Lurie in
several ways: first, we allow stratified spaces that have locally weakly
contractible strata, rather than being locally of singular shape, we remove
all noetherianity assumptions and we consider more general coefficients (e.g.
compactly assembled or stable presentable $\infty$-categories). Furthermore,
our approach shows that the exodromy equivalence is functorial for every
morphism of stratified spaces. As an application, we construct, under suitable
finiteness assumptions, a higher Artin derived stack of hyperconstructible
hypersheaves and prove that every perversity function gives rise to an open
substack of perverse hypersheaves. Using the derived structure, we provide the
construction of a new cohomological Hall algebra, generalizing the ones of
character varieties studied in the past by Schiffmann-Vasserot, Davison and
Mistry.

###### Key words and phrases:

Monodromy, exodromy, constructible sheaves, hyperconstructible, hypersheaves,
stratified spaces, exit paths, moduli stack, perverse sheaves, Hall algebras,
CoHA

###### 2020 Mathematics Subject Classification:

55P65,55P65,55U25,14D23,32S40,32S60

###### Contents

  1. 1 Introduction
  2. 2 Review
  3. 3 The categorical framework
  4. 4 Exodromy adjunction for constructible hypersheaves
  5. 5 The exodromy equivalence
  6. 6 Consequences
  7. 7 Moduli spaces

##  1\. Introduction

Let $X$ be a locally contractible topological space. In classical terms, the
monodromy correspondence is an equivalence between the abelian category
$\mathrm{LC}(X;\mathrm{Ab})$ of locally constant sheaves of abelian groups on
$X$ and the abelian category $\operatorname{Fun}(\Pi_{1}(X),\mathrm{Ab})$ of
representations of the first fundamental groupoid of $X$. This equivalence can
be seen as a concrete way of bridging the topological world, incarnated by the
locally constant sheaves, and the algebraic world, incarnated by the
representations of the first fundamental groupoid of $X$. The advent of higher
categorical techniques [HTT] allowed to improve this correspondence by
replacing $\Pi_{1}(X)$ with the _homotopy type_ $\Pi_{\infty}(X)$ of $X$,
which in many senses is still an object of algebraic nature. There is a
natural map $\Pi_{\infty}(X)\to\Pi_{1}(X)$, which induces an equivalence

$\operatorname{Fun}(\Pi_{1}(X),\mathrm{Ab})\longrightarrow\operatorname{Fun}(\Pi_{\infty}(X),\mathrm{Ab})\
,$

but interestingly this is no longer true if we replace $\mathrm{Ab}$ by its
derived $\infty$-category $\mathcal{D}(\mathrm{Ab})$111Notice that throughout
the main body of the paper, we will rather write $\mathrm{Mod}_{\mathbb{Z}}$
instead of $\mathcal{D}(\mathrm{Ab})$.. On the other hand, what is still true
is that there is a canonical equivalence

$\operatorname{Fun}(\Pi_{\infty}(X),\mathcal{D}(\mathrm{Ab}))\simeq\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{D}(\mathrm{Ab}))\
,$

where the right hand side denotes the $\infty$-category of hypersheaves that
are locally the hypersheafification of a constant presheaf. Notice that these
two categories are sensitive to the whole homotopy type of $X$, rather than
just to its first fundamental groupoid.

There is a second natural generalization of the monodromy correspondence,
whose fundamental idea is due to MacPherson. When $X$ is equipped with a
stratification $P$, that is a continuous morphism $X\to P$ where $P$ is a
poset endowed with the Alexandroff topology (cf. Section 2.1), one can
introduce a stratified variant of $\Pi_{1}(X)$, noted $\Pi_{1}^{\Sigma}(X,P)$.
This is a non-full subcategory of $\Pi_{1}(X)$ which is no longer a groupoid;
its objects are the same (that is, the points of $X$), but the only morphisms
that are allowed are those that _exit_ from lower strata to go to upper ones.
For this reason, the category $\Pi_{1}^{\Sigma}(X,P)$ is called the _category
of exit paths of $(X,P)$_. In [T, Theorem 5.7], D. Treumann studied this
category and established an equivalence

$\operatorname{Fun}\big{(}\Pi_{1}^{\Sigma}(X,P),\mathrm{Ab}\big{)}\simeq\operatorname{Cons}_{P}(X;\mathrm{Ab})\
,$

where the right hand side denotes the abelian category of $\mathrm{Ab}$-valued
constructible sheaves. Following the influential work [BGH], this equivalence
is nowadays referred to as the _exodromy equivalence_. It was generalized to
the higher categorical world by J. Lurie in [HA, Appendix A]. His first step
is to define a version $\Pi_{\infty}^{\Sigma}(X,P)$ of $\Pi_{1}^{\Sigma}(X,P)$
and establish a major result (Theorem A.6.4 in _loc. cit._) asserting that if
the stratification $(X,P)$ is _conical_ , then $\Pi_{\infty}^{\Sigma}(X,P)$ is
indeed an $\infty$-category. Then, he proves:

######  Theorem 1.1 (J. Lurie, [HA, Theorem A.9.3]).

Let $X$ be a paracompact topological space which is locally of singular shape
and equipped with a conical $P$-stratification, where $P$ is a partially
ordered set satisfying the ascending chain condition. Then there is a natural
equivalence

$\Psi_{X,P}\colon\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{S}\big{)}\longrightarrow\operatorname{Cons}_{P}(X;\mathcal{S})\
,$

where $\mathcal{S}$ denotes the $\infty$-category of homotopy types (a.k.a.
$\infty$-groupoids or animated sets).

The goal of this paper is to improve this result in several ways:

  1. (1)

replace the condition that $X$ is locally of singular shape with the weaker
condition that the strata of $(X,P)$ are locally weakly contractible;

  2. (2)

remove the ascending chain condition on the poset $P$;

  3. (3)

allow more general coefficients than $\mathcal{S}$; in particular every
presentable $\infty$-category that is either stable or a retract of a
compactly generated $\infty$-category is permitted.

  4. (4)

provide an explicit formula for the inverse of $\Psi_{X,P}$.

We refer to 5.17 for the precise statement. Let us observe that improvements
(1) and (2) are related, and the latter has already been addressed by D. Lejay
in [L]. The key idea is to replace the $\infty$-category
$\operatorname{Cons}_{P}(X;\mathcal{S})$ with the $\infty$-category
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{S})$ of $P$-_hyper_
constructible _hyper_ sheaves. Indeed, the singular shape assumption
guarantees that every locally constant sheaf is automatically a hypersheaf.
Working with hypersheaves from the very beginning, we can relax this
assumption and replace it with the weaker one of having locally weakly
contractible strata.

Improvements (3) and (4) are also related. The proof contained in [HA]
explicitly relies on model categories, and is heavily built on the fact that
functors from $\Pi_{\infty}^{\Sigma}(X,P)$ to $\mathcal{S}$ can be realized as
left fibrations over $\Pi_{\infty}^{\Sigma}(X,P)$. These two facts together
make it difficult to adapt the proof _verbatim_ to different coefficients than
$\mathcal{S}$. On the other hand, it is also difficult to bootstrap on Lurie’s
result to deduce the same result for arbitrary presentable
$\infty$-categories, although it would be possible to deal with the case of
compactly generated $\infty$-categories in this way.

We propose therefore a new proof which is purely $\infty$-categorical and that
we briefly summarize below. Our approach has also several pleasant
consequences. First and foremost, we obtain a _better functoriality_ of the
exodromy equivalence. Notably, if $f\colon(X,P)\to(Y,Q)$ is a morphism between
conically stratified spaces, then the square

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

is made $2$-commutative by a natural transformation $\psi_{f}$ that is a
priori not invertible. We say that $f$ is _exodromic_ if $\psi_{f}$ is an
equivalence. In [HA, Proposition A.9.6] it is shown that if the stratification
on $X$ coincides with the stratification $f^{-1}(Q)$ induced by $f$, then $f$
is exodromic. Our approach allows to improve this result:

######  Corollary 1.2 (See 6.8).

If $f\colon(X,P)\to(Y,Q)$ is a morphism between conically stratified spaces
with locally weakly contractible strata, then $f$ is exodromic.

As a consequence of this improved functoriality, we obtain the following:

######  Corollary 1.3 (See 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.

Taking $P$ to be the trivial stratification, we deduce that $\Pi_{\infty}(X)$
is a localization of $\Pi_{\infty}^{\Sigma}(X,Q)$. This result, which is of
course extremely intuitive at the geometric level, was only known to the best
of our knowledge in the conically smooth situation [AFTb, Proposition 1.2.13].

The key new ingredient in our proof is to interpolate between
$\mathrm{Open}(X)^{\mathrm{op}}$ and $\Pi_{\infty}^{\Sigma}(X,P)$ as follows.
Given a conically stratified space $(X,P)$, let $\mathrm{E}_{X}$ be the
$\infty$-category informally defined as:

  * •

its objects are pairs $(U,x)$, where $U$ is an open subset of $X$ and $x$ is
an object in $\Pi_{\infty}^{\Sigma}(U,P)$, where $U$ is seen as a
$P$-stratified space in the natural way;

  * •

given two objects $(U,x)$ and $(V,y)$, the mapping space
$\operatorname{Map}_{\mathrm{E}_{X}}((U,x),(V,y))$ is non-empty if and only if
$V\subset U$, and in that case it coincides with
$\operatorname{Map}_{\Pi_{\infty}^{\Sigma}(U,P)}(x,y)$.

Put otherwise, $\mathrm{E}_{X}$ is the Grothendieck construction of the
functor $\Pi_{\infty}^{\Sigma}\colon\mathrm{Open}(X)\to\mathrm{Cat}_{\infty}$.
We refer to Section 3 and to 5.3 for the precise definition of
$\mathrm{E}_{X}$. This $\infty$-category comes with natural functors

$\displaystyle{\mathrm{E}_{X}}$$\displaystyle{\mathrm{Open}(X)^{\mathrm{op}}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X,P)\
.}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\lambda_{X}}$

Then the functor $\Psi_{X,P}$ of Lurie can be identified with
$\pi_{X,\ast}\circ\lambda_{X}^{\ast}$ where $\pi_{X,\ast}$ denotes the right
Kan extension along $\pi_{X,\ast}$, and $\lambda_{X}^{\ast}$ denotes the
restriction along $\lambda_{X}$. The main bulk of the paper is dedicated to
show, for more general coefficients than just $\mathcal{S}$ and independently
from Lurie’s result, that this functor and its left adjoint
$\Phi_{X,P}\coloneqq\lambda_{X,!}\circ\pi_{X}^{\ast}$ realize the exodromy
equivalence.

Let $F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ be a hypersheaf. A simple
inspection reveals that $\Phi_{X,P}(F)$ evaluated at one point $x\in X$ seen
as an object in $\Pi_{\infty}^{\Sigma}(X,P)$ can be written as

$\Phi_{X,P}(F)(x)\simeq\operatorname*{colim}_{(U,y,\gamma)}F(U)\ ,$

where the colimit ranges over all morphisms $\gamma\colon y\to x$ in
$\Pi_{\infty}^{\Sigma}(X,P)$ and over all possible open neighborhood $U$ of
$y$. If we could simply limit ourselves to the case where $\gamma$ is the
identity of $x$, this would simply produce the stalk $F_{x}$ of $F$ at the
point $x$. However, this is typically false and $\Phi_{X,P}(F)(x)$ should be
rather thought of as an “average over all the possible stalks at points close
to $x$”. Remarkably, when $F$ is $P$-hyperconstructible, we can prove that
there is indeed an equivalence

$\Phi_{X,P}(F)(x)\simeq F_{x}\ .$

See 6.10. Although this is formally obtained as a consequence of our results,
in many ways one should think of this equivalence as the key technical
ingredient needed in the proof of our main theorem. Interestingly, the left
hand side only depends on the equivalence class of $x$ inside
$\Pi_{\infty}^{\Sigma}(X,P)$, a statement that neatly encodes the _parallel
transport_ on the right hand side.

In the rest of the paper, we explore the consequences of the exodromy
correspondence. First, we obtain a couple of structural results for
constructible hypersheaves:

######  Theorem 1.4 (See Corollaries 5.20, 5.22, 6.3 and 6.5).

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. Then:

  1. (1)

_Stability under limits and colimits:_ the $\infty$-category
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ of
$P$-hyperconstructible hypersheaves is presentable and closed under limits and
colimits inside $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$;

  2. (2)

_Recognition criterion:_ a hypersheaf
$F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ is $P$-hyperconstructible if
and only if for every pair of open subsets $U\subseteq V$ of $X$ for which the
induced morphism $\Pi_{\infty}^{\Sigma}(U,P)\to\Pi_{\infty}^{\Sigma}(V,P)$ is
a categorical equivalence, the restriction map

$F(V)\longrightarrow F(U)$

is an equivalence in $\mathcal{E}$.

  3. (3)

_Tensor decomposition:_ the canonical equivalence
$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\simeq\mathrm{Sh}^{\mathrm{hyp}}(X)\otimes\mathcal{E}$
restricts to an equivalence

$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\simeq\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{S})\otimes\mathcal{E}\
;$

  4. (4)

_Categorical Künneth formula:_ given a second conically stratified space
$(Y,Q)$ with locally weakly contractible strata, there is a canonical
equivalence

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

Observe that point (1) implies that the inclusion
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\hookrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
admits both a left and a right adjoint. Combined with 1.2, it follows that for
every morphism $f\colon(X,P)\to(Y,Q)$ of conically stratified spaces with
locally weakly contractible strata, the pullback functor

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

admits a left adjoint $f_{\sharp}^{\mathrm{hyp}}$, a statement that is
generically false for the $\infty$-category
$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ (with the due exceptions, see
[HPT, §2.2]). Points (2), (3) and (4) are generalizations to the setting of
hyperconstructible hypersheaves of the analogous results already obtained in
[HPT] for locally hyperconstant hypersheaves. As an immediate consequence of
these results, we recover and generalize some of the results on Morita
cohomology of J. V. Holstein [Ha, Hb]:

######  Corollary 1.5.

Let $X$ be a locally weakly contractible topological space and let $A$ be a
$\mathbb{E}_{\infty}$-ring spectrum. Assume that $X$ is connected and let
$x\in X$ be a point. Then there exists a canonical equivalence

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

where
$\mathrm{C}_{\ast}(\Omega_{x}(X);A)\coloneqq\Pi_{\infty}(\Omega_{x}(X))\otimes
A$ are the $A$-valued chains on the based loop space of $X$ at $x$.

Last but not least, we combine the results of this paper together with the
recent finiteness result for $\Pi_{\infty}^{\Sigma}(X,P)$ of M. Volpe [V,
Corollary 2.13], to prove the following result:

######  Theorem 1.6 (cf. Theorems 7.8 & 7.32).

Let $X$ be a complex analytic space equipped with a Whitney stratification
$(X,P)$. Assume that for every $p\in P$, the stratum $X_{p}$ is homotopically
compact.222This means that $\Pi_{\infty}(X)$ is a compact object in
$\mathcal{S}$. This is a very mild condition that includes many cases of
interest, see 7.3. Then there exists a higher Artin derived stack
$\mathbf{Cons}_{P}(X)$ parametrizing hyperconstructible hypersheaves on
$(X,P)$. Furthermore, for every perversity function $\mathfrak{p}\colon
P\to\mathbb{Z}$, there exists an open substack
${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$ of $\mathbf{Cons}_{P}(X)$
parametrizing $\mathfrak{p}$-perverse hypersheaves on $X$.

In the main body of the paper we actually prove a finer result which does not
require neither a complex analytic structure nor the stratification to be
Whitney. The analysis carried out suggests that Volpe’s finitness result can
be strengthened and we provide a precise statement of the expected result in
7.10. Recall also that a similar result was obtained by Nitsure and Sabbah in
[NS]. The novelty in our case is the derived structure and the fact that
$\mathbf{Cons}_{P}(X)$ and ${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$ are
sensitive to the higher homotopical information of $(X,P)$. As an example of
application of the derived structure, we obtain the following result:

######  Theorem 1.7.

Let $X$ be a Riemann surface and let $(X,P)$ be a stratification on $X$ made
by a finite number of points. Let $\mathfrak{p}$ be the middle perversity
function. Then the stable $\infty$-category
$\mathrm{Coh}^{\mathrm{b}}({}^{\mathfrak{p}}\mathbf{Perv}_{P}(X))$ carries an
$\mathbb{E}_{1}$-monoidal structure _à la Hall_ , whose underlying tensor
product is given by pull-push along the 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)}$$\scriptstyle{q}$$\scriptstyle{p}$

Similarly, the Borel-Moore homology and the $G$-theory of
${}^{\mathfrak{p}}\mathbf{Perv}_{P}(X)$ carry a canonical Hall multiplication.

Implicit in the above theorem is the fact that the map $p$ is derived locally
complete intersection and that $q$ is representable by proper algebraic spaces
(see 7.34). This result generalizes [PS, Theorem 1.4], where the case of local
systems was dealt with, and generalizes the cohomological Hall algebras of
character varieties studied in [M]. This raises new interesting questions in
geometric representation theory, that are outlined in 7.36 and to which we
will come back in a future work. Finally, we expect $\mathbf{Cons}_{P}(X)$ to
carry a canonical shifted symplectic form, at least under some orientation
condition on $(X,P)$ that should conjecturally be related to Poincaré duality
for the intersection homology of $(X,P)$ [AFTa], and tightly related to the
paper in preparation [APT].

#### Acknowledgments.

We are grateful to Clark Barwick, Federico Binda, Damien Calaque, Denis-
Charles Cisinski, Andrea Gagna, Julian V. Holstein, Damien Lejay, Alexandru
Oancea, Marco Robalo, Francesco Sala, Olivier Schiffmann and Alberto Vezzani
for useful conversations about this paper. We especially thank Peter J. Haine,
Guglielmo Nocera, Tony Pantev and Marco Volpe for their continuous support and
interest in this work. We wish to thank the Oberwolfach MFO institute that
hosted the Research in Pairs “2027r: The geometry of the Riemann-Hilbert
correspondence”, where the initial bulk of this paper was conceived. We also
wish to thank the CNRS for the fundings PEPS “Jeunes Chercheurs Jeunes
Chercheuses” and the ANR CatAG from which both authors benefited during the
writing of this paper.

##  2\. Review

###  2.1. Stratified spaces

The main reference for the definitions below is [HA, Appendix A].

If $P$ be a poset, we endow $P$ with the topology whose open subsets are the
closed upward subsets $Q\subset P$. That is for every $a\in Q$ and $b\in P$
such that $b\geqslant a$, we have $b\in Q$.

######  Definition 2.1.

Let $X$ be a topological space. Let $P$ be a poset. A stratification of $X$ by
$P$ is a continuous morphism $f\colon X\to P$. For a subset $S\subset P$, we
let $f|_{S}\colon X_{S}\to S$ be the induced stratification. For $a\in P$, the
subset $X_{a}$ is the stratum of $(X,P)$ over $a$.

######  Remark 2.2.

We abuse notations by denoting a stratification of $X$ by $P$ as $(X,P)$
instead of $f\colon X\to P$ and refer to $(X,P)$ as a stratified space. The
collection of stratified spaces organize into a category in an obvious manner.

######  Example 2.3.

Let $f\colon Y\to Q$ be a stratified space. Put
$C(Y)\coloneqq\ast\sqcup\left(Y\times\mathbb{R}_{>0}\right)$. The set $C(Y)$
is endowed with the topology whose open subsets are the subsets $U\subset
C(Y)$ such that $U\cap\left(Y\times\mathbb{R}_{>0}\right)$ is open and if
$\ast\in U$, then

$C_{\varepsilon}(Y):=\ast\sqcup\left(Y\times(0,\varepsilon)\right)\subset U$

for some $\varepsilon>0$. Let $Q^{\vartriangleleft}$ be the poset obtained
from $Q$ by adding a smallest element $-\infty$. We define a continuous map
$g\colon C(Y)\to Q^{\vartriangleleft}$ by sending $\ast$ to $-\infty$ and
$(y,t)\in Y\times\mathbb{R}_{>0}$ to $f(y)$. We refer to
$(C(Y),Q^{\vartriangleleft})$ as the cone of $(Y,Q)$.

Following [HA, A.6.2], we introduce the

######  Definition 2.4.

Let $(X,P)$ be a stratified space. We define $\mathrm{Exit}(X,P)$ as the
simplicial subset of $\operatorname{Sing}X$ formed by the simplices
$\sigma\colon|\Delta^{n}|\to X$ such that there exists a chain
$a_{1}\leqslant\cdots\leqslant a_{n}$ of elements of $P$ such that for every
$(t_{0},\dots,t_{i},0,\dots,0)\in|\Delta^{n}|$ with $t_{i}>0$, we have
$\sigma(t_{0},\dots,t_{i},0,\dots,0)\in X_{a_{i}}$.

The simplicial set $\mathrm{Exit}(X,P)$ is the exit-path simplicial set of
$(X,P)$.

######  Notation 2.5.

If $X$ is trivially stratified $\Pi_{\infty}^{\Sigma}(X)$ is the homotopy type
$\Pi_{\infty}(X)$ of $X$. In virtue of this observation, we think of
$\Pi_{\infty}^{\Sigma}(X)$ as the _stratified homotopy type of $X$_ [D, H],
and for this reason if $\mathrm{Exit}(X,P)$ is an $\infty$-category we write
$\Pi_{\infty}^{\Sigma}(X,P)$ instead of $\mathrm{Exit}(X,P)$.

The following lemma follows immediately from the definition of the exit-paths.

######  Lemma 2.6.

Let $(X,P)$ be a stratified space. Let $S\subset P$ be a subset. Assume that
$\mathrm{Exit}(X,P)$ and $\mathrm{Exit}(X_{S},S)$ are $\infty$-categories.
Then, the natural inclusion $\mathrm{Exit}(X_{S},S)\to\mathrm{Exit}(X,P)$ is
fully-faithful.

######  Lemma 2.7.

Let $(Y,Q)$ be a stratified space such that
$\mathrm{Exit}(C(Y),Q^{\vartriangleleft})$ is an $\infty$-category. Then,
$\ast$ is an initial object in $\mathrm{Exit}(C(Y),Q^{\vartriangleleft})$.

###### Proof.

Since $\ast$ is the only point of its stratum,
$\operatorname{Map}_{\mathrm{Exit}(C(Y),Q^{\vartriangleleft})}(*,*)$ is
contractible. Let $(y,\varepsilon)\in Y\times\mathbb{R}_{>0}$. We are going to
construct a deformation retract of

$\operatorname{Map}_{\mathrm{Exit}(C(Y),Q^{\vartriangleleft})}(*,(y,\varepsilon))$

on the simplicial subset spanned by the exit path $\gamma\colon[0,1]\to C(Y)$
sending $0$ to $\ast$ and $t$ to $(y,t\varepsilon)$. To do this, we represent
the above mapping space as $\operatorname{Hom}^{R}(\ast,(y,\varepsilon))$ and
construct an homotopy

$H\colon\operatorname{Hom}^{R}(\ast,(y,\varepsilon))\times\Delta^{1}\longrightarrow\operatorname{Hom}^{R}(\ast,(y,\varepsilon))$

between the map constant to $\gamma$ and the identity of
$\operatorname{Hom}^{R}(\ast,(y,\varepsilon))$. At the cost of writing
$\operatorname{Hom}^{R}(\ast,(y,\varepsilon))$ as a colimit of its simplices
$\sigma\colon\Delta^{n}\longrightarrow\operatorname{Hom}^{R}(\ast,(y,\varepsilon))$,
it is enough to construct an homotopy $H$ between the $n$-simplex constant to
$\gamma$ and $\sigma$. That is, we have to construct

$H\colon\Delta^{n}\times\Delta^{1}\longrightarrow\operatorname{Hom}^{R}(\ast,(y,\varepsilon))$

such that $H|_{\Delta^{n}\times\Delta^{\\{0\\}}}$ is constantly equal to
$\gamma$ and $H|_{\Delta^{n}\times\Delta^{\\{1\\}}}=\sigma$. Set
$Z\coloneqq|\Delta^{n}|\times[0,1]$ and
$Z_{\lambda}\coloneqq|\Delta^{n}|\times\\{\lambda\\}$ for every
$\lambda\in[0,1]$. By definition, the above construction amounts to the
construction of a continuous map

$H\colon Z\star\Delta^{0}\longrightarrow C(Y)$

satisfying the following conditions :

  1. (1)

$H|_{Z_{0}\star\Delta^{0}}$ is constantly equal to $\gamma$ and
$H|_{Z_{1}\star\Delta^{0}}=\sigma$.

  2. (2)

$H$ sends the base of $Z\star\Delta^{0}$ to $\ast$ and the tip of
$Z\star\Delta^{0}$ to $(y,\varepsilon)$.

  3. (3)

$H$ sends $Z\times]0,1]$ in the stratum of $(y,\varepsilon)$.

For $(u,t)\in Z_{1}\times]0,1]$, put

$\sigma(u,t)=(y(u,t),\varepsilon(u,t))\in Y\times\mathbb{R}_{>0}$

By definition $\sigma(u,1)=(y,\varepsilon)$. For $(u,\lambda,t)\in
Z\star\Delta^{0}$, we define

$H(u,\lambda,t)=\begin{cases}\big{(}y(u,t/\lambda),\lambda\varepsilon(u,t/\lambda)\big{)}&\text{if
}0<t\leqslant\lambda\ ,\\\ \gamma(t)&\text{if }\lambda\leqslant t\ ,\\\
\ast&\text{if }t=0\ .\end{cases}$

Observe that $H$ satisfies conditions (1), (2), (3). To conclude the proof of
2.7, we have to show that $H$ is continuous. Away from $t=0$, the map $H$ is
continuous as the glueing of two continuous maps along $t=\lambda$. Let
$(u_{0},\lambda_{0},0)\in Z\star\Delta^{0}$. We want to check that $H$ is
continuous at $(u_{0},\lambda_{0},0)$. If $\lambda_{0}>0$, we have to check
that $H$ is continuous on the open set $\lambda>t$, which is true since
$\sigma$ is continuous. The continuity of $H$ at $(u_{0},0,0)$ follows from
the definition of the topology of $C(Y)$ and the observation that
$\varepsilon$ is bounded. ∎

######  Definition 2.8.

Let $(X,P)$ be a stratified space. We say that $(X,P)$ is conically stratified
if for every point $x\in X$ lying over $a\in P$, there exists a topological
space $Z$, a stratified space $(Y,P_{>a})$ and a morphism of stratified spaces
$(Z\times C(Y),P_{\geqslant a})\to(X,P)$ inducing an homeomorphism between
$Z\times C(Y)$ and an open neighbourhood of $x$ in $X$.

######  Remark 2.9.

The open sub-stratified space $(Z\times C(Y),P_{\geqslant a})$ of $(X,P)$ in
2.8 is a conical chart of $(X,P)$ at $x$. By definition of the topology of
$C(Y)$, the set of conical charts of $(X,P)$ at $x$ form a fundamental system
of open neighbourhoods of $x$ in $X$.

Conically stratified spaces enjoy the following stability property:

######  Lemma 2.10.

Let $(X,P)$ be a conically stratified space. Let $S\subset P$ be a subset.
Then $(X_{S},S)$ is a conically stratified space.

###### Proof.

Let $x\in X_{S}$ and let $s\in S$ be the stratum of $x$. We have to show that
$(X_{S},S)$ is conical at $x$. The question is local on $X_{S}$. At the cost
of replacing $X$ by the open set $X_{\geqslant s}$, we can thus suppose that
$s$ is the minimum of $P$. Since $X$ is conical at $x$, we can suppose further
that $(X,P)$ is of the form $(Z\times C(Y),P)$ where $Z$ is a topological
space and where $(Y,P_{>s})$ is a stratified space. Then $X_{S}=Z\times
C(Y_{S_{>s}})$ and 2.10 is proved. ∎

The following theorem due to Lurie [HA, A.6.4] provides a wide range of
stratified spaces whose exit-paths form an $\infty$-category:

######  Theorem 2.11.

Let $(X,P)$ be a conically stratified space. Then $\mathrm{Exit}(X,P)$ is an
$\infty$-category.

######  Remark 2.12.

In view of 2.11 and 2.5, the $\infty$-category of exit-paths of a conically
stratified space $(X,P)$ will be denoted as $\Pi_{\infty}^{\Sigma}(X,P)$.

######  Definition 2.13.

Let $(X,P)$ be a conically stratified space. We say that $(X,P)$ is locally
weakly contractible if every point $x\in X$ admits a fundamental system of
open neighbourhoods $U$ such that $x$ is an initial object of
$\Pi_{\infty}^{\Sigma}(U,P)$.

######  Lemma 2.14.

Let $f\colon X\to X^{\prime}$ be a weak homotopy equivalence and let $(Y,Q)$
be a stratified space. Then the induced map

$\mathrm{Exit}(X\times Y,Q)\longrightarrow\mathrm{Exit}(X^{\prime}\times Y,Q)$

is a categorical equivalence of simplicial sets.

###### Proof.

Unraveling the definitions, we find canonical _isomorphisms_ of simplicial
sets

$\mathrm{Exit}(X\times
Y,Q)\simeq\operatorname{Sing}(X)\times\mathrm{Exit}(Y,Q)\quad\text{and}\quad\mathrm{Exit}(X^{\prime}\times
Y,Q)\simeq\operatorname{Sing}(X^{\prime})\times\mathrm{Exit}(Y,Q)\ .$

Since the map $f$ induces a categorical equivalence
$\operatorname{Sing}(X)\to\operatorname{Sing}(X^{\prime})$, the conclusion now
follows from [HTT, Corollary 2.2.5.4]. ∎

######  Proposition 2.15.

Let $(X,P)$ be a conically stratified space. The following conditions are
equivalent:

  1. (1)

$(X,P)$ is locally weakly contractible.

  2. (2)

The strata of $(X,P)$ are locally weakly contractible.

######  Remark 2.16.

Let $(X,P)$ be a conically stratified space whose strata are CW-complexes.
Then 2.15 states that $(X,P)$ is locally weakly contractible if and only if
its strata are locally contractible.

###### Proof of 2.15.

2.6 shows that $(1)$ implies $(2)$. Assume now that $(2)$ holds. Let $x\in X$
be a point and let $a$ be its image in $P$. Since $X$ is conical, we can
suppose that $X=Z\times C(Y)$ where $(Y,Q)$ is a stratified space with $a$ the
minimum of $P=Q^{\vartriangleleft}$. Let $\mathcal{W}_{x}$ be a fundamental
system of weakly contractible open neighborhoods of $x$ in $Z$. Then
$\\{W\times C_{\varepsilon}(Y)\\}_{W\in\mathcal{W}_{x},\varepsilon>0}$ is a
fundamental system of open neighborhoods of $x$ inside $X$. For
$W\in\mathcal{W}_{x}$ and $\varepsilon>0$, we have an isomorphism of
simplicial sets

$\mathrm{Exit}(W\times
C_{\varepsilon}(Y),P)\simeq\operatorname{Sing}(W)\times\mathrm{Exit}(C_{\varepsilon}(Y),Q^{\vartriangleleft})$

On the other hand, if $K$ and $L$ are non empty simplicial sets such that
$K\times L$ is an $\infty$-category, so are $K$ and $L$. Thus, 2.11 ensures
that $\mathrm{Exit}(C_{\varepsilon}(Y),Q^{\vartriangleleft})$ is an
$\infty$-category. Since $W$ is weakly contractible, we obtain an equivalence
of $\infty$-categories

$\mathrm{Exit}(W\times
C_{\varepsilon}(Y),P)\simeq\mathrm{Exit}(C_{\varepsilon}(Y),Q^{\vartriangleleft})$

sending $x$ to $\ast\in C(Y)$. We conclude using 2.7. ∎

###  2.2. Local excellence at strata

######  Definition 2.17.

Let $(X,P)$ be a conically stratified space and let $S\subset P$ be a
subposet. We say that an open neighbourhood $U$ of $X_{S}$ inside $X$ is
_excellent at $S$_ if the functor

$\mathrm{Exit}(X_{S},S)\longrightarrow\mathrm{Exit}(U,P)$

is a final.

######  Definition 2.18.

Let $(X,P)$ be a conically stratified space and let $S\subset P$ be a
subposet. We say that $(X,P)$ is _excellent at $S$_ if the collection of
excellent at $S$ open neighbourhoods of $X_{S}$ inside $X$ forms a fundamental
system of neighbourhoods of $X_{S}$ inside $X$.

######  Definition 2.19.

Let $(X,P)$ be a conically stratified space and let $S\subset P$ be a
subposet. We say that $(X,P)$ is _locally excellent at $S$_ if every point
$x\in X_{S}$ admits a fundamental system of open neighbourhoods $U$ such that
the stratified space $(U,P)$ is excellent at $S$.

######  Proposition 2.20.

Let $(X,P)$ be a conically stratified space. Then $(X,P)$ is locally excellent
at every stratum.

###### Proof.

Let $a\in P$ and let $x\in X$ be a point lying over $a$. Since $(X,P)$ is
conically stratified, the point $x$ admits a fundamental system of open
neighbourhoods of the form $Z\times C(Y)$ where $(Y,Q)$ is a stratified space,
where $Z$ is an open set of $X_{a}$ containing $x$ and where $a$ is the
minimum of $P_{\geqslant a}=Q^{\vartriangleleft}$. To prove 2.20, it is thus
enough to prove that the conically stratified space $(Z\times
C(Y),Q^{\vartriangleleft})$ is excellent at $a$. In that case, $\\{Z\times
C_{\varepsilon}(Y)\\}_{\varepsilon>0}$ is a fundamental system of open
neighborhoods of $Z$. On the other hand, there is an isomorphism of simplicial
sets

$\mathrm{Exit}(Z\times
C_{\varepsilon}(Y),Q^{\vartriangleleft})\simeq\operatorname{Sing}(Z)\times\mathrm{Exit}(C_{\varepsilon}(Y),Q^{\vartriangleleft})$

Hence, the functor

$\mathrm{Exit}(Z\times\ast,\\{a\\})\longrightarrow\mathrm{Exit}(Z\times
C_{\varepsilon}(Y),Q^{\vartriangleleft})$

reads as a product

(2.21)
$\operatorname{Sing}(Z)\times\ast\longrightarrow\operatorname{Sing}(Z)\times\mathrm{Exit}(C_{\varepsilon}(Y),Q^{\vartriangleleft})$

From 2.7, the functor
$\ast\to\mathrm{Exit}(C_{\varepsilon}(Y),Q^{\vartriangleleft})$ is final.
Thus, Eq. 2.21 is final as a consequence of [HTT, 4.1.1.13]. ∎

###  2.3. Hypersheaves, hyperconstancy and hyperconstructibility

######  Definition 2.22.

Let $X$ be a topological space. Let $\mathcal{E}$ be a presentable
$\infty$-category. We denote by $\mathrm{Sh}(X;\mathcal{E})$ the full
subcategory of
$\operatorname{Fun}(\mathrm{Open}(X)^{\mathrm{op}},\mathcal{E})$ spanned by
$\mathcal{E}$-valued presheaves satisfying descent.

######  Remark 2.23.

If $\mathcal{E}=\mathcal{S}$, we simply denote $\mathrm{Sh}(X;\mathcal{E})$ by
$\mathrm{Sh}(X)$. In general, we have
$\mathrm{Sh}(X;\mathcal{E})\simeq\mathrm{Sh}(X)\otimes\mathcal{E}$, where the
tensor product denotes the tensor product of presentable $\infty$-categories
introduced in [HA, §4.8].

######  Definition 2.24.

The category of hypersheaves on $X$ is the subcategory
$\mathrm{Sh}^{\mathrm{hyp}}(X)\subseteq\mathrm{Sh}(X)$ spanned by objects
which are local with respect to $\infty$-connective morphisms in
$\mathrm{Sh}(X)$. For a presentable $\infty$-category $\mathcal{E}$, we put

$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(X)\otimes\mathcal{E}\simeq\operatorname{Fun^{R}}(\mathrm{Sh}^{\mathrm{hyp}}(X)^{\mathrm{op}},\mathcal{E})$

######  Remark 2.25.

The canonical inclusion $\mathrm{Sh}^{\mathrm{hyp}}(X)\subset\mathrm{Sh}(X)$
admits a left adjoint
$(-)^{\mathrm{hyp}}:\mathrm{Sh}(X)\to\mathrm{Sh}^{\mathrm{hyp}}(X)$ referred
to as the hypersheafification functor. Tensoring with $\mathcal{E}$ thus
induces a functor

$(-)^{\mathrm{hyp}}\colon\mathrm{Sh}(X;\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$

admitting a fully-faithful right adjoint. Hence, we can think of objects of
$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ as $\mathcal{E}$-valued
preseheaves satisfying hyperdescent.

Let $f\colon X\to Y$ be a morphism of topological spaces. Let $\mathcal{E}$ be
a presentable $\infty$-category. Let

$f^{\ast}\colon\mathrm{Sh}(Y;\mathcal{E})\leftrightarrows\mathrm{Sh}(X;\mathcal{E})\colon
f_{\ast}$

be the canonical adjunction. The functor $f_{*}$ preserve hypersheaves and
thus restricts to a functor

$f_{*}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})$

Furthermore, the functor $f_{*}$ admits

$f^{\ast,\mathrm{hyp}}\coloneqq(-)^{\mathrm{hyp}}\circ
f^{*}:\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E}).$

as left adjoint.

In the next definition, $\Gamma_{X}:X\to\ast$ denotes the tautological
morphism.

######  Definition 2.26.

Fix $F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$. We say that $F$ is
hyperconstant if $F$ lies in the essential image of
$\Gamma_{X}^{\ast,\mathrm{hyp}}\colon\mathcal{E}\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$.

We say that $F$ is locally hyperconstant if there exists a cover of $X$ by
open subsets $i\colon U\to X$ such that $i^{\ast,\mathrm{hyp}}(F)$ is
hyperconstant. We denote by $\mathrm{LC}(X;\mathcal{E})$ the full subcategory
of $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ spanned by locally
hyperconstant hypersheaves.

Following [L], we now introduce the main player of this paper:

######  Definition 2.27.

Let $(X,P)$ be a stratified space. Let $\mathcal{E}$ be a presentable
$\infty$-category. An hypersheaf
$F\colon\mathrm{Open}(X)^{\mathrm{op}}\to\mathcal{E}$ with value in
$\mathcal{E}$ is hyperconstructible if for every $p\in P$, the hypersheaf
$j_{p}^{\ast,\mathrm{hyp}}(F)$ is locally hyperconstant on $X_{p}$, where
$j_{p}\colon X_{p}\to X$ denotes the canonical inclusion. We denote by
$\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ the full-subcategory
of $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ spanned by hyperconstructible
hypersheaves on $(X,P)$.

###  2.4. Change of coefficients and hyperconstructibility

Let $X$ be a topological space. Let $f\colon\mathcal{E}\to\mathcal{D}$ be a
cocontinuous functor between presentable $\infty$-categories with right
adjoint $g\colon\mathcal{D}\to\mathcal{E}$. Then, there is an adjunction

(2.28)
$f\circ-\colon\operatorname{Fun}(\mathrm{Open}(X)^{\mathrm{op}},\mathcal{E})\leftrightarrows\operatorname{Fun}(\mathrm{Open}(X)^{\mathrm{op}},\mathcal{D})\colon
g\circ-$

whose left adjoint induces a cocontinuous functor

$f^{\mathrm{hyp}}\coloneqq(-)^{\mathrm{hyp}}\circ
f\circ-\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\leftrightarrows\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{D})\
.$

Furthermore, for every continuous morphism of topological spaces $i:Y\to X$,
the following diagram

$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{D})}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{D})\
.}$$\scriptstyle{i^{\ast,\mathrm{hyp}}}$$\scriptstyle{f^{\mathrm{hyp}}}$$\scriptstyle{f^{\mathrm{hyp}}}$$\scriptstyle{i^{\ast,\mathrm{hyp}}}$

commutes. In particular, if $(X,P)$ is a stratified space, $f^{\mathrm{hyp}}$
restricts to a functor

$f^{\mathrm{hyp}}\colon\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{D})\
.$

######  Remark 2.29.

Note that $f^{\mathrm{hyp}}$ may not have a right adjoint in general. We will
show that when Exodromy holds, a right adjoint exists and admits an explicit
description. See 6.16.

######  Lemma 2.30.

Let $X$ be a locally weakly contractible topological space. Let
$f\colon\mathcal{E}\to\mathcal{D}$ be a cocontinuous functor between
presentable $\infty$-categories with right adjoint
$g:\mathcal{D}\to\mathcal{E}$. Then, the commutative diagram

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

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

$\Gamma_{X}^{\ast,\mathrm{hyp}}\circ g\longrightarrow
g\circ\Gamma_{X}^{\ast,\mathrm{hyp}}\ $

is an equivalence. In particular, $g\circ-$ restricts to a well-defined
functor

$g\circ-\colon\mathrm{LC}(X,\mathcal{D})\longrightarrow\mathrm{LC}(X,\mathcal{E})\
$

###### Proof.

Let $d\in\mathcal{D}$ be an object. We have to show that

$\Gamma_{X}^{\ast,\mathrm{hyp}}g(d)\longrightarrow
g\circ\Gamma_{X}^{\ast,\mathrm{hyp}}(d)$

is an equivalence of hypersheaves on $X$. It is enough to check that the above
morphism is an equivalence above any weakly contractible open subset $U$ of
$X$. From [HPT, Theorem 2.13], we have

$(\Gamma_{X}^{\ast,\mathrm{hyp}}\circ
g(d))(U):=\big{(}\Gamma_{X}^{-1}(g(d))\big{)}^{\mathrm{hyp}}(U)\simeq(\Gamma_{X}^{-1}\circ
g(d))(U)\simeq g(d)$

From [HPT, Theorem 2.13] again, we have

$(g\circ\Gamma_{X}^{\ast,\mathrm{hyp}}(d))(U)=g\big{(}(\Gamma_{X}^{\ast,\mathrm{hyp}}(d))(U)\big{)}\simeq
g(d)$

and 2.30 follows. ∎

##  3\. The categorical framework

Let $\mathcal{X}$ be an $\infty$-category and let
$\mathsf{A}\colon\mathcal{X}\to\mathrm{Cat}_{\infty}$ be a functor with values
in small $\infty$-categories. Throughout this section we either assume
$\mathcal{X}$ to be a small $\infty$-category with a terminal object
$\mathbf{1}_{\mathcal{X}}$ or to be presentable. In the latter case, we assume
$\mathsf{A}$ to be an accessible functor. We let

$\pi_{\mathsf{A}}\colon\mathcal{A}\longrightarrow\mathcal{X}^{\mathrm{op}}$

be the associated _cartesian_ fibration. Writing $\mathbf{1}_{\mathcal{X}}$
for the terminal object of $\mathcal{X}$, [HTT, Corollary 3.3.4.3] provides a
canonical _localization_ functor

$\lambda_{\mathsf{A}}\colon\mathcal{A}\longrightarrow\mathsf{A}(\mathbf{1}_{\mathcal{X}}).$

######  Example 3.1.

In this paper we are mainly interested in the following situations:

  1. (1)

Let $X$ be a topological space. Take $\mathcal{X}\coloneqq\mathrm{Open}(X)$
and $\mathsf{A}\coloneqq\Pi_{\infty}$ the functor
$\mathsf{Open}(X)\to\mathcal{S}$ sending an open set $U$ to its homotopy type
$\Pi_{\infty}(U)$. We will also take $\mathcal{X}\coloneqq\mathrm{PSh}(X)$ and
$A\colon\mathrm{PSh}(X)\to\mathcal{S}$ the colimit-preserving functor obtained
as left Kan extension of $\Pi_{\infty}$ along the Yoneda embedding. From [HA,
A.3.10], $\mathsf{A}$ carries $\infty$-connective morphisms in
$\mathrm{PSh}(X)$ to equivalences in $\mathcal{S}$. If we finally put
$\mathcal{X}\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(X)$, the functor $\mathsf{A}$
thus factors as a colimit-preserving functor issued from
$\mathrm{Sh}^{\mathrm{hyp}}(X)$.

  2. (2)

Let $(X,P)$ be a conically stratified topological space. Take
$\mathcal{X}\coloneqq\mathrm{Open}(X)$ and
$\mathsf{A}\coloneqq\Pi_{\infty}^{\Sigma}$ the functor
$\mathsf{Open}(X)\to\mathrm{Cat}_{\infty}$ sending an open set $U$ to
$\Pi_{\infty}^{\Sigma}(U,P|_{U})$. We will also take
$\mathcal{X}\coloneqq\mathrm{PSh}(X)$ and
$\mathsf{A}\colon\mathrm{PSh}(X)\to\mathcal{S}$ the colimit-preserving functor
obtained as left Kan extension of $\Pi_{\infty}^{\Sigma}$ along the Yoneda
embedding. Similarly as in situation $(1)$, $\mathsf{A}$ carries
$\infty$-connective morphisms in $\mathrm{PSh}(X)$ to equivalences in
$\mathrm{Cat}_{\infty}$. If we finally put
$\mathcal{X}\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(X)$, the functor $\mathsf{A}$
thus factors as a colimit-preserving functor issued from
$\mathrm{Sh}^{\mathrm{hyp}}(X)$. We will abuse notations by using the notation
$\Pi_{\infty}^{\Sigma}$ to also refer to the functors $\mathsf{A}$ and its
factorization through $\mathrm{Sh}^{\mathrm{hyp}}(X)$.

######  Remark 3.2.

Situation (1) is a special case of situation (2) in the same way monodromy is
a special case of exodromy. Still, the proof of the latter is a reduction to
the trivial stratification case. It will therefore be useful to analyze the
situation (1) on its own.

###  3.1. Size issues

The main point of this paper is to show that the exodromy adjunction is
realized by push-pull along the correspondence
$\pi_{\mathsf{A}}\times\lambda_{\mathsf{A}}\colon\mathcal{A}\to\mathcal{X}^{\mathrm{op}}\times\mathsf{A}(\mathbf{1}_{\mathcal{X}})$.
When $\mathcal{X}$ is a presentable $\infty$-category, there may be size-
theoretical issues in defining the right Kan extension along
$\pi_{\mathsf{A}}$ and the left Kan extension along $\lambda_{\mathsf{A}}$.
The following lemmas show this is not an actual problem:

######  Lemma 3.3.

For every $x\in\mathcal{X}$, the canonical functor

$\mathsf{A}(x)\longrightarrow\mathcal{A}\times_{\mathcal{X}^{\mathrm{op}}}(\mathcal{X}^{\mathrm{op}})_{x/}$

is limit-final. In particular, for every presentable $\infty$-category
$\mathcal{E}$, the pullback functor

$\pi_{\mathsf{A}}^{\ast}\colon\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})\longrightarrow\operatorname{Fun}(\mathcal{A},\mathcal{E})$

admits a right adjoint, given by right Kan extension along $\pi_{\mathsf{A}}$.

###### Proof.

The second half of the statement is a direct consequence of the first half and
the fact that $\mathsf{A}(x)$ is a small $\infty$-category, which implies that
the right Kan extension along $p$ is indeed well-defined. To prove the first
half, we use Quillen’s theorem A [HTT, Theorem 4.1.3.1]. Write
$\mathcal{D}\coloneqq\mathcal{A}\times_{\mathcal{X}^{\mathrm{op}}}(\mathcal{X}^{\mathrm{op}})_{x/}$.
We can represent an object $\mathbf{a}\in\mathcal{D}$ as a pair
$\mathbf{a}=(a,f)$ where $a$ belongs to $\mathcal{A}$ and $f\colon
x\to\pi_{\mathsf{A}}(a)$ is a morphism in $\mathcal{X}^{\mathrm{op}}$. We have
to prove that for every object $\mathbf{a}\in\mathcal{D}$, the
$\infty$-category

$\mathsf{A}(x)\times_{\mathcal{D}}\mathcal{D}_{/\mathbf{a}}$

is weakly contractible. By definition, an object of
$\mathsf{A}(x)\times_{\mathcal{D}}\mathcal{D}_{/\mathbf{a}}$ is a lift of $f$
with target $a$. In particular,
$\mathsf{A}(x)\times_{\mathcal{D}}\mathcal{D}_{/\mathbf{a}}$ identifies with
$\mathcal{A}_{/a}\times_{\mathcal{X}^{\mathrm{op}}_{/\pi_{\mathsf{A}}(a)}}\\{f\\}$.
From [HTT, 2.4.1.9], the latter category admits a cartesian lift of $f$ as
final object. It is thus weakly contractible. ∎

It is slightly more technical to deal with the left Kan extension along
$\lambda_{\mathsf{A}}$. It is useful to introduce the following definition:

######  Definition 3.4.

Let $\kappa$ be a regular cardinal. We say that a functor
$f\colon\mathcal{C}\to\mathcal{D}$ is $\kappa$-cofiltered if for every
essentially $\kappa$-small $\infty$-category $I$, the functor $f$ has the
right lifting property against the inclusion $I\hookrightarrow I^{\lhd}$. When
$\kappa=\omega$, we simply say that $f$ is cofiltered instead of
$\omega$-cofiltered.

The proof of the following lemma is straightforward and it is left to the
reader:

######  Lemma 3.5.

Let $\kappa$ be a regular cardinal. Then:

  1. (1)

the collection of $\kappa$-filtered functors is closed under composition and
pullback in $\mathrm{Cat}_{\infty}$;

  2. (2)

the functor $\Gamma_{\mathcal{C}}\colon\mathcal{C}\to*$ is $\kappa$-filtered
if and only if $\mathcal{C}$ is $\kappa$-filtered.

######  Lemma 3.6.

Choose a regular cardinal $\kappa\gg 0$ such that $\mathcal{X}$ is
$\kappa$-presentable and $\mathsf{A}$ is $\kappa$-accessible. Let

$\mathcal{A}^{(\kappa)}\coloneqq(\mathcal{X}^{\kappa})^{\mathrm{op}}\times_{\mathcal{X}^{\mathrm{op}}}\mathcal{A}.$

Then for every $\alpha\in\mathsf{A}(\mathbf{1}_{\mathcal{X}})$ the canonical
functor

$\mathcal{A}^{(\kappa)}\times_{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}\mathsf{A}(\mathbf{1}_{\mathcal{X}})_{/\alpha}\longrightarrow\mathcal{A}\times_{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}\mathsf{A}(\mathbf{1}_{\mathcal{X}})_{/\alpha}$

is colimit-final. In particular, for every presentable $\infty$-category
$\mathcal{E}$, the pullback functor

$\lambda_{\mathsf{A}}^{\ast}\colon\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})\longrightarrow\operatorname{Fun}(\mathcal{A},\mathcal{E})$

admits a left adjoint, given by left Kan extension along
$\lambda_{\mathsf{A}}$.

###### Proof.

The second half of the statement is a direct consequence of the first half and
the fact that
$\mathcal{A}^{(\kappa)}\times_{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}\mathsf{A}(\mathbf{1}_{\mathcal{X}})_{/\alpha}$
is a small $\infty$-category, which implies that the left Kan extension along
$\lambda_{\mathsf{A}}$ is indeed well-defined. To prove the first half, we use
Quillen’s theorem A. Write for simplicity

$\mathcal{A}_{/\alpha}\coloneqq\mathcal{A}\times_{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}\mathsf{A}(\mathbf{1}_{\mathcal{X}})_{/\alpha}\quad\text{and}\quad\mathcal{A}^{(\kappa)}_{/\alpha}\coloneqq\mathcal{A}^{(\kappa)}\times_{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}\mathsf{A}(\mathbf{1}_{\mathcal{X}})_{/\alpha}.$

Fix an object $\mathbf{\beta}\in\mathcal{A}_{/\alpha}$, which we represent as
a pair $\mathbf{\beta}=(\beta,f)$, where $\beta\in\mathcal{A}$ and
$f\colon\lambda_{\mathsf{A}}(\beta)\to\alpha$ is a morphism in
$\mathsf{A}(\mathbf{1}_{\mathcal{X}})$. Write
$\mathcal{A}_{\mathbf{\beta}/\\!\\!/\alpha}\coloneqq(\mathcal{A}_{/\alpha})_{\mathbf{\beta}/}$.
Then, we have to prove that $\infty$-category

$\mathcal{A}^{(\kappa)}_{\mathbf{\beta}/\\!\\!/\alpha}\coloneqq\mathcal{A}^{(\kappa)}_{/\alpha}\times_{\mathcal{A}_{/\alpha}}\mathcal{A}_{\mathbf{\beta}/\\!\\!/\alpha}$

is weakly contractible. We claim that it is cofiltered. To see this, start by
writing $x\coloneqq p(\beta)$, so that $\beta$ can be seen as an element of
$\mathsf{A}(x)$, and $\mathbf{\beta}$ as an element of
$\mathsf{A}(x)_{/\alpha}\coloneqq\mathsf{A}(x)\times_{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}\mathsf{A}(\mathbf{1}_{\mathcal{X}})_{/\alpha}$.
We can thus set
$\mathsf{A}(x)_{\mathbf{\beta}/\\!\\!/\alpha}\coloneqq(\mathsf{A}(x)_{/\alpha})_{\mathbf{\beta}/}$.
Observe now that since $\pi_{\mathsf{A}}$ is a cartesian fibration, there is
an induced commutative diagram

$\displaystyle{\mathcal{A}^{(\kappa)}_{\mathbf{\beta}/\\!\\!/\alpha}}$$\displaystyle{\mathcal{A}^{(\kappa)}_{\beta/}}$$\displaystyle{\mathsf{A}(x)_{\mathbf{\beta}/\\!\\!/\alpha}}$$\displaystyle{\mathsf{A}(x)_{\beta/}\
.}$

Inspection immediately reveals that it is a pullback square. Moreover,
$\mathsf{A}(x)_{\mathbf{\beta}/\\!\\!/\alpha}$ has an initial object, and it
is in particular cofiltered. Thus, 3.5 reduces us to check that the map
$\mathcal{A}^{(\kappa)}_{\beta/}\to\mathsf{A}(x)_{\beta/}$ is cofiltered.
Unraveling the definitions, we see that it is enough to check that for every
finite category $I$ the canonical map

$\operatorname*{colim}_{y\in\mathcal{X}^{\kappa}_{/x}}\operatorname{Fun}(I,\mathsf{A}(y)_{\beta/})\longrightarrow\operatorname{Fun}(I,\mathsf{A}(x)_{\beta/})$

is an equivalence. This, however, is guaranteed from the assumption that
$\mathcal{X}$ is $\kappa$-filtered and that the functor $\mathsf{A}$ is
$\kappa$-accessible. ∎

Fix a presentable $\infty$-category $\mathcal{E}$. 3.3 provides a functor

$\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\coloneqq\pi_{\mathsf{A},\ast}\circ\,\lambda_{\mathsf{A}}^{\ast}\colon\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})\longrightarrow\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E}).$

From 3.3, this functor is concretely described by the formula

(3.7)
$\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}(F)(x)\simeq\lim_{\mathsf{A}(x)}F|_{\mathsf{A}(x)}\
,$

where the restriction is performed along the map
$\mathsf{A}(x)\to\mathsf{A}(\mathbf{1}_{\mathcal{X}})$ induced by the
canonical morphism $x\to\mathbf{1}_{\mathcal{X}}$. On the other hand, 3.6
shows the existence of a second functor

$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\coloneqq\lambda_{A,!}\circ\pi^{\ast}_{\mathcal{A}}\colon\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})\longrightarrow\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E}).$

By construction, $\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$ is left adjoint
to $\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$.

###  3.2. Change of coefficients

Let $\mathcal{E}$ be a presentable $\infty$-category. For every presentable
$\infty$-category $\mathcal{E}^{\prime}$ and for every
$\mathcal{C}\in\mathrm{Cat}_{\infty}$, there is a canonical equivalence

$\operatorname{Fun}(\mathcal{C},\mathcal{E})\otimes\mathcal{E}^{\prime}\simeq\operatorname{Fun}(\mathcal{C},\mathcal{E}\otimes\mathcal{E}^{\prime})\
.$

This gives an evident functoriality in $\mathcal{E}^{\prime}$ with respect to
morphisms in $\mathcal{P}\mathrm{r}^{\mathrm{L}}$. To see that the exodromy
adjunction is compatible with this functoriality, consider first the following
lemma:

######  Lemma 3.8.

Let $\mathcal{X}$ be an $\infty$-category with a terminal object
$1_{\mathcal{X}}$. Let $\mathsf{A}\colon\mathcal{X}\to\mathrm{Cat}_{\infty}$
be a functor. For every presentable $\infty$-categories $\mathcal{E}$ and
$\mathcal{E}^{\prime}$, there is a canonical equivalence

$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}\otimes\mathcal{E}^{\prime}}\simeq\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\otimes\mathrm{id}_{\mathcal{E}^{\prime}}\
.$

###### Proof.

Since $\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}\otimes\mathcal{E}^{\prime}}$
commutes with colimits, it is enough to observe that for every
$F\in\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})$ and
$E^{\prime}\in\mathcal{E}^{\prime}$, one has

$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}(F\otimes
E^{\prime})\coloneqq\lambda_{\mathsf{A},!}(\pi_{\mathsf{A}}^{\ast}(F\otimes
E^{\prime}))\simeq\lambda_{\mathsf{A},!}(\pi_{\mathsf{A}}^{\ast}(F)\otimes
E^{\prime})\simeq\lambda_{\mathsf{A},!}(\pi_{\mathsf{A}}^{\ast}(F))\otimes
E^{\prime}\ ,$

where the last equivalence follows from the fact that the external tensor
product
$\otimes\colon\mathcal{E}\times\mathcal{E}^{\prime}\to\mathcal{E}\otimes\mathcal{E}^{\prime}$
commutes with the colimits computing the left Kan extension in the first
variable. ∎

######  Corollary 3.9.

Let $\mathcal{X}$ be an $\infty$-category with a terminal object
$1_{\mathcal{X}}$. Let $\mathsf{A}\colon\mathcal{X}\to\mathrm{Cat}_{\infty}$
be a functor. Let $f\colon\mathcal{E}\to\mathcal{E}^{\prime}$ be a morphism in
$\mathcal{P}\mathrm{r}^{\mathrm{L}}$. Then the diagram

$\displaystyle{\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E}^{\prime})}$$\displaystyle{\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E}^{\prime})}$$\scriptstyle{f\circ-}$$\scriptstyle{\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}}$$\scriptstyle{f\circ-}$$\scriptstyle{\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}^{\prime}}}$

is canonically commutative.

###### Proof.

This follows from 3.8 and the observation that under the identification
$\operatorname{Fun}(\mathcal{C},\mathcal{E})\simeq\operatorname{Fun}(\mathcal{C},\mathcal{S})\otimes\mathcal{E}$
for $\mathcal{C}\in\mathrm{Cat}_{\infty}$, the functor $f\circ-$ corresponds
to $\mathrm{id}_{\operatorname{Fun}(\mathcal{C},\mathcal{S})}\otimes f$. ∎

###  3.3. The sheaf condition

Assume now that $\mathcal{X}$ is presentable and that the functor $\mathsf{A}$
commutes with colimits. Then the adjunction
$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\dashv\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$
can be refined thanks to the following observation:

######  Lemma 3.10.

Assume that $\mathsf{A}\colon\mathcal{X}\to\mathrm{Cat}_{\infty}$ commutes
with colimits. Then $\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$ factors
through the full subcategory
$\mathcal{X}\otimes\mathcal{E}\simeq\operatorname{Fun^{R}}(\mathcal{X}^{\mathrm{op}},\mathcal{E})$
of $\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})$.

###### Proof.

Fix a functor $F\colon\mathsf{A}(\mathbf{1}_{\mathcal{X}})\to\mathcal{E}$ and
let $I\to\mathcal{X}^{\mathrm{op}}$ be a diagram, noted $i\mapsto x_{i}$, and
let $x$ denote the limit of this diagram. By assumption, the canonical map

$\operatorname*{colim}_{i\in
I^{\mathrm{op}}}\mathsf{A}(x_{i})\longrightarrow\mathsf{A}(x)$

is an equivalence, the colimit being computed in $\mathrm{Cat}_{\infty}$. It
follows that the canonical map

$\operatorname{Fun}(\mathsf{A}(x),\mathcal{E})\longrightarrow\lim_{i\in
I}\operatorname{Fun}(\mathsf{A}(x_{i}),\mathcal{E})$

is an equivalence as well. Applying [PY, §8.2], we deduce a canonical
equivalence

$\lim_{\mathsf{A}(x)}F|_{\mathsf{A}(x)}\simeq\lim_{i\in
I}\lim_{\mathsf{A}(x_{i})}F|_{\mathsf{A}(x_{i})}\ .$

Thus, $\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}(F)$ commutes with limits in
$\mathcal{X}^{\mathrm{op}}$. ∎

######  Corollary 3.11.

Assume that $\mathsf{A}\colon\mathcal{X}\to\mathrm{Cat}_{\infty}$ commutes
with colimits. Then for any presentable $\infty$-category $\mathcal{E}$, the
functors $\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$ and
$\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$ induce an adjunction

$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\colon\mathcal{X}\otimes\mathcal{E}\leftrightarrows\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})\colon\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\
.$

######  Example 3.12.

Let us place ourselves again in the context of 3.1. Let $(X,P)$ be a conically
stratified space. To lighten the notation, we write

$\Phi_{X,P}^{\mathrm{hyp},\mathcal{E}}\coloneqq\Phi_{\mathrm{Sh}^{\mathrm{hyp}}(X),\Pi_{\infty}^{\Sigma}}^{\mathcal{E}}\quad\text{and}\quad\Psi_{X,P}^{\mathrm{hyp},\mathcal{E}}\coloneqq\Psi_{\mathrm{Sh}^{\mathrm{hyp}}(X),\Pi_{\infty}^{\Sigma}}^{\mathcal{E}}\
.$

When the stratification is trivial we further simplify these notations by
removing the subscript $P$, so that in this case
$\Pi_{\infty}=\Pi_{\infty}^{\Sigma}$. When $X$ and $\mathcal{E}$ are clear
from the context we also remove the corresponding decoration. Given
$F\in\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})$, 3.10 shows
that the functor $\Psi(F)$ belongs to
$\operatorname{Fun^{R}}(\mathrm{Sh}^{\mathrm{hyp}}(X)^{\mathrm{op}};\mathcal{E})\simeq\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$.
The previous discussion shows that for an open subset $U$ of $X$, $\Psi(F)(U)$
is canonically given by the formula

(3.13)
$\Psi(F)(U)\simeq\lim_{\Pi_{\infty}^{\Sigma}(U,P|_{U})}F|_{\Pi_{\infty}^{\Sigma}(U,P|_{U})}.$

Finally, we have obvious variants

$\displaystyle\Phi_{X,P}^{\mathrm{psh},\mathcal{E}}\coloneqq\Phi_{\mathrm{PSh}(X),\Pi_{\infty}^{\Sigma}}^{\mathcal{E}}\
,\qquad\Psi_{X,P}^{\mathrm{psh},\mathcal{E}}\coloneqq\Psi_{\mathrm{PSh}(X),\Pi_{\infty}^{\Sigma}}^{\mathcal{E}}\
,$
$\displaystyle\Phi_{X,P}^{\mathcal{E}}\coloneqq\Phi_{\mathrm{Open}(X),\Pi_{\infty}^{\Sigma}}^{\mathcal{E}}\
,\qquad\Psi_{X,P}^{\mathcal{E}}\coloneqq\Psi_{\mathrm{Open}(X),\Pi_{\infty}^{\Sigma}}^{\mathcal{E}}\
,$

obtained replacing $\mathrm{Sh}^{\mathrm{hyp}}(X)$ by $\mathrm{PSh}(X)$ and
$\mathrm{Open}(X)$ respectively.

###  3.4. Functoriality

Let $f\colon\mathcal{X}\to\mathcal{Y}$ be a functor. We assume that
$\mathcal{X}$ and $\mathcal{Y}$ are either small with final objects, or
presentable (importantly, we allow $\mathcal{X}$ to be of the first kind and
$\mathcal{Y}$ to be of the second). Furthermore, we assume that
$f(\mathbf{1}_{\mathcal{X}})\simeq\mathbf{1}_{\mathcal{Y}}$, and if both
$\mathcal{X}$ and $\mathcal{Y}$ are presentable then we additionally assume
that $f$ is a left adjoint. Let

$\mathsf{B}\colon\mathcal{Y}\longrightarrow\mathrm{Cat}_{\infty}\quad\text{and}\quad\mathsf{A}\colon\mathcal{X}\longrightarrow\mathrm{Cat}_{\infty}$

be two functors and let

$\gamma\colon\mathsf{B}\circ f\longrightarrow\mathsf{A}$

be a natural transformation. We can summarize this information in the
following diagram:

(3.14)
$\displaystyle{\mathsf{B}(\mathbf{1}_{\mathcal{Y}})}$$\displaystyle{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}$$\displaystyle{\mathcal{B}}$$\displaystyle{\mathcal{B}_{f}}$$\displaystyle{\mathcal{A}}$$\displaystyle{\mathcal{Y}^{\mathrm{op}}}$$\displaystyle{\mathcal{X}^{\mathrm{op}}\
,}$$\scriptstyle{\gamma_{\mathbf{1}}}$$\scriptstyle{\lambda_{\mathsf{B}}}$$\scriptstyle{\pi_{\mathsf{B}}}$$\scriptstyle{\lambda_{f}}$$\scriptstyle{q}$$\scriptstyle{\pi_{f}}$$\scriptstyle{p}$$\scriptstyle{\pi_{\mathsf{A}}}$$\scriptstyle{\lambda_{\mathsf{A}}}$$\scriptstyle{f}$

where $\mathcal{B}_{f}$ and the maps $\pi_{f}$ and $q$ are defined by
declaring that the bottom left square is a pullback, $p$ is induced by the
natural transformation $\gamma$ and
$\gamma_{\mathbf{1}}\coloneqq\gamma_{\mathbf{1}_{\mathcal{X}}}$ is the value
of $\gamma$ on the final object of $\mathcal{X}$.

######  Lemma 3.15.

For every presentable $\infty$-category $\mathcal{E}$, the diagram

$\displaystyle{\operatorname{Fun}(\mathcal{Y}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{B},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{B}_{f},\mathcal{E})}$$\scriptstyle{f^{\ast}}$$\scriptstyle{\pi_{\mathsf{B}}^{\ast}}$$\scriptstyle{\pi_{f}^{\ast}}$$\scriptstyle{q^{\ast}}$

is vertically right adjointable.

###### Proof.

The existence of the right adjoints to $\pi_{\mathsf{A}}^{\ast}$ and
$\pi_{\mathsf{B}}^{\ast}$ follows from 3.3. In order to check that the Beck-
Chevalley transformation

$f^{\ast}\circ\pi_{\mathsf{B},\ast}\longrightarrow\pi_{f,\ast}\circ q^{\ast}$

is an equivalence, it is enough to fix
$F\in\operatorname{Fun}(\mathcal{B},\mathcal{E})$ and
$x\in\mathcal{X}^{\mathrm{op}}$. Then 3.3 identifies
$f^{\ast}(\pi_{\mathsf{B},\ast}(F))(x)$ and $\pi_{f,\ast}(q^{\ast}(F))(x)$
with

$\lim_{\mathsf{B}(f(x))}F|_{\mathsf{B}(f(x))}.$

The conclusion follows. ∎

######  Notation 3.16.

Assume that $\mathcal{X}$ and $\mathcal{Y}$ are presentable and that
$f\colon\mathcal{X}\to\mathcal{Y}$ commutes with colimits. For any presentable
$\infty$-category $\mathcal{E}$ set $f_{\mathcal{E}}\coloneqq
f\otimes\mathrm{id}_{\mathcal{E}}\colon\mathcal{X}\otimes\mathcal{E}\to\mathcal{Y}\otimes\mathcal{E}$.
We let $g_{\mathcal{E}}$ denote the right adjoint to $f_{\mathcal{E}}$.

######  Corollary 3.17.

Fix a presentable $\infty$-category $\mathcal{E}$.

  1. (1)

Assume that $\mathcal{X}$ and $\mathcal{Y}$ are small. Then the diagrams

$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{Y}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})}$$\scriptstyle{\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}}$$\scriptstyle{\Psi_{\mathcal{X},\mathsf{B}\circ
f}^{\mathcal{E}}}$$\scriptstyle{f^{\ast}}$ and
$\displaystyle{\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{Y}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\scriptstyle{f_{!}}$$\scriptstyle{\Phi^{\mathcal{E}}_{\mathcal{X},\mathsf{B}\circ
f}}$$\scriptstyle{\Phi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}}$

are canonically commutative.

  2. (2)

Assume that $\mathcal{X}$ and $\mathcal{Y}$ are presentable and that
$\mathsf{B}$ and $f$ commute with colimits. Then the diagrams

$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\displaystyle{\mathcal{Y}\otimes\mathcal{E}}$$\displaystyle{\mathcal{X}\otimes\mathcal{E}}$$\scriptstyle{\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}}$$\scriptstyle{\Psi_{\mathcal{X},\mathsf{B}\circ
f}^{\mathcal{E}}}$$\scriptstyle{g_{\mathcal{E}}}$ and
$\displaystyle{\mathcal{X}\otimes\mathcal{E}}$$\displaystyle{\mathcal{Y}\otimes\mathcal{E}}$$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\scriptstyle{f_{\mathcal{E}}}$$\scriptstyle{\Phi^{\mathcal{E}}_{\mathcal{X},\mathsf{B}\circ
f}}$$\scriptstyle{\Phi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}}$

are canonically commutative.

###### Proof.

It is enough to prove the commutativity of the left triangle of $(1)$.
Breaking $\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}$ into its components, we
have to show that the squares of the following diagram commute:

$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{B},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{Y}^{\mathrm{op}},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{B}_{f},\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathcal{X}^{\mathrm{op}},\mathcal{E})\
.}$$\scriptstyle{\lambda_{\mathsf{B}}^{\ast}}$$\scriptstyle{\pi_{\mathsf{B},\ast}}$$\scriptstyle{q^{\ast}}$$\scriptstyle{f^{\ast}}$$\scriptstyle{\lambda_{f}^{\ast}}$$\scriptstyle{\pi_{f,\ast}}$

For the left square, this just follows obviously from the definition of the
functors. For the right square, this is a consequence of 3.15. ∎

######  Example 3.18.

Let $(X,P)$ be a conically stratified topological space.

  1. (1)

Take $\mathcal{X}\coloneqq\mathrm{Open}(X)$,
$\mathcal{Y}\coloneqq\mathrm{PSh}(X)$ and take $f$ to be the Yoneda embedding.
We take $\mathsf{A}\coloneqq\Pi_{\infty}^{\Sigma}$, the exit paths
$\infty$-functor and we take $\mathsf{B}$ to be the left Kan extension of
$\mathsf{A}$ along $f$. Then

$f^{\ast}\colon\operatorname{Fun^{R}}(\mathrm{PSh}(X)^{\mathrm{op}},\mathcal{E})\longrightarrow\operatorname{Fun}(\mathrm{Open}(X)^{\mathrm{op}},\mathcal{E})$

is an equivalence as a consequence of [HTT, Theorem 5.1.5.6]. Thus 3.10 $(1)$
and 3.17 yield a natural equivalence

$\Psi_{X,P}^{\mathcal{E}}\simeq\Psi_{X,P}^{\mathrm{psh},\mathcal{E}}\ .$

3.17 also identifies $\Phi_{X,P}^{\mathrm{psh},\mathcal{E}}$ with the left Kan
extension of $\Phi_{X,P}^{\mathcal{E}}$ along the Yoneda embedding.

  2. (2)

Take $\mathcal{X}\coloneqq\mathrm{PSh}(X)$,
$\mathcal{Y}\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(X)$ and take
$f\coloneqq(-)^{\mathrm{hyp}}$ to be the hypersheafification functor. In this
case, 3.17 yields natural equivalences

$\Psi_{X,P}^{\mathrm{psh},\mathcal{E}}\simeq\Psi_{X,P}^{\mathrm{hyp},\mathcal{E}}\qquad\text{and}\qquad\Phi_{X,P}^{\mathrm{psh},\mathcal{E}}\simeq\Phi_{X,P}^{\mathrm{hyp},\mathcal{E}}\circ(-)^{\mathrm{hyp}}\
.$

In particular, we see that for every $F\in\mathrm{PSh}(X;\mathcal{E})$, the
functor $\Phi_{X,P}^{\mathcal{E}}$ takes the natural transformation $F\to
F^{\mathrm{hyp}}$ to an equivalence.

######  Construction 3.19.

Associated to the top right square in the diagram (3.14) there is, for every
presentable $\infty$-category $\mathcal{E}$, a Beck-Chevalley transformation

$\mathsf{BC}\colon\lambda_{f,!}\circ
p^{\ast}\longrightarrow\gamma_{\mathbf{1}}^{\ast}\circ\lambda_{\mathsf{A},!}\
,$

which induces the following transformation

$\phi_{\mathsf{A},\mathsf{B},f,\gamma}\colon\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\circ
f_{\mathcal{E}}\simeq\Phi_{\mathcal{X},\mathsf{B}\circ
f}^{\mathcal{E}}\simeq\lambda_{f,!}\circ\pi_{f}^{\ast}\simeq\lambda_{f,!}p^{\ast}\pi_{\mathsf{A}}^{\ast}\xrightarrow{\mathsf{BC}}\gamma_{\mathbf{1}}^{\ast}\circ\lambda_{A,!}\circ\pi_{A}^{\ast}\simeq\gamma_{\mathbf{1}}^{\ast}\circ\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\
.$

In turn, associated to $\phi_{\mathsf{A},\mathsf{B},f,\gamma}$ there is a
natural exchange transformation

$\psi_{\mathsf{A},\mathsf{B},f,\gamma}\colon
f_{\mathcal{E}}\circ\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\xrightarrow{\eta_{\mathcal{Y}}}\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}f_{\mathcal{E}}\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\xrightarrow{\phi_{\mathsf{A},\mathsf{B},f,\gamma}}\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\gamma_{\mathbf{1}}^{\ast}\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\xrightarrow{\varepsilon_{\mathcal{X}}}\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\circ\gamma_{\mathbf{1}}^{\ast}\
,$

where $\eta_{\mathcal{Y}}$ denotes the unit of the adjunction
$\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\dashv\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}$
and $\varepsilon_{\mathcal{X}}$ denotes the counit of the adjunction
$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\dashv\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$.
In the rest of the paper we will often write $\phi_{f}$ and $\psi_{f}$ instead
of $\phi_{\mathsf{A},\mathsf{B},f,\gamma}$ and
$\psi_{\mathsf{A},\mathsf{B},f,\gamma}$, respectively.

######  Remark 3.20.

  1. (1)

The triangular identities imply that the exchange transformation associated to
$\psi_{\mathsf{A},\mathsf{B},f,\gamma}$ is once again
$\phi_{\mathsf{A},\mathsf{B},f,\gamma}$. In particular, if one knows that both
$\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\dashv\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}$
and
$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\dashv\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$
are equivalences, then $\phi_{\mathsf{A},\mathsf{B},f,\gamma}$ is an
equivalence if and only if $\psi_{\mathsf{A},\mathsf{B},f,\gamma}$ is one.
Furthermore, triangular identities also imply that the square

$\displaystyle{f}$$\displaystyle{\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}f}$$\displaystyle{f\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}}$$\displaystyle{\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\gamma_{\mathbf{1}}^{\ast}\Phi_{\mathcal{X},\mathsf{B}}^{\mathcal{E}}}$$\scriptstyle{\eta_{\mathcal{Y}}f}$$\scriptstyle{f\eta_{\mathcal{X}}}$$\scriptstyle{\Psi_{\mathcal{Y}}\phi_{f}}$$\scriptstyle{\psi_{f}\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}}$

is canonically commutative.

  2. (2)

Suppose that the natural transformation $\psi_{f}$ is an equivalence, so that
it renders the square

$\displaystyle{\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})}$$\displaystyle{\mathcal{X}\otimes\mathcal{E}}$$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\displaystyle{\mathcal{Y}\otimes\mathcal{E}}$$\scriptstyle{\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}}$$\scriptstyle{\gamma_{\mathbf{1}}^{\ast}}$$\scriptstyle{f_{\mathcal{E}}}$$\scriptstyle{\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}}$

commutative. Then this square is horizontally right adjointable _if and only
if_ the natural transformation $\phi_{f}$ is an equivalence as well. Thus, if
both $\phi_{f}$ and $\psi_{f}$ are equivalences, it follows formally that
$f_{\mathcal{E}}$ takes the unit of the adjunction
$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\dashv\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$
to the unit of the adjunction
$\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\dashv\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}$.
Similarly, $\gamma_{\mathbf{1}}^{\ast}$ takes the counit of the adjunction
$\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\dashv\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}$
to the counit of the adjunction
$\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\dashv\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}$.

######  Corollary 3.21.

Assume that $\mathcal{X}$ and $\mathcal{Y}$ are presentable, that $\mathsf{B}$
commutes with colimits, that $f\colon\mathcal{X}\to\mathcal{Y}$ has a left
adjoint $h\colon\mathcal{Y}\to\mathcal{X}$ and that the induced transformation

$\mathsf{B}\xrightarrow{\eta}\mathsf{B}\circ f\circ
h\xrightarrow{\gamma}A\circ h$

is an equivalence, where $\eta$ is the unit of the adjunction $h\dashv f$.
Then the natural transformation

$\psi_{\mathsf{A},\mathsf{B},f,\gamma}\colon
f_{\mathcal{E}}\circ\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}\longrightarrow\Psi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}\circ\gamma_{\mathbf{1}}^{\ast}$

is an equivalence.

###### Proof.

Indeed, since $f$ has a left adjoint $h$, the same goes for $f_{\mathcal{E}}$
and moreover under the equivalences
$\mathcal{X}\otimes\mathcal{E}\simeq\operatorname{Fun^{R}}(\mathcal{X}^{\mathrm{op}},\mathcal{E})$
and
$\mathcal{Y}\otimes\mathcal{E}\simeq\operatorname{Fun^{L}}(\mathcal{Y}^{\mathrm{op}},\mathcal{E})$,
the functor $f_{\mathcal{E}}$ corresponds to $h^{\ast}$. Unwinding the
definitions and using 3.3, we see that for every
$F\in\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})$ and
every $y\in\mathcal{Y}^{\mathrm{op}}$, the map
$\psi_{\mathsf{A},\mathsf{B},f,\gamma}(F)$ at $y$ is given by

$\lim_{\mathsf{A}(h(y))}F|_{\mathsf{A}(h(y))}\longrightarrow\lim_{\mathsf{B}(y)}(F\circ
u)|_{\mathsf{B}(y)}\ .$

Thus, the conclusion follows from the fact that the functor
$\mathsf{B}(y)\to\mathsf{A}(h(y))$ is an equivalence. ∎

######  Example 3.22.

  1. (1)

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Take
$\mathcal{X}\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(X)$ and
$\mathcal{Y}\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(Y)$ and
$f^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X)\to\mathrm{Sh}^{\mathrm{hyp}}(Y)$
the induced geometric morphism. We let $\mathsf{A}$ and $\mathsf{B}$ be the
exit paths functors

$\Pi_{\infty}^{\Sigma}\colon\mathrm{Sh}^{\mathrm{hyp}}(X)\longrightarrow\mathrm{Cat}_{\infty}\qquad\text{and}\qquad\Pi_{\infty}^{\Sigma}\colon\mathrm{Sh}^{\mathrm{hyp}}(Y)\longrightarrow\mathrm{Cat}_{\infty},$

respectively. The functoriality of the exit path construction yields a natural
transformation $\gamma\colon\mathsf{B}\circ
f^{\ast,\mathrm{hyp}}\to\mathsf{A}$. In this setup, we denote by

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

the natural transformations of 3.19. Taking $\mathcal{X}=\mathrm{PSh}(X)$ and
$\mathcal{Y}=\mathrm{PSh}(Y)$ or $\mathcal{X}=\mathrm{Open}(X)$ and
$\mathcal{Y}=\mathrm{Open}(Y)$ we obtain obvious variants of this
construction, that we denote respectively by $\phi_{f}^{\mathrm{psh}}$,
$\psi_{f}^{\mathrm{psh}}$, $\phi_{f}$ and $\psi_{f}$.

  2. (2)

In the setting of the previous point, assume furthermore that the underlying
morphism of topological spaces $f\colon Y\to X$ is an open immersion. Then
$f^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X)\to\mathrm{Sh}^{\mathrm{hyp}}(Y)$
admits a fully faithful _left_ adjoint, denoted $f_{!}^{\mathrm{hyp}}$.
Furthermore, if $U$ is an open subset of $Y$, seen as an object in
$\mathrm{Sh}^{\mathrm{hyp}}(Y)$, then
$f_{!}^{\mathrm{hyp}}(U)\in\mathrm{Sh}^{\mathrm{hyp}}(X)$ represents again
$U$, seen as an open in $X$. It follows that the assumptions of 3.21 are
satisfied and therefore that $\psi_{f}^{\mathrm{hyp}}$ is an equivalence in
this case.

###  3.5. A glimpse of higher-functoriality

The natural transformations $\phi_{f}$ and $\psi_{f}$ of 3.19 enjoy themselves
a form of higher functoriality. We fix two composable morphisms
$f\colon\mathcal{X}\to\mathcal{Y}$ and $h\colon\mathcal{Y}\to\mathcal{Z}$ in
$\mathrm{Cat}_{\infty}$. We further fix three functors

$\mathsf{A}\colon\mathcal{X}\longrightarrow\mathrm{Cat}_{\infty},\qquad\mathsf{B}\colon\mathcal{Y}\longrightarrow\mathrm{Cat}_{\infty},\qquad\mathsf{C}\colon\mathcal{Z}\longrightarrow\mathrm{Cat}_{\infty}\
,$

together with natural transformations

$\gamma\colon\mathsf{B}\circ
f\longrightarrow\mathsf{A}\qquad\text{and}\qquad\delta\colon\mathsf{C}\circ
h\longrightarrow\mathsf{B}\ .$

As in the previous section, we assume that $\mathcal{X}$, $\mathcal{Y}$,
$\mathcal{Z}$ are either small with a final objects or presentable. As for $f$
and $h$, we assume that they preserve final objects and, if both their source
and target are presentable, that they commute with colimits.

Fix a presentable $\infty$-category $\mathcal{E}$. In this situation, 3.19
provides three natural transformations

$\phi_{h}\colon\Phi^{\mathcal{E}}_{\mathcal{Z},\mathsf{C}}\circ
h_{\mathcal{E}}\longrightarrow\delta_{\mathbf{1}}^{\ast}\circ\Phi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}\
,\qquad\phi_{f}\colon\Phi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}\circ
f_{\mathcal{E}}\longrightarrow\gamma_{\mathbf{1}}^{\ast}\circ\Phi^{\mathcal{E}}_{\mathcal{X},\mathsf{A}}$

as well as

$\phi_{h\\!f}\colon\Phi^{\mathcal{E}}_{\mathcal{Z},\mathsf{C}}\circ
h_{\mathcal{E}}\circ
f_{\mathcal{E}}\longrightarrow\delta_{\mathbf{1}}^{\ast}\circ\gamma_{\mathbf{1}}^{\ast}\circ\Phi^{\mathcal{E}}_{\mathcal{X},\mathsf{A}}\
.$

We have:

######  Proposition 3.23.

The triangle

(3.24) $\displaystyle{\Phi^{\mathcal{E}}_{\mathcal{Z},\mathsf{C}}\circ
h_{\mathcal{E}}\circ
f_{\mathcal{E}}}$$\displaystyle{\delta_{\mathbf{1}}^{\ast}\circ\Phi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}\circ
f_{\mathcal{E}}}$$\displaystyle{\delta_{\mathbf{1}}^{\ast}\circ\gamma_{\mathbf{1}}^{\ast}\circ\Phi^{\mathcal{E}}_{\mathcal{X},\mathsf{A}}}$$\scriptstyle{\phi_{h}\circ
f_{\mathcal{E}}}$$\scriptstyle{\phi_{h\\!f}}$$\scriptstyle{\phi_{f}}$

is canonically commutative.

###### Proof.

Consider the following diagram

$\displaystyle{\mathsf{A}(\mathbf{1}_{\mathcal{X}})}$$\displaystyle{\mathsf{B}(\mathbf{1}_{\mathcal{Y}})}$$\displaystyle{\mathcal{A}}$$\displaystyle{\mathsf{C}(\mathbf{1}_{\mathcal{Z}})}$$\displaystyle{\mathcal{B}}$$\displaystyle{\mathcal{B}_{f}}$$\displaystyle{\mathcal{C}}$$\displaystyle{\mathcal{C}_{h}}$$\displaystyle{\mathcal{C}_{h\\!f}}$$\displaystyle{\mathcal{Z}^{\mathrm{op}}}$$\displaystyle{\mathcal{Y}^{\mathrm{op}}}$$\displaystyle{\mathcal{X}^{\mathrm{op}}}$$\scriptstyle{\gamma_{\mathbf{1}}}$$\scriptstyle{\lambda_{\mathsf{A}}}$$\scriptstyle{\pi_{\mathsf{A}}}$$\scriptstyle{\delta_{\mathbf{1}}}$$\scriptstyle{\lambda_{\mathsf{B}}}$$\scriptstyle{\pi_{\mathsf{B}}}$$\scriptstyle{\pi_{f}}$$\scriptstyle{\lambda_{f}}$$\scriptstyle{r}$$\scriptstyle{u}$$\scriptstyle{\pi_{\mathsf{C}}}$$\scriptstyle{\lambda_{\mathsf{C}}}$$\scriptstyle{\lambda_{h}}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{\pi_{h}}$$\scriptstyle{\lambda_{h\\!f}}$$\scriptstyle{\pi_{h\\!f}}$$\scriptstyle{t}$$\scriptstyle{s}$$\scriptstyle{h}$$\scriptstyle{f}$

Using 3.17 twice, we find canonical equivalences

$\alpha\colon\Phi^{\mathcal{E}}_{\mathcal{Z},\mathsf{C}}\circ
h_{\mathcal{E}}\circ
f_{\mathcal{E}}\simeq\lambda_{h\\!f,!}\circ\pi_{h\\!f}^{\ast},\qquad\beta\colon\Phi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}\circ
f_{\mathcal{E}}\simeq\lambda_{f,!}\circ\pi_{f}^{\ast}\ .$

Write $\sigma_{1}$ for the square whose edges are
$(\lambda_{f},\delta_{\mathbf{1}},\lambda_{h\\!f},s)$ and $\sigma_{2}$ for the
square with edges $(\lambda_{\mathsf{A}},\gamma_{\mathbf{1}},\lambda_{f},u)$.
We also let $\sigma_{12}$ denote the rectangle obtained juxtaposing
$\sigma_{1}$ and $\sigma_{2}$. Write

$\mathsf{BC}(\sigma_{1})\colon\lambda_{h\\!f,!}\circ
s^{\ast}\longrightarrow\delta_{\mathbf{1}}^{\ast}\circ\lambda_{f,!}$

for the Beck-Chevalley transformation associated to the square $\sigma_{1}$,
and define similarly $\mathsf{BC}(\sigma_{2})$ and $\mathsf{BC}(\sigma_{12})$.
Unwinding 3.19 and under the equivalences $\alpha$ and $\beta$ above, we are
called to prove that the diagram

$\displaystyle{\lambda_{h\\!f,!}\circ\pi_{h\\!f}^{\ast}}$$\displaystyle{\lambda_{f,!}\circ\pi_{f}^{\ast}}$$\displaystyle{\lambda_{\mathsf{A},!}\circ\pi_{\mathsf{A}}^{\ast}}$$\scriptstyle{\mathsf{BC}(\sigma_{1})}$$\scriptstyle{\mathsf{BC}(\sigma_{12})}$$\scriptstyle{\mathsf{BC}(\sigma_{2})}$

is canonically commutative. Since $\sigma_{12}$ is the composite square of
$\sigma_{1}$ and $\sigma_{2}$, this follows from the well-known componibility
property of Beck-Chevalley transformations. ∎

######  Remark 3.25.

The above proposition is just a shadow of the actual functorial dependence of
$\Phi^{\mathcal{E}}_{\mathcal{X},\mathsf{A}}$ on the pair
$(\mathcal{X},\mathsf{A})$. To make this construction into an actual functor
requires, as clearly shown by the above proof, to use the
$(\infty,2)$-functoriality property of the Beck-Chevalley transformation.
Since this is not needed for the current paper, we leave the details for the
interested reader.

######  Corollary 3.26.

In the same setting of 3.23, the triangle

$\displaystyle{h_{\mathcal{E}}\circ
f_{\mathcal{E}}\circ\Psi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}}$$\displaystyle{h_{\mathcal{E}}\circ\Psi^{\mathcal{E}}_{\mathcal{Y},\mathsf{B}}\circ\gamma_{\mathbf{1}}^{\ast}}$$\displaystyle{\Psi^{\mathcal{E}}_{\mathcal{Z},\mathsf{C}}\circ\delta_{\mathbf{1}}^{\ast}\circ\gamma_{\mathbf{1}}^{\ast}}$$\scriptstyle{h_{\mathcal{E}}\circ\psi_{f}}$$\scriptstyle{\psi_{h\\!f}}$$\scriptstyle{\psi_{h}}$

is canonically commutative.

###### Proof.

We consider the case where $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$ are
presentable, as every other case is dealt with similarly. We start with the
following diagram:

$\displaystyle{\mathcal{X}\otimes\mathcal{E}}$$\displaystyle{\mathcal{Y}\otimes\mathcal{E}}$$\displaystyle{\mathcal{Z}\otimes\mathcal{E}}$$\displaystyle{\operatorname{Fun}(\mathsf{A}(\mathbf{1}_{\mathcal{X}}),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathsf{B}(\mathbf{1}_{\mathcal{Y}}),\mathcal{E})}$$\displaystyle{\operatorname{Fun}(\mathsf{C}(\mathbf{1}_{\mathcal{Z}}),\mathcal{E})}$$\scriptstyle{\Phi_{\mathcal{X},\mathsf{A}}^{\mathcal{E}}}$$\scriptstyle{f_{\mathcal{E}}}$$\scriptstyle{h_{\mathcal{E}}}$$\scriptstyle{\Phi_{\mathcal{Y},\mathsf{B}}^{\mathcal{E}}}$$\scriptstyle{\Phi^{\mathcal{E}}_{\mathcal{Z},\mathsf{C}}}$$\scriptstyle{\gamma_{\mathbf{1}}^{\ast}}$$\scriptstyle{\delta_{\mathbf{1}}^{\ast}}$

It is made $2$-commutative by the transformations $\phi_{f}$, $\phi_{h}$. 3.23
identifies their composite $\phi_{f}\circ(\phi_{h}\circ f_{\mathcal{E}})$ with
$\phi_{h\\!f}$. Since $\psi_{f}=\mathsf{BC}(\phi_{f})$,
$\psi_{h}=\mathsf{BC}(\phi_{h})$ and $\psi_{h\\!f}=\mathsf{BC}(\phi_{h\\!f})$,
the conclusion follows directly from the componibility property of the Beck-
Chevalley transformations. ∎

###  3.6. Monodromy, revisited

Let us explain how to recover the monodromy equivalence as a particular
instance of 3.11. Let $X$ be a topological space and let $\mathcal{E}$ be a
presentable $\infty$-category. From [HPT, Construction 2.1], we have a functor

$\Pi_{\infty}^{\mathcal{E}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathcal{E}\
.$

When $\mathcal{E}=\mathcal{S}$ is the $\infty$-category of spaces, this
functor – denoted simply by $\Pi_{\infty}$ – is the left Kan extension of the
homotopy type functor, sending an open $U\in\mathrm{Open}(X)$ to its homotopy
type $\Pi_{\infty}(U)$. For a general $\mathcal{E}$, we have a canonical
identification
$\Pi_{\infty}^{\mathcal{E}}\simeq\Pi_{\infty}\otimes\mathrm{id}_{\mathcal{E}}$.
As observed in _loc. cit._ , $\Pi_{\infty}^{\mathcal{E}}$ has a right adjoint

$\Pi^{\infty}_{\mathcal{E}}\colon\mathcal{E}\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\
.$

Assume now that $X$ is locally weakly contractible. Then, [HPT, Proposition
2.5] implies that $\Pi^{\infty}_{\mathcal{E}}$ is canonically identified with
the constant hypersheaf functor $\Gamma_{X}^{\ast,\mathrm{hyp}}$.

If $\mathcal{E}=\mathcal{S}$, the functor $\Pi_{\infty}$ can be canonically
lifted to a functor

$\widetilde{\Pi}_{\infty}\colon\mathrm{Sh}^{\mathrm{hyp}}(X)\longrightarrow\mathcal{S}_{/\Pi_{\infty}(X)}\
,$

which in turn admits a right adjoint $\widetilde{\Pi}^{\infty}$ sending
$f\colon K\to\Pi_{\infty}(X)$ to the hypersheaf defined by

(3.27)
$\widetilde{\Pi}^{\infty}(K)(U)\coloneqq\operatorname{Map}_{/\Pi_{\infty}(X)}(\Pi_{\infty}(U),K)\
.$

Put
$\widetilde{\Pi}_{\infty}^{\mathcal{E}}:=\widetilde{\Pi}_{\infty}\otimes\mathrm{id}_{\mathcal{E}}$
and denote by $\widetilde{\Pi}^{\infty}_{\mathcal{E}}$ the right adjoint of
$\widetilde{\Pi}_{\infty}^{\mathcal{E}}$.

######  Lemma 3.28.

Let $X$ be a weakly contractible and locally weakly contractible topological
space. Let $\mathcal{E}$ be a presentable $\infty$-category. Then,
$\widetilde{\Pi}^{\infty}_{\mathcal{E}}$ factors through the full subcategory
$\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$.

###### Proof.

Since $X$ is weakly contractible, $\Pi_{\infty}(X)\simeq*$, and therefore
$\widetilde{\Pi}_{\infty}$ identifies with $\Pi_{\infty}$. By tensoring with
$\mathcal{E}$, we deduce that $\widetilde{\Pi}_{\infty}^{\mathcal{E}}$
identifies with $\Pi_{\infty}^{\mathcal{E}}$. Hence,
$\widetilde{\Pi}^{\infty}_{\mathcal{E}}$ identifies with
$\Pi^{\infty}_{\mathcal{E}}$. Since $X$ is locally weakly contractible, [HPT,
Proposition 2.5] supplies a canonical identification of
$\Pi^{\infty}_{\mathcal{E}}$ with
$\Gamma_{X}^{\ast,\mathrm{hyp}}\colon\mathcal{E}\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$.
The conclusion follows. ∎

######  Lemma 3.29.

Let $X$ be a locally weakly contractible topological space and let
$\mathcal{E}$ be a presentable $\infty$-category. Then
$\Phi_{X}^{\mathrm{hyp},\mathcal{E}}$ and
$\widetilde{\Pi}_{\infty}^{\mathcal{E}}$ are canonically equivalent as
functors from $\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ to
$\operatorname{Fun}(\Pi_{\infty}(X),\mathcal{E})$. In particular,
$\Psi_{X}^{\mathrm{hyp},\mathcal{E}}$ and
$\widetilde{\Pi}^{\infty}_{\mathcal{E}}$ are canonically equivalent.

###### Proof.

It is enough to show that $\Phi_{X}^{\mathrm{hyp},\mathcal{E}}$ and
$\widetilde{\Pi}_{\infty}^{\mathcal{E}}$ are canonically equivalent. From 3.8,
we have
$\Phi_{X}^{\mathrm{hyp},\mathcal{E}}\simeq\Phi_{X}^{\mathrm{hyp}}\otimes\mathrm{id}_{\mathcal{E}}$.
Since
$\widetilde{\Pi}_{\infty}^{\mathcal{E}}=\widetilde{\Pi}_{\infty}\otimes\mathrm{id}_{\mathcal{E}}$,
we are left to show that $\Phi_{X}^{\mathrm{hyp}}$ and
$\widetilde{\Pi}_{\infty}$ are canonically equivalent. To do this, it is
enough to show that $\Psi_{X}^{\mathrm{hyp}}$ and $\widetilde{\Pi}^{\infty}$
are canonically equivalent. Under the straightening equivalence
$\mathcal{S}_{/\Pi_{\infty}(X)}\simeq\operatorname{Fun}(\Pi_{\infty}(X),\mathcal{S})$,
the formula Eq. 3.27 for $\widetilde{\Pi}^{\infty}$ identifies canonically
with the formula Eq. 3.7 for $\Psi^{\mathrm{hyp}}_{X}$. Thus
$\Psi_{X}^{\mathrm{hyp}}$ and $\widetilde{\Pi}^{\infty}$ are canonically
equivalent. ∎

######  Corollary 3.30.

Let $X$ be a locally weakly contractible topological space. Then for every
presentable $\infty$-category $\mathcal{E}$ the functor
$\Psi_{X}^{\mathrm{hyp},\mathcal{E}}\colon\operatorname{Fun}(\Pi_{\infty}(X),\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
factors through the full subcategory
$\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$.

###### Proof.

Let $F\colon\Pi_{\infty}(X)\to\mathcal{E}$ be a functor. Since $X$ is locally
weakly contractible, it is enough to prove that for every weakly contractible
open subset $U$ of $X$ the restriction
$\Psi_{X}^{\mathrm{hyp},\mathcal{E}}(F)|_{U}$ is locally hyperconstant. Using
3.21 in the form of 3.22-(2), we find a canonical equivalence

$\Psi_{X}^{\mathrm{hyp},\mathcal{E}}(F)|_{U}\simeq\Psi_{U}^{\mathrm{hyp},\mathcal{E}}(F|_{\Pi_{\infty}(U)}).$

Hence, we are left to suppose that $X$ is weakly contractible and locally
weakly contractible. In that case, 3.30 follows from the combination of 3.29
and 3.28. ∎

In particular, 3.30 implies that the adjunction
$\Phi_{X}^{\mathrm{hyp},\mathcal{E}}\dashv\Psi_{X}^{\mathrm{hyp},\mathcal{E}}$
restricts to an adjunction

(3.31)
$\Phi_{X}^{\mathrm{hyp},\mathcal{E}}\colon\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})\leftrightarrows\operatorname{Fun}(\Pi_{\infty}(X),\mathcal{E})\colon\Psi_{X}^{\mathrm{hyp},\mathcal{E}}\
.$

At this point, we have:

######  Corollary 3.32.

Let $X$ be a locally weakly contractible topological space. Let $\mathcal{E}$
be a presentable $\infty$-category. Then the adjunction (3.31) is an
equivalence.

###### Proof.

Combine 3.29 with the monodromy equivalence [HPT, Corollary 3.9]. ∎

##  4\. Exodromy adjunction for constructible hypersheaves

Fix a conically stratified space $(X,P)$. We specialize the discussion of
Section 3 to the context of 3.1-(2). In other words, we take
$\mathcal{X}\coloneqq\mathrm{Sh}^{\mathrm{hyp}}(X)$ and consider the functor

$\Pi_{\infty}^{\Sigma}\colon\mathrm{Sh}^{\mathrm{hyp}}(X)\longrightarrow\mathrm{Cat}_{\infty}$

left Kan extended from the functor sending an open $U$ of $X$ to the
$\infty$-category of exit paths $\Pi_{\infty}^{\Sigma}(U,P|_{U})$. Van
Kampen’s theorem for exit paths (see [HA, Theorem A.7.1]) shows that
$\Pi_{\infty}^{\Sigma}$ is a colimit preserving functor. Thus, 3.11 provides
for every presentable $\infty$-category $\mathcal{E}$ an adjunction

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

which we refer to as the exodromy adjunction. When the stratification on $X$
is trivial, Corollaries 3.30 and 3.32 show that this adjunction restricts to
an equivalence between
$\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})$ and the
$\infty$-category of locally hyperconstant hypersheaves on $X$. Our goal in
this section is to show that the functor
$\Psi_{X,P}^{\mathrm{hyp},\mathcal{E}}$ factors through the full subcategory
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ of constructible
hypersheaves on $X$, and therefore that the exodromy adjunction restricts to
an adjunction

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

Since the case of trivial stratification is already known, the main point is
to show that the functor $\Psi_{X,P}^{\mathrm{hyp},\mathcal{E}}$ is compatible
with the restriction to strata.

Throughout this section, we fix a presentable $\infty$-category $\mathcal{E}$.
To lighten the notation, we will suppress the $\mathcal{E}$ superscript in the
exodromy adjunction.

###  4.1. Criteria for $\psi_{f}^{\mathrm{hyp}}$ to be an equivalence

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. 3.19
associated to $f$ a natural transformation

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

Our goal is to prove that $\psi_{f}^{\mathrm{hyp}}$ is an equivalence when $f$
is the inclusion of a single stratum. For this, we start laying out general
locality properties of $\psi_{f}^{\mathrm{hyp}}$.

######  Notation 4.1.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. For
every open $U\subseteq X$ we let $Y_{U}\coloneqq U\times_{X}Y$ and we denote
by $f_{U}\colon(Y_{U},Q)\to(U,P)$ the induced morphism.

The formation of the hypersheaf associated to a functor from the Exit Paths is
local. This is justified by the following

######  Lemma 4.2.

Let $(X,P)$ be a conically stratified space. Let $j:U\to X$ be the inclusion
of an open subset of $X$. Then, the natural transformation

$\psi_{j}^{\mathrm{hyp}}\colon
j^{\ast,\mathrm{hyp}}\circ\Psi_{X,P}^{\mathrm{hyp}}\longrightarrow\Psi_{U,P}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(j)^{\ast}\
$

is an equivalence.

###### Proof.

Since $j:U\to X$ is the inclusion of an open subset of $X$, the functor
$j^{\ast,\mathrm{hyp}}:\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(U;\mathcal{E})$
is computed as the presheaf pullback $j^{-1}$. On the other hand, $j^{-1}$
admits a fully-faithul left adjoint $j_{!}$. Hence,
$j_{\sharp}^{\mathrm{hyp}}:=\mathrm{hyp}\circ j_{!}$ is left adjoint to
$j^{\ast,\mathrm{hyp}}$. Since the hypersheafification commutes with $j^{-1}$,
the unit transformation $\mathrm{id}\to j^{\ast,\mathrm{hyp}}\circ
j_{\sharp}^{\mathrm{hyp}}$ is an equivalence. Then, 4.2 follows from 3.21. ∎

######  Lemma 4.3.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Let
$i:U\to X$ be the inclusion of an open subset of $X$. Let $j:f^{-1}(U)\to Y$
be the induced inclusion. Then, for every
$F\in\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})$, the
transformation $j^{\ast,\mathrm{hyp}}(\psi_{f}^{\mathrm{hyp}}(F))$ is
canonically equivalent to
$\psi_{f_{U}}^{\mathrm{hyp}}(\Pi_{\infty}^{\Sigma}(i)^{\ast}(F))$.

###### Proof.

Apply 3.26 and 4.2 twice. ∎

######  Corollary 4.4.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Let
$\mathcal{B}$ be a basis for the topology of $X$. Assume that for every
$U\in\mathcal{B}$, the transformation

$\psi_{f_{U}}^{\mathrm{hyp}}\colon
f_{U}^{\ast,\mathrm{hyp}}\circ\Psi_{U,P}^{\mathrm{hyp}}\longrightarrow\Psi_{Y_{U},Q}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(f_{U})^{\ast}$

is an equivalence. Then, $\psi_{f}^{\mathrm{hyp}}$ is an equivalence.

###### Proof.

Fix $F\in\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})$. We have
to show that $\psi_{f}^{\mathrm{hyp}}(F)$ is an equivalence of hypersheaves on
$Y$. This statement is local on $Y$. In particular, for $U\in\mathcal{B}$, if
$i:U\to X$ and $j:f^{-1}(U)\to Y$ are the inclusions, it is enough to show
that $j^{\ast,\mathrm{hyp}}(\psi_{f}^{\mathrm{hyp}}(F))$ is an equivalence.
Then, 4.4 follows from 4.3. ∎

######  Lemma 4.5.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Then,
$\psi_{f}^{\mathrm{hyp}}$ is canonically equivalent to
$\mathrm{hyp}(\psi_{f}^{\mathrm{psh}})$.

###### Proof.

Apply 3.26 and 3.18-$(2)$ twice. ∎

######  Corollary 4.6.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Fix
$F\in\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})$. Assume that
the collection $\mathcal{B}=\mathcal{B}(f,F)$ of open subsets $V\subseteq Y$
for which the transformation

$\psi^{\mathrm{psh}}_{f}(F)(V)\colon
f^{-1}\big{(}\Psi_{X,P}^{\mathrm{psh}}(F)\big{)}(V)\to\Psi_{Y,Q}^{\mathrm{psh}}\big{(}\Pi_{\infty}^{\Sigma}(f)^{\ast}(F)\big{)}(V)$

is an equivalence is a basis for the topology of $Y$. Then,
$\psi_{f}^{\mathrm{hyp}}(F)$ is an equivalence.

###### Proof.

Let $j\colon\mathcal{B}\hookrightarrow\mathrm{Open}(Y)$ be the canonical
inclusion. Consider the commutative square

(4.7)
$\displaystyle{f^{-1}\big{(}\Psi_{X,P}^{\mathrm{psh}}(F)\big{)}}$$\displaystyle{\Psi_{Y,Q}^{\mathrm{psh}}\big{(}\Pi_{\infty}^{\Sigma}(f)^{\ast}(F)\big{)}}$$\displaystyle{j_{*}j^{-1}f^{-1}\big{(}\Psi_{X,P}^{\mathrm{psh}}(F)\big{)}}$$\displaystyle{j_{*}j^{-1}\Psi_{Y,Q}^{\mathrm{psh}}\big{(}\Pi_{\infty}^{\Sigma}(f)^{\ast}(F)\big{)}}$$\scriptstyle{\psi^{\mathrm{psh}}_{f}(F)}$$\scriptstyle{j_{*}j^{-1}\psi^{\mathrm{psh}}_{f}(F)}$

From 3.18, the presheaf
$\Psi_{Y,Q}^{\mathrm{psh}}\big{(}\Pi_{\infty}^{\Sigma}(f)^{\ast}(F)\big{)}$ is
a hypersheaf on $Y$. From [HPT, Proposition 1.13], the adjunction
$j^{-1}\dashv j_{*}$ induces an equivalence between
$\mathrm{Sh}^{\mathrm{hyp}}(Y;\mathcal{E})$ and the full subcategory
$\mathrm{Sh}^{\mathrm{hyp}}(\mathcal{B};\mathcal{E})$ of
$\mathrm{PSh}(\mathcal{B};\mathcal{E})$ spanned by the functors
$F:\mathcal{B}^{\mathrm{op}}\to\mathcal{E}$ such that $j_{*}(F)$ is a
hypersheaf on $Y$. Hence, the right vertical arrow of Eq. 4.7 is an
equivalence. Since $j^{-1}\psi_{f}^{\mathrm{psh}}(F)$ is an equivalence by
assumption, the bottom arrow is an equivalence as well. Thus,
$j_{*}j^{-1}f^{-1}\big{(}\Psi_{X,P}^{\mathrm{psh}}(F)\big{)}$ is a hypersheaf
on $Y$. Then, [HPT, Remark 1.14] implies that the left vertical arrow exhibits
its target as the hypersheafification of its source. Hence,

$j_{*}j^{-1}\psi_{f}^{\mathrm{psh}}(F)\simeq\mathrm{hyp}(\psi_{f}^{\mathrm{psh}}(F))\simeq\psi_{f}^{\mathrm{hyp}}(F)$

where the last equivalence follows from 4.5. 4.6 thus follows. ∎

######  Corollary 4.8.

Let $f\colon Y\to X$ be a morphism between locally weakly contractible
topological spaces. Then for every presentable $\infty$-category
$\mathcal{E}$, both transformations

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

are equivalences once restricted to locally hyperconstant hypersheaves.

###### Proof.

Combining 3.20-(1) and 3.32 we see that it is enough to show that $\psi_{f}$
is an equivalence. Using 4.4, we can reduce ourselves to the case where $X$ is
weakly contractible, so that the conclusion follows from 3.29 and [HPT,
Proposition 2.5]. ∎

###  4.2. Restriction to strata for $\Psi$

######  Proposition 4.9.

Let $(X,P)$ be a conically stratified space. Let $a\in P$ and let $i_{a}\colon
X_{a}\to X$ be the inclusion of the associated stratum. Then the natural
transformation

$\psi_{a}^{\mathrm{hyp}}\colon
i_{a}^{\ast,\mathrm{hyp}}\circ\Psi_{X,P}^{\mathrm{hyp}}\longrightarrow\Psi_{X_{a}}^{\mathrm{hyp}}\circ\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}$

is an equivalence.

###### Proof.

Let $\mathcal{B}$ be the poset of open subsets $U$ of $X_{a}$ such that the
set $\mathcal{B}^{\operatorname{exc}}(U)$ of open neighbourhoods $V$ of $U$ in
$X$ where $\Pi_{\infty}^{\Sigma}(U,\\{a\\})\to\Pi_{\infty}^{\Sigma}(V,P)$ is
final is a fundamental system of open neighbourhoods of $U$ in $X$. From 2.20,
the set $\mathcal{B}$ is a basis of $X_{a}$. Fix
$F\in\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})$. From 4.6, we
are left to show that for every $U\in\mathcal{B}$, the morphism

$\psi^{\mathrm{psh}}_{i_{a}}(F)(U)\colon
i_{a}^{-1}\big{(}\Psi_{X,P}^{\mathrm{psh}}(F)\big{)}(U)\to\Psi_{X_{a},\\{a\\}}^{\mathrm{psh}}\big{(}\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}(F)\big{)}(U)$

is an equivalence. On the one hand

$i_{a}^{-1}(\Psi_{X,P}^{\mathrm{psh}}(F))(U)\simeq\operatorname*{colim}_{V\in\mathcal{B}^{\operatorname{exc}}(U)}\Psi_{X,P}^{\mathrm{psh}}(F)(V)\simeq\operatorname*{colim}_{V\in\mathcal{B}^{\operatorname{exc}}(U)}\lim_{\Pi_{\infty}^{\Sigma}(V,P)}F|_{\Pi_{\infty}^{\Sigma}(V,P)}\
.$

Since for every $V\in\mathcal{B}^{\operatorname{exc}}(U)$, the functor
$\Pi_{\infty}^{\Sigma}(U,\\{a\\})\to\Pi_{\infty}^{\Sigma}(V,P)$ is final, we
deduce

$i_{a}^{-1}(\Psi_{X,P}^{\mathrm{psh}}(F))(U)\simeq\operatorname*{colim}_{V\in\mathcal{B}^{\operatorname{exc}}(U)}\lim_{\Pi_{\infty}^{\Sigma}(U,\\{a\\})}F|_{\Pi_{\infty}^{\Sigma}(U,\\{a\\})}\simeq\lim_{\Pi_{\infty}^{\Sigma}(U,\\{a\\})}F|_{\Pi_{\infty}^{\Sigma}(U,\\{a\\})}\
.$

On the other hand,

$\Psi_{X_{a}}^{\mathrm{hyp}}\big{(}\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}(F)\big{)}(U)\simeq\lim_{\Pi_{\infty}^{\Sigma}(U,\\{a\\})}F|_{\Pi_{\infty}^{\Sigma}(U,\\{a\\})}\
.$

This concludes the proof of 4.9. ∎

######  Corollary 4.10.

Let $(X,P)$ be a conically stratified space with locally weakly contractible
strata. Then the functor
$\Psi_{X,P}^{\mathrm{hyp}}\colon\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
factors through the full subcategory
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.

###### Proof.

This is an immediate consequence of 4.9 and 3.30. ∎

##  5\. The exodromy equivalence

###  5.1. A criterion for $\phi_{f}$ to be an equivalence

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces and
let $\mathcal{E}$ be a presentable $\infty$-category. As usual $\mathcal{E}$
will be fixed throughout this section, and we therefore suppress it from the
notations. Our first goal is to provide a general method to establish when the
natural transformation

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

is an equivalence.

######  Lemma 5.1.

Let $f\colon(Y,Q)\to(X,P)$ be a morphism of conically stratified spaces. Then
the following statements hold;

  1. (1)

the transformations $\phi_{f}^{\mathrm{psh}}$ and
$\phi_{f}^{\mathrm{hyp}}\circ\mathrm{hyp}$ are canonically equivalent.

  2. (2)

for every diagram $F:I\to\mathrm{PSh}(X;\mathcal{E})$ with $I$ a small
$\infty$-category, the canonical morphism

$\phi_{f}^{\mathrm{psh}}(\operatorname*{colim}_{i\in
I}j(F_{i}))\longrightarrow\operatorname*{colim}_{i\in I}\phi_{f}(F_{i})$

is an equivalence, where
$j:\mathrm{PSh}(X;\mathcal{E})\to\operatorname{Fun}(\mathrm{PSh}(X;\mathcal{E})^{\mathrm{op}};\mathcal{E})$
is the Yoneda embedding.

###### Proof.

For the first point, apply 3.18-$(2)$ and 3.23 twice. For the second point, we
have

$\phi_{f}^{\mathrm{psh}}(\operatorname*{colim}_{i\in
I}j(F_{i}))\simeq\operatorname*{colim}_{i\in
I}\phi_{f}^{\mathrm{psh}}(j(F_{i}))\simeq\operatorname*{colim}_{i\in
I}\phi_{f}(F_{i})$

where the first equivalence follows from the fact that
$\phi_{f}^{\mathrm{psh}}$ is built out of left adjoint functors and where the
second equivalence follows from the observation that
$\Phi_{X,P}^{\mathrm{psh}}$ identifies with the left Kan extension of
$\Phi_{X,P}$ along the Yoneda embedding. ∎

######  Corollary 5.2.

If $\phi_{f}$ is an equivalence, the same goes for $\phi_{f}^{\mathrm{psh}}$
and $\phi_{f}^{\mathrm{hyp}}$.

###### Proof.

The first statement comes from 5.1-$(2)$ and the fact that every presheaf is a
small colimit of representable objects. The second statements then follows
from the observation from 5.1-$(1)$. ∎

The advantage of $\phi_{f}$ is that it can be explicitly computed in terms of
open subsets of $Y$, rather than having to deal with all hypersheaves on $Y$.
To fully exploit this, let us specialize the main construction of Section 3.4
to the current setting.

######  Notation 5.3.

For a conically stratified space $(X,P)$, let

$\pi_{X}\colon\mathrm{E}_{X}\longrightarrow\mathrm{Open}(X)^{\mathrm{op}}$

be the cartesian fibration classifying the exit paths $\infty$-functor
$\Pi_{\infty}^{\Sigma}\colon\mathrm{Open}(X)\to\mathrm{Cat}_{\infty}$. We let

$\lambda_{X}\colon\mathrm{E}_{X}\longrightarrow\Pi_{\infty}^{\Sigma}(X,P)$

be the canonical localization functor. Given a morphism $f\colon(Y,Q)\to(X,P)$
of conically stratified spaces, we consider the following diagram:

(5.4)
$\displaystyle{\Pi_{\infty}^{\Sigma}(Y,Q)}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X,P)}$$\displaystyle{\mathrm{E}_{Y}}$$\displaystyle{\mathrm{E}_{f}}$$\displaystyle{\mathrm{E}_{X}}$$\displaystyle{\mathrm{Open}(Y)^{\mathrm{op}}}$$\displaystyle{\mathrm{Open}(X)^{\mathrm{op}}}$$\scriptstyle{\Pi_{\infty}^{\Sigma}(f)}$$\scriptstyle{\lambda_{Y}}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{\pi_{f}}$$\scriptstyle{\lambda_{f}}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{f^{-1}}$

With these notations, 3.18-(1) shows that
$\Psi_{X,P}^{\mathrm{psh}}\simeq\pi_{X,\ast}\circ\lambda_{X}^{\ast}$. In turn,
we obtain:

######  Corollary 5.5.

  1. (1)

There is a canonical identification
$\Phi_{X,P}^{\mathrm{psh}}\simeq\lambda_{X,!}\circ\pi_{X}^{\ast}$.

  2. (2)

The Beck-Chevalley transformation

$(f^{-1})^{\ast}\circ\pi_{Y,\ast}\longrightarrow\pi_{f,\ast}\circ q^{\ast}$

is an equivalence.

  3. (3)

If the Beck-Chevalley transformation

$\lambda_{f,!}\circ
p^{\ast}\longrightarrow\Pi_{\infty}^{\Sigma}(f)^{\ast}\circ\lambda_{X,!}$

is an equivalence, then so is $\phi_{f}^{\mathrm{hyp}}$.

###### Proof.

Point (1) follows from the identification
$\Psi_{X,P}^{\mathrm{psh}}\simeq\pi_{X,\ast}\circ\lambda_{X}^{\ast}$ passing
to left adjoints. Point (2) is just a reformulation of 3.15 in this specific
situation. As for point (3), observe that by construction $\phi_{f}$ is the
composition

$\lambda_{Y,!}\circ\pi_{Y}^{\ast}\circ(f^{-1})_{!}\simeq\lambda_{f,!}\circ\pi_{f}^{\ast}\simeq\lambda_{f,!}\circ
p^{\ast}\circ\pi_{X}^{\ast}\longrightarrow\Pi_{\infty}^{\Sigma}(f)^{\ast}\circ\lambda_{X,!}\circ\pi_{X}^{\ast}\
.$

Therefore, if the Beck-Chevalley transformation of the statement is an
equivalence, the same goes for $\phi_{f}$. Thus, the conclusion follows from
5.2. ∎

###  5.2. Restriction to a closed union of strata for $\Phi$

Assume that $f\colon(Y,Q)\to(X,P)$ is the inclusion of a closed union of
strata. The following lemma is a straightforward consequence of Quillen’s
theorem A and its proof is left to the reader:

######  Lemma 5.6.

Let

$\displaystyle{\mathcal{C}_{0}}$$\displaystyle{\mathcal{C}}$$\displaystyle{\mathcal{D}_{0}}$$\displaystyle{\mathcal{D}}$$\scriptstyle{g}$$\scriptstyle{i}$$\scriptstyle{f}$$\scriptstyle{j}$

be a pullback square in $\mathrm{Cat}_{\infty}$. Assume that:

  1. (1)

the functors $i$ and $j$ are fully faithful;

  2. (2)

for $d\in\mathcal{D}$, $d_{0}\in\mathcal{D}_{0}$, if
$\operatorname{Map}_{\mathcal{D}}(d,j(d_{0}))\neq\emptyset$, then $d$ belongs
to the essential image of $j$.

Then, for any $\mathcal{E}$ presentable $\infty$-category the Beck-Chevalley
transformation $g_{!}\circ i^{*}\to j^{*}\circ f_{!}$ is an equivalence.

######  Corollary 5.7.

Let $(X,P)$ be a conically stratified space. Let $S\subset P$ be a closed
downwards subset and let $f\colon X_{S}\to X$ be the inclusion of the
corresponding closed union of strata. The upper right square of the diagram
(5.4)

$\displaystyle{\mathrm{E}_{f}}$$\displaystyle{\mathrm{E}_{X}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X_{S},S)}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X,P)}$$\scriptstyle{p}$$\scriptstyle{\lambda_{f}}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{f}$

induces the Beck-Chevalley transformation

$\beta\colon\lambda_{f!}\circ p^{*}\longrightarrow f^{*}\circ\lambda_{X!},$

which is an equivalence.

###### Proof.

We check that the conditions of 5.6 are satisfied. First of all we observe
that for every open subset $U$ of $X$, one has $U\cap X_{S}\simeq U_{S}$,
where we consider $U$ equipped with the induced stratification $P|_{U}$. From
2.6, the functor

$f_{U}\colon\mathrm{Exit}(U_{S},S|_{U})\longrightarrow\mathrm{Exit}(U,P|_{U})$

is fully faithful. It follows that both $f$ and $p$ are fully faithful
functors. That the above square is a pullback is then obvious. Finally, the
condition (3) from 5.6 holds since $S\subset P$ is closed downwards. ∎

Combining Corollaries 5.5-(3) and 5.7 we immediately obtain the following:

######  Corollary 5.8.

Let $(X,P)$ be a conically stratified space. Let $S\subset P$ be a closed
downwards subset and let $f\colon X_{S}\to X$ be the inclusion of the
corresponding closed union of strata. Then the natural transformation

$\phi_{S}^{\mathrm{hyp}}\colon\Phi_{X_{S},S}^{\mathrm{hyp}}\circ
f^{\ast,\mathrm{hyp}}\longrightarrow\Pi_{\infty}^{\Sigma}(f)^{\ast}\circ\Phi_{X,P}^{\mathrm{hyp}}$

is an equivalence.

###  5.3. Restriction to an open union of strata for $\Phi$

Dealing with the inclusion of an open union of strata is more complicated. We
start collecting some general $\infty$-categorical facts:

######  Definition 5.9.

We say that a functor $f\colon\mathcal{C}\to\mathcal{D}$ is weakly cofiltered
if for every object $d\in\mathcal{D}$ the $\infty$-category
$\mathcal{C}_{d/}\coloneqq\mathcal{C}\times_{\mathcal{D}}\mathcal{D}_{d/}$ is
cofiltered.

######  Remark 5.10.

If $f\colon\mathcal{C}\to\mathcal{D}$ is cofiltered in the sense of 3.4 it is
also weakly cofiltered: indeed, 3.5 shows that for every $d\in\mathcal{D}$ the
induced map $\mathcal{C}_{d/}\to\mathcal{D}_{d/}$ is cofiltered and, on the
other hand, $\mathcal{D}_{d/}$ is obviously cofiltered since it has an initial
object. On the other hand, Quillen’s theorem A implies that if
$f\colon\mathcal{C}\to\mathcal{D}$ is weakly cofiltered, then it is also
colimit-final.

######  Lemma 5.11.

Let $\mathcal{X}$ be an $\infty$-category. Let
$\mathsf{A},\mathsf{B}\colon\mathcal{X}\to\mathrm{Cat}_{\infty}$ be functors
and let

$\pi_{\mathsf{A}}\colon\mathcal{A}\longrightarrow\mathcal{X}^{\mathrm{op}}\
,\qquad\pi_{\mathsf{B}}\colon\mathcal{B}\longrightarrow\mathcal{X}^{\mathrm{op}}$

be the associated cartesian fibrations. Let $f\colon\mathsf{A}\to\mathsf{B}$
be a natural transformation. If for every $x\in\mathcal{X}$ the functor
$f_{x}\colon\mathsf{A}(x)\to\mathsf{B}(x)$ is weakly cofiltered, then the
induced functor $\mathcal{A}\to\mathcal{B}$ is colimit-final.

###### Proof.

Let $b\in\mathcal{B}$ be an object. In virtue of Quillen’s theorem A, we have
to prove that the $\infty$-category

$\mathcal{A}_{b/}\coloneqq\mathcal{A}\times_{\mathcal{B}}\mathcal{B}_{b/}$

is weakly contractible. We claim that it is cofiltered. Set
$x\coloneqq\pi_{\mathsf{B}}(b)$, so that we can review $b$ as an element in
$\mathcal{B}_{x}\simeq\mathsf{B}(x)$. Write
$\mathsf{A}(x)_{b/}\coloneqq\mathsf{A}(x)\times_{\mathsf{B}(x)}\mathsf{B}(x)_{b/}$
and let

$j\colon\mathsf{A}(x)_{b/}\longrightarrow\mathcal{A}_{b/}$

be the natural functor. Let now $F\colon I\to\mathcal{A}_{b/}$ be a finite
diagram. Composing $F$ with the canonical projection
$\mathcal{A}_{b/}\to\mathcal{X}_{x/}$ and using the fact that both
$\pi_{\mathsf{A}}$ and $\pi_{\mathsf{B}}$ are cartesian fibrations, we deduce
the existence of a functor

$F^{\prime}\colon
I\to\mathsf{A}(x)_{b/}\coloneqq\mathsf{A}(x)\times_{\mathsf{B}(x)}\mathsf{B}(x)_{b/}$

together with a natural transformation $\gamma\colon j\circ F^{\prime}\to F$.
By assumption the functor $f_{x}\colon\mathsf{A}(x)\to\mathsf{B}(x)$ is weakly
cofiltered, and therefore $\mathsf{A}(x)_{b/}$ is cofiltered. It follows that
$F^{\prime}$ admits an extension $\widetilde{F}^{\prime}\colon
I^{\lhd}\to\mathsf{A}(x)_{b/}$. At this point, the natural transformation
$\gamma$ allows us to prolong $\widetilde{F}^{\prime}$ into an extension
$\widetilde{F}\colon I^{\lhd}\to\mathcal{A}_{b/}$ of $F$. The conclusion
follows. ∎

Let $(X,P)$ be a conically stratified space. Fix an object
$x\in\Pi_{\infty}^{\Sigma}(X,P)$. Define
$\Pi_{\infty}^{\Sigma}(X,P)_{/x}^{\simeq}$ as the full subcategory of
$\Pi_{\infty}^{\Sigma}(X,P)_{/x}$ spanned by its final objects. For every
functor $\mathcal{C}\to\Pi_{\infty}^{\Sigma}(X,P)$ we set

$\mathcal{C}_{/x}\coloneqq\mathcal{C}\times_{\Pi_{\infty}^{\Sigma}(X,P)}\Pi_{\infty}^{\Sigma}(X,P)_{/x}\quad\text{and}\quad\mathcal{C}_{/x}^{\simeq}\coloneqq\mathcal{C}\times_{\Pi_{\infty}^{\Sigma}(X,P)}\Pi_{\infty}^{\Sigma}(X,P)_{/x}^{\simeq}\
.$

The following lemma contains the key geometrical argument of this section:

######  Lemma 5.12.

The functor
$(\mathrm{E}_{X})_{/x}^{\simeq}\longrightarrow(\mathrm{E}_{X})_{/x}$ is
cofinal.

###### Proof.

We apply 5.11 taking $\mathcal{X}=\mathrm{Open}(X)$, and $\mathsf{A}$ and
$\mathsf{B}$ to be the functors given by

$\mathsf{A}(U)\coloneqq\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}\quad\text{and}\quad\mathsf{B}(U)\coloneqq\Pi_{\infty}^{\Sigma}(U,P)_{/x}\
.$

By construction, the cartesian fibration classified by $\mathsf{B}$ is given
by the natural projection
$(\mathrm{E}_{X})_{/x}\to\mathrm{Open}(X)^{\mathrm{op}}$, and similarly the
cartesian fibration classified by $\mathsf{A}$ is given by
$(\mathrm{E}_{X})_{/x}^{\simeq}\to\mathrm{Open}(X)^{\mathrm{op}}$. Thus, 5.11
shows that it is enough to show that for every open $U$ of $X$, the functor

$\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}\longrightarrow\Pi_{\infty}^{\Sigma}(U,P)_{/x}$

is weakly cofiltered. Let $\mathbf{\gamma}\in\Pi_{\infty}^{\Sigma}(U,P)_{/x}$
and write
$\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}:=(\Pi_{\infty}^{\Sigma}(U,P)_{/x})_{\mathbf{\gamma}/}$.
We show that the $\infty$-category

$\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}^{\simeq}\coloneqq\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}\times_{\Pi_{\infty}^{\Sigma}(U,P)_{/x}}\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}$

is cofiltered. Let therefore $I$ be a finite category and consider the lifting
problem

$\displaystyle{I}$$\displaystyle{\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}^{\simeq}}$$\displaystyle{I^{\lhd}}$$\scriptstyle{g}$$\scriptstyle{j}$$\scriptstyle{\overline{g}}$

Since $\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}^{\simeq}$ is
fully faithful inside $\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}$,
we can rewrite the above lifting problem as

$\displaystyle{I}$$\displaystyle{\Pi_{\infty}^{\Sigma}(U,P)_{\mathbf{\gamma}/\\!\\!/x}}$$\displaystyle{I^{\lhd}}$$\scriptstyle{g}$$\scriptstyle{j}$$\scriptstyle{\overline{g}}$

where $g$ sends an object $i\in I$ to a morphism
$\mathbf{\gamma}\to\mathbf{\delta}_{i}$ in $\Pi_{\infty}^{\Sigma}(U,P)_{/x}$
with $\mathbf{\delta}_{i}\in\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}$ and
where $\overline{g}$ is required to send the vertex $v_{0}$ of $I^{\lhd}$ to a
morphism $\mathbf{\gamma}\to\mathbf{\delta}$ in
$\Pi_{\infty}^{\Sigma}(U,P)_{/x}$ with
$\mathbf{\delta}\in\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}$. This is
equivalent to a lifting problem

$\displaystyle{I^{\lhd}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(U,P)_{/x}}$$\displaystyle{(I^{\lhd})^{\lhd}}$
$\scriptstyle{g^{\prime}}$$\scriptstyle{j^{\lhd}}$$\scriptstyle{\overline{g}^{\prime}}$

where $g^{\prime}$ sends the vertex $v_{1}$ of the exterior cone
$(I^{\lhd})^{\lhd}$ to $\mathbf{\gamma}$ and the other objects in
$\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}$, and where $\overline{g}^{\prime}$
is required to send $v_{0}$ in $\Pi_{\infty}^{\Sigma}(U,P)_{/x}^{\simeq}$.
Unraveling the definitions and denoting by $v_{2}$ the final object of
$((I^{\lhd})^{\lhd})^{\rhd}$, we can further reduce this lifting problem to
the following one

$\displaystyle{(I^{\lhd})^{\rhd}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X,P)}$$\displaystyle{((I^{\lhd})^{\lhd})^{\rhd}}$$\displaystyle{\phantom{\Pi_{\infty}^{\Sigma}(X,P)}\
,}$$\scriptstyle{j^{\lhd}}$$\scriptstyle{h}$$\scriptstyle{\overline{h}}$

where we ask for $\overline{h}$ to represent a continuous morphism

$\overline{\mathrm{h}}\colon|((I^{\lhd})^{\lhd})^{\rhd}|\longrightarrow X,$

with the following properties:

  1. (1)

the morphism $\mathrm{h}$ takes the segment $[v_{1},v_{2}]$ to the exit path
$\gamma$ underlying $\mathbf{\gamma}$.

  2. (2)

the morphism $\mathrm{h}$ takes the complement of $v_{1}$ to the same stratum
of $x$;

  3. (3)

the morphism $\mathrm{h}$ takes the double cone $(I^{\lhd})^{\lhd}$ inside the
open $U$.

Observe that there is a canonical equivalence

$((I^{\lhd})^{\lhd})^{\rhd}\simeq(\Delta^{1}\star
I)\star\Delta^{0}\simeq\Delta^{1}\star(I\star\Delta^{0})\simeq((I^{\rhd})^{\lhd})^{\lhd}\
,$

where $\star$ denotes the join operation. Set $J\coloneqq I^{\rhd}$. By
definition of $\Pi_{\infty}^{\Sigma}(X,P)$, we can represent $h$ by an
explicit continuous morphism

$\mathrm{h}\colon|J^{\lhd}|\longrightarrow X,$

with the property that it takes the segment $[v_{1},v_{2}]$ to $\gamma$, that
the complement of $v_{1}$ is sent in the same stratum as $x$, and that the
cone over $I$ with vertex $v_{1}$ is sent in $U$. Since $\mathrm{h}$ is
continuous, the preimage of $U$ is an open of $|J^{\lhd}|$ containing the cone
over $I$ with vertex $v_{1}$. Then, the inclusion of the exterior cone

$i\coloneqq|j^{\lhd}|\colon|J^{\lhd}|\hookrightarrow|(J^{\lhd})^{\lhd}|$

admits a retraction $r$ sending $v_{0}$ inside $[v_{1},v_{2}]$ close enough to
$v_{1}$ and satisfying the following extra properties:

  1. (i)

$r$ sends $|(I^{\lhd})^{\lhd}|$ inside $\mathrm{h}^{-1}(U)$;

  2. (ii)

the vertex $v_{1}$ is the unique point $s\in|(J^{\lhd})^{\lhd}|$ such that
$r(s)=v_{1}$.

Define

$\overline{\mathrm{h}}\coloneqq\mathrm{h}\circ r.$

It is then straightforward to verify that $\overline{\mathrm{h}}$ satisfies
the conditions (1),(2) and (3) listed above. ∎

######  Corollary 5.13.

Let $(X,P)$ be a conically stratified space. Let $S\subset P$ be a closed
upwards subset and let $f\colon X_{S}\to X$ be the inclusion of the
corresponding open union of strata. The upper right square of the diagram
(5.4)

(5.14)
$\displaystyle{\mathrm{E}_{f}}$$\displaystyle{\mathrm{E}_{X}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X_{S},S)}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X,P)}$$\scriptstyle{p}$$\scriptstyle{\lambda_{f}}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{f}$

induces the Beck-Chevalley transformation

$\beta\colon\lambda_{f!}\circ p^{*}\longrightarrow f^{*}\circ\lambda_{X!}$

which is an equivalence.

###### Proof.

As in the proof of 5.7, $f$ and $p$ are fully faithful and the above square is
a pullback. Let now $x\in\Pi_{\infty}^{\Sigma}(X_{S},S)$ be an object.
Committing a slight abuse of notation, we still denote by $x$ its image in
$\Pi_{\infty}^{\Sigma}(X,P)$ via the functor $f$. Put

$(\mathrm{E}_{f})_{/x}\coloneqq\mathrm{E}_{f}\times_{\Pi_{\infty}^{\Sigma}(X_{S},S)}\Pi_{\infty}^{\Sigma}(X_{S},S)_{/x}$

and consider the commutative diagram

$\displaystyle{(\mathrm{E}_{f})_{/x}}$$\displaystyle{\mathrm{(}\mathrm{E}_{X})_{/x}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X_{S},S)_{/x}}$$\displaystyle{\Pi_{\infty}^{\Sigma}(X,P)_{/x}.}$

Unraveling the definitions, we reduce ourselves to check that
$(\mathrm{E}_{f})_{/x}\longrightarrow(\mathrm{E}_{X})_{/x}$ is cofinal.
Observe that since $p$ is fully faithful, the same goes for this map. By
definition, $(\mathrm{E}_{f})_{/x}$ is the full subcategory of triples
$(U,y,\gamma)$ in $(\mathrm{E}_{X})_{/x}$ such that $y$ lies in $X_{S}$. Using
the same notations of 5.12, we find the following commutative diagram of
$\infty$-categories over $\Pi_{\infty}^{\Sigma}(X,P)_{/x}$ :

$\displaystyle{(\mathrm{E}_{f})_{/x}^{\simeq}}$$\displaystyle{\mathrm{(}\mathrm{E}_{X})_{/x}^{\simeq}}$$\displaystyle{(\mathrm{E}_{f})_{/x}}$$\displaystyle{(\mathrm{E}_{X})_{/x}.}$

Since the vertical functors and the bottom horizontal one are fully faithful,
the same goes for the top horizontal one. Furthermore, if
$(U,y,\gamma)\in(\mathrm{E}_{X})_{/x}^{\simeq}$ then $\gamma\colon y\to x$ is
an equivalence in $\Pi_{\infty}^{\Sigma}(X,P)$. Since $x$ belongs to $X_{S}$,
the same goes for $y$. In other words, $(U,y,\gamma)$ belongs to
$(\mathrm{E}_{f})_{/x}^{\simeq}$. This shows that the top horizontal arrow is
also essentially surjective, hence an equivalence. As a consequence, to prove
that the bottom horizontal functor is cofinal it is enough to prove that both
vertical functors are cofinal.

We now show that it is enough to prove that the right vertical functor is
cofinal. By Quillen’s Theorem A and the above discussion, it is enough to show
that for $\mathbf{y}=(U,y,\gamma)\in(\mathrm{E}_{f})_{/x}$, the fully faithful
functor
$((\mathrm{E}_{f})_{/x})_{\mathbf{y}/}\to((\mathrm{E}_{X})_{/x})_{\mathbf{y}/}$
is an equivalence. Let
$\mathbf{z}\coloneqq(V,z,\delta)\in(\mathrm{E}_{X})_{/x}$ and let
$\mathbf{y}\to\mathbf{z}$ be a morphism in $(\mathrm{E}_{X})_{/x}$. Observe
that this morphism produces an exit path $y\to z$. Since $S$ is closed
upwards, this implies that $z$ lies in $X_{S}$. Thus, we have
$\mathbf{z}\in(\mathrm{E}_{f})_{/x}$. Hence, the fully faithful functor
$((\mathrm{E}_{f})_{/x})_{\mathbf{y}/}\to((\mathrm{E}_{X})_{/x})_{\mathbf{y}/}$
is an equivalence and the sought-after reduction is complete. We then conclude
the proof of 5.13 using 5.12. ∎

Combining Corollaries 5.5-(3) and 5.13 we immediately obtain the following:

######  Corollary 5.15.

Let $(X,P)$ be a conically stratified space. Let $S\subset P$ be a closed
upwards subset and let $f\colon X_{S}\to X$ be the inclusion of the
corresponding open union of strata. Then the natural transformation

$\phi_{S}^{\mathrm{hyp}}\colon\Phi_{X_{S},S}^{\mathrm{hyp}}\circ
f^{\ast,\mathrm{hyp}}\longrightarrow\Pi_{\infty}^{\Sigma}(f)^{\ast}\circ\Phi_{X,P}^{\mathrm{hyp}}$

is an equivalence.

######  Corollary 5.16.

Let $(X,P)$ be a conically stratified space. Let $a\in P$ and let $i_{a}\colon
X_{a}\to X$ be the inclusion of the corresponding stratum. Then the natural
transformation

$\phi_{a}^{\mathrm{hyp}}\colon\Phi_{X_{a}}^{\mathrm{hyp}}\circ
i_{a}^{\ast,\mathrm{hyp}}\longrightarrow\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}\circ\Phi_{X,P}^{\mathrm{hyp}}$

is an equivalence.

###### Proof.

It follows by first applying first 5.15 to $(X,P)$ and the subset
$P_{\geqslant a}\subseteq P$ and subsequently 5.8 to $(X_{\geqslant
a},P_{\geqslant a})$ and the subset $\\{a\\}\subseteq P_{\geqslant a}$. ∎

###  5.4. The equivalence

We are now ready for the main theorem:

######  Theorem 5.17.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. Assume that:

  1. (1)

for every $a\in P$, the associated stratum $X_{a}$ is locally weakly
contractible;

  2. (2)

the hyper-restrictions

$\big{\\{}i_{a}^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\longrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X_{a};\mathcal{E})\big{\\}}_{a\in
P}$

are jointly conservative.

Then the exodromy adjunction

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

is an equivalence.

###### Proof.

We have to prove that for every functor
$F\colon\Pi_{\infty}^{\Sigma}(X,P)\to\mathcal{E}$ and every hyperconstructible
hypersheaf $G\in\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$, the unit and
counit

$\eta_{G}\colon
G\longrightarrow\Psi_{X,P}^{\mathrm{hyp}}(\Phi_{X,P}^{\mathrm{hyp}}(G))\quad\text{and}\quad\varepsilon_{F}\colon\Phi_{X,P}^{\mathrm{hyp}}(\Psi_{X,P}^{\mathrm{hyp}}(F))\longrightarrow
F$

are equivalences. By assumption, it is enough to check that for every $a\in
P$, the (hyper-)restrictions $i_{a}^{\ast,\mathrm{hyp}}(\eta_{G})$ and
$\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}(\varepsilon_{F})$ are equivalences.
Consider the following square:

$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})}$$\displaystyle{\operatorname{Fun}\big{(}\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E}\big{)}}$$\displaystyle{\mathrm{Sh}^{\mathrm{hyp}}(X_{a};\mathcal{E})}$$\displaystyle{\operatorname{Fun}\big{(}\Pi_{\infty}(X_{a}),\mathcal{E}\big{)}\
.}$$\scriptstyle{i_{a}^{\ast,\mathrm{hyp}}}$$\scriptstyle{\Phi_{X,P}^{\mathrm{hyp}}}$$\scriptstyle{\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}}$$\scriptstyle{\Phi_{X_{a}}^{\mathrm{hyp}}}$

5.16 shows that the natural transformation $\phi_{a}^{\mathrm{hyp}}$ makes
this into a commutative diagram. Moreover, combining 3.20-(2) and 4.9 we
deduce that it is horizontally right adjointable. Therefore we obtain
canonical identifications

$i_{a}^{\ast,\mathrm{hyp}}(\eta_{G})\simeq\eta_{i_{a}^{\ast,\mathrm{hyp}}(G)}\qquad\text{and}\qquad\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}(\varepsilon_{F})\simeq\varepsilon_{\Pi_{\infty}^{\Sigma}(i_{a})^{\ast}(F)}\
,$

where the latter are the unit and the counit of the adjunction
$\Phi_{X_{a}}^{\mathrm{hyp}}\dashv\Psi_{X_{a}}^{\mathrm{hyp}}$. Observe now
that since $G$ is hyperconstructible, $i_{a}^{\ast,\mathrm{hyp}}(G)$ is
locally constant. Thus the conclusion follows directly from 3.32. ∎

######  Remark 5.18.

The joint conservativity of the hyper-restrictions to strata is satisfied in
the following two cases of interest:

  1. (1)

the category $\mathcal{E}$ is compactly generated. See [HPT, Corollary 5.16].

  2. (2)

the poset $P$ is noetherian and the category $\mathcal{E}$ is stable and
presentable. See [HPT, Corollary 5.21].

Let us emphasize some immediate consequences of the above theorem.

######  Corollary 5.19.

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:

  1. (1)

the $\infty$-category $\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$ is
presentable;

  2. (2)

if $\mathcal{E}$ is an $\infty$-topos, then so is
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$.

The following corollary generalizes at the same time [HA, Lemma A.9.14] and
[HPT, Corollary 3.2]

######  Corollary 5.20.

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 full subcategory
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})\hookrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
is closed under small limits and small colimits.

###### Proof.

For colimits it is enough to observe that if $a\in P$ and if $i_{a}\colon
X_{a}\to X$ is the inclusion of the corresponding stratum, then
$i_{a}^{\ast,\mathrm{hyp}}\colon\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X_{a};\mathcal{E})$
commutes with colimits. Thus, we are reduced to show that
$\mathrm{LC}^{\mathrm{hyp}}(X;\mathcal{E})$ is closed under colimits in
$\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$. This was shown in [HPT, Corollary
3.2]. Concerning limits, it is enough to observe that
$\Psi_{X,P}^{\mathrm{hyp}}\colon\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
is fully faithful and right adjoint to $\Phi_{X,P}^{\mathrm{hyp}}$. Thus
$\Psi_{X,P}^{\mathrm{hyp}}$ commutes with small limits. On the other hand 5.17
shows that the essential image of $\Psi_{X,P}^{\mathrm{hyp}}$ is
$\mathrm{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$, whence the conclusion. ∎

######  Remark 5.21.

  1. (1)

Combining Corollaries 5.19 and 5.20 it follows that if the assumptions of 5.17
are satisfied then the inclusion
$\iota:\mathrm{Cons}_{P}(X;\mathcal{E})\hookrightarrow\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
admits both a left adjoint $\mathrm{L}_{P}^{\mathrm{hyp}}$ and a right adjoint
$\mathrm{R}_{P}^{\mathrm{hyp}}$.

  2. (2)

The functor
$\Psi_{X,P}^{\mathrm{hyp}}:\operatorname{Fun}(\Pi_{\infty}^{\Sigma}(X,P),\mathcal{E})\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$
factors as $\iota\circ\Psi_{X,P}^{\mathrm{hyp}}$. Hence, if the assumptions of
5.17 are satisfied, it admits both $\Phi_{X,P}^{\mathrm{hyp}}$ and
$\Phi_{X,P}^{\mathrm{hyp}}|_{\mathrm{Cons}_{P}(X;\mathcal{E})}\circ\mathrm{L}_{P}^{\mathrm{hyp}}$
as left adjoints. Thus, for every
$F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$, the unit adjunction
$F\to\iota\circ L_{P}^{\mathrm{hyp}}(F)$ induces an equivalence
$\Phi_{X,P}^{\mathrm{hyp}}(F)\simeq\Phi_{X,P}^{\mathrm{hyp}}L_{P}^{\mathrm{hyp}}(F)$.

######  Corollary 5.22.

Let $(X,P)$ be a conically stratified space and let $\mathcal{E}$ be a
presentable $\infty$-category. Under the assumptions of 5.17, for a hypersheaf
$F\in\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$ the following statements are
equivalent:

  1. (1)

$F$ is hyperconstructible on $(X,P)$;

  2. (2)

for every open inclusion $U\subset V$ for which the induced map
$\Pi_{\infty}^{\Sigma}(U,P)\to\Pi_{\infty}^{\Sigma}(V,P)$ is a categorical
equivalence, the restriction map $F(V)\to F(U)$ is an equivalence;

  3. (3)

for every conical chart $Z\times C(Y)$ of $(X,P)$, $F$ satisfies the following
two conditions:

    1. (i)

for every open subset $W\subset Z$ and every
$0<\varepsilon<\varepsilon^{\prime}\leqslant 1$, the restriction map

$F(W\times C_{\varepsilon^{\prime}}(Y))\longrightarrow F(W\times
C_{\varepsilon}(Y))$

is an equivalence;

    2. (ii)

for every inclusion $U\subseteq V$ of weakly contractible open subsets of $Z$,
the restriction map

$F(V\times C(Y))\longrightarrow F(U\times C(Y))$

is an equivalence.

###### Proof.

The implication (1) $\Rightarrow$ (2) is a direct consequence of 5.17. For the
implication (2) $\Rightarrow$ (3), it is enough to observe that, on the one
hand, the inclusion $W\times C_{\varepsilon}(Y)\hookrightarrow W\times
C_{\varepsilon}(Y)$ is a stratified homotopy equivalence (and hence it
satisfies the assumption of point (2)); and on the other hand, for an
inclusion $U\subseteq V$ of weakly contractible open subsets of $Z$, the
induced map $\Pi_{\infty}^{\Sigma}(U\times
C(Y))\to\Pi_{\infty}^{\Sigma}(V\times C(Y))$ is also a categorical
equivalence, thanks to 2.14. We are therefore left to prove that (3) implies
(1). The question is local, so we can replace $X$ by a conical chart of the
form $Z\times C(Y)$. Letting $i\colon Z\to Z\times C(Y)$ be the natural
inclusion, we only have to check that $i^{\ast,\mathrm{hyp}}(F)$ is locally
hyperconstant. Consider first the presheaf-theoretic pullback $i^{-1}(F)$. For
every open subset $W$ of $Z$, one has

$i^{-1}(F)(W)\simeq\operatorname*{colim}_{0<\varepsilon\leqslant 1}F(W\times
C_{\varepsilon}(Y))\ .$

Our assumption (i) guarantees that the transition maps in the above colimit
diagram are equivalences, and hence that the canonical restriction map

$F(W\times C(Y))\longrightarrow i^{-1}(F)(W)$

is an equivalence. In particular, it follows that $i^{-1}(F)$ is a hypersheaf,
and therefore that it coincides with $i^{\ast,\mathrm{hyp}}(F)$.

At this point, in order to prove that $i^{-1}(F)\simeq
i^{\ast,\mathrm{hyp}}(F)$ is locally hyperconstant it is enough, in virtue of
[HPT, Proposition 3.1], to prove that for every inclusion $U\subset V$ of
weakly contractible open subsets of $Z$, the restriction map

$i^{-1}(F)(V)\longrightarrow i^{-1}(F)(U)$

is an equivalence. As we showed above, this amounts to check that the
restriction map

$F(V\times C(Y))\longrightarrow F(U\times C(Y))$

is an equivalence. Since this holds by assumption (ii), the conclusion
follows. ∎

For later use, let us remark that in the previous proof we established the
following fact:

######  Corollary 5.23.

Let $Z$ be a locally weakly contractible topological space and let $(Y,Q)$ be
a stratified space so that $(X,P)\coloneqq(Z\times C(Y),Q^{\vartriangleleft})$
is conically stratified. Let $\mathcal{E}$ be a presentable $\infty$-category
and assume that the assumptions of 5.17 are satisfied by $(X,P)$ and
$\mathcal{E}$. Let $F\in\operatorname{Cons}_{P}^{\mathrm{hyp}}(X;\mathcal{E})$
and let $U$ be an open subset of $Z$. Then the canonical map

$F(U\times C(Y))\longrightarrow i^{\ast,\mathrm{hyp}}(F)(U)$