# An enhanced six-functor formalism for diamonds and v-stacks

Daniel Gulotta, David Hansen and Jared Weinstein

###### Abstract

This article extends the six functor formalism for diamonds [11] to a very
general class of stacky morphisms between v-stacks, using $\infty$-categorical
techniques developed by Liu-Zheng [8].

###### Contents

  1. 1 Introduction
    1. 1.1 Context and motivation
    2. 1.2 Main results
    3. 1.3 Comments and conventions
    4. 1.4 Acknowledgments
  2. 2 Statement of main result
    1. 2.1 The notion of an enhanced operation map
    2. 2.2 The main theorem
  3. 3 Enhanced operations for qcqs v-sheaves
    1. 3.1 Prespatial diamonds
    2. 3.2 Enhanced operations for qcqs v-sheaves
    3. 3.3 Descent and codescent
  4. 4 Running DESCENT
    1. 4.1 Decent v-stacks and fine morphisms
    2. 4.2 A fragment of DESCENT
    3. 4.3 First iteration of DESCENT
    4. 4.4 Second iteration of DESCENT
    5. 4.5 Third iteration of DESCENT
    6. 4.6 Endgame
    7. 4.7 Cohomological smoothness for stacky maps

##  1 Introduction

###  1.1 Context and motivation

Grothendieck’s six functors refer to the operations
$f^{*},Rf_{*},Rf_{!},Rf^{!}$, $\otimes$, and internal Hom in a suitable
derived category of sheaves on some category of geometric objects. In the
original context, the category was schemes, but for some applications one may
wish to extend the formalism to Artin stacks. Considerable problems arise in
defining $Rf_{!}$ when $f$ is a stacky morphism, owing to issues of homotopy
coherence. These problems were overcome in the work of Lu-Zheng [8], which
applies $\infty$-categorical techniques to extend the full formalism to Artin
stacks.

This article solves the analogous problem in the setting of perfectoid spaces
and diamonds. Fix a prime $p$ and a ring $\Lambda$ killed by some integer
prime to $p$. In a very important paper [11], Scholze laid foundations for the
six functor formalism in the étale cohomology of diamonds. In particular, he
defined a triangulated category of étale sheaves
$D_{\mathrm{\acute{e}t}}(X,\Lambda)$ for any small v-stack $X$, agreeing with
the left-completion of $D(X_{\mathrm{\acute{e}t}},\Lambda)$ when e.g. $X$ is
an analytic adic space, and equipped with the usual $\otimes$ and
$R\mathscr{H}\mathrm{om}$ operations. Moreover, for any morphism $f:X\to Y$ he
constructed a pair of adjoint functors $f^{\ast}$ and $Rf_{\ast}$, and for
morphisms $f$ which are _compactifiable, representable in locally spatial
diamonds, and locally of finite dim.trg_ he constructed the exceptional
pushforward and pullback functors $Rf_{!}$ and $Rf^{!}$ with all their
expected properties: composability, base change, the projection formula, etc.
Finally, he showed that there is a reasonable notion of “cohomologically
smooth” morphisms, for which $Rf^{!}$ is an invertible twist of $f^{\ast}$.

For applications it is important to extend Scholze’s construction of $Rf_{!}$
and $Rf^{!}$ to some $f$ which are stacky (i.e., not representable).111A
simple example is the structure map
$f\colon[\ast/\underline{G(\mathbf{Q}_{p})}]\to\ast$, where $G/\mathbf{Q}_{p}$
is a linear algebraic group. Here one expects $Rf_{!}$ to be group homology
(whereas $Rf_{\ast}$ is group cohomology). See [3, Example 4.2.7]. In
particular, the results on the Kottwitz conjecture proved in [3] make crucial
use of the $!$-functor formalism with its expected properties for certain
stacky maps of Artin v-stacks.222In fact, despite numerous attempts, DH and JW
were never able to arrange the arguments in [3] in such a way as to avoid the
use of a stacky $!$-functor formalism. In the present paper, we extend the
!-functor formalism of [11] to a specific class of stacky maps, as required by
[3]. We expect this extended formalism to have many other applications.

###  1.2 Main results

In this section we give a precise statement of our main result. We first
define the category of geometric objects to which our formalism applies.

######  Definition 1.1.

A small v-stack $X$ is _decent_ if it satisfies the following two conditions:

  1. (1)

For any locally separated locally spatial diamond $T$ with a map $T\to X\times
X$, the pullback $T\times_{X\times X,\Delta}X$ is a locally separated locally
spatial diamond.

  2. (2)

There exists a locally separated locally spatial diamond $U$ together with a
surjective map $f:U\to X$ which is representable in locally spatial diamonds
and which locally on $U$ is separated and cohomologically smooth.

Any choice of $f:U\to X$ as in (2) will be called a _chart_ of $X$.

######  Remark 1.2.

i. In (1) it is enough to quantify over all separated spatial $T$. Moreover,
(1) implies that for any locally separated locally spatial diamonds $T,S$ with
maps $T\to X\leftarrow S$, the fiber product $T\times_{X}S$ is a locally
separated locally spatial diamond. Indeed, this fiber product can be written
as $(T\times S)\times_{X\times X,\Delta}X$, so the claim follows by observing
that $T\times S$ is a locally separated locally spatial diamond.

ii. In (2), we can assume without loss of generality that $U$ is a separated
locally spatial diamond and that $f$ is separated and cohomologically smooth,
for instance by replacing a given chart $f$ by the composition $\coprod
U_{i}\to U\overset{f}{\to}X$ for some open cover $U=\bigcup U_{i}$ by
separated locally spatial diamonds such that all restrictions $f|_{U_{i}}$ are
separated and cohomologically smooth. If these conditions hold, we say $f:U\to
X$ is a _clean chart_.

We will see (Proposition 4.4) that decent v-stacks are Artin v-stacks in the
sense of [1, Definition IV.1.1]. Generally, the condition of being decent is a
very mild restriction. For instance, all locally separated locally spatial
diamonds are decent; in particular, for any analytic adic space $X$ over
$\mathrm{Spa}\,\mathbf{Z}_{p}$, the associated diamond $X^{\lozenge}$ is
decent. More generally, if $X$ is any locally separated v-sheaf such that
there exists a separated locally spatial diamond $U$ with a surjective
cohomologically smooth map $U\to X$, then $X$ is decent. This applies, for
instance, to $X=\operatorname{Spd}\mathbf{Z}_{p}$ and
$X=\operatorname{Div}^{1}$. Neither of these is a diamond, but in both cases,
$U=X\times\operatorname{Spd}\mathbf{F}_{p}(\\!(t^{1/p^{\infty}})\\!)$ is
representable by an analytic adic space over $\mathbf{Z}_{p}$ (and is thus a
locally spatial diamond), and $U\to X$ is surjective and cohomologically
smooth.

Continuing this line of thought, one can check that all Artin v-stacks
appearing in [1] and [3] are decent, and in fact it takes some work to find an
example of an Artin v-stack which is not decent.

######  Definition 1.3.

i. A morphism $f:X\to Y$ between decent v-stacks is _fine_ if there exists a
commutative diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

where the vertical maps are charts and $g$ is locally on $W$ compactifiable of
finite dim.trg.

ii. A morphism $f:X\to Y$ between decent v-stacks is _$\ell$ -cohomologically
smooth_ if there exists a commutative diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$
(1.1)

where the vertical maps are charts and $g$ is locally on $W$ compactifiable of
finite dim.trg. and $\ell$-cohomologically smooth in the sense of [11].

We will see that these classes of morphisms are quite reasonable: they are
stable under composition and base change, membership can be tested smooth-
locally on the source and target, etc. We note that the name “fine” is chosen
to hint at the idea of being “locally of finite type”, and also because these
morphisms are “good enough” for all practical purposes.

The main result of this paper is the following theorem.

######  Theorem 1.4.

Let $\Lambda$ be a ring killed by some integer $n$ prime to $p$. If $f:X\to Y$
is any fine map of decent v-stacks, there is natural functor
$Rf_{!}:D_{\mathrm{\acute{e}t}}(X,\Lambda)\to
D_{\mathrm{\acute{e}t}}(Y,\Lambda)$ satisfying the following properties:

  1. (1)

When $f$ is separated and representable in locally spatial diamonds, $Rf_{!}$
coincides with the functor constructed in [11].

  2. (2)

There is a natural isomorphism of functors $R(f\circ g)_{!}\cong Rf_{!}\circ
Rg_{!}$ whenever fine morphisms $f$ and $g$ are composable. More precisely,
the assignments $X\mapsto D_{\text{\rm\'{e}t}}(X,\Lambda)$ and $f\mapsto
Rf_{!}$ upgrade to a pseudo-functor from the 2-category of decent v-stacks
with fine morphisms, to the 2-category of triangulated categories.

  3. (3)

The projection formula: there is a natural isomorphism $Rf_{!}(A\otimes
f^{*}B)\cong Rf_{!}A\otimes B$ for $A\in D_{\text{\rm\'{e}t}}(X,\Lambda)$ and
$B\in D_{\text{\rm\'{e}t}}(Y,\Lambda)$.

  4. (4)

Proper base change: For a cartesian square of decent v-stacks

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{g}}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

with $f$ fine, there is a proper base change isomorphism $g^{*}Rf_{!}\cong
Rf^{\prime}_{!}\tilde{g}^{*}$.

  5. (5)

$Rf_{!}$ admits a right adjoint $Rf^{!}$.

A heuristic for the construction of $Rf_{!}$ involves _cohomological
codescent._ To explain the idea, let $f:X\to Y$ be a fine map between decent
v-stacks, and let $g:U\to X$ be a chart. Set

$g_{n}:U_{n}=\underbrace{U\times_{X}U\cdots\times_{X}U}_{n+1}\to X,$

so $U_{n}$ is a locally separated locally spatial diamond, and $g_{n}$ and
$f\circ g_{n}$ are all fine and 0-truncated. Suppose that we already have
access to $Rh_{!}$ for fine 0-truncated maps $h$. We would then like to define
$Rf_{!}$ by the formula

$Rf_{!}A=\mathrm{colim}_{n\in\boldsymbol{\Delta}}R(f\circ
g_{n})_{!}Rg_{n}^{!}A.$ (1.2)

where $\boldsymbol{\Delta}$ is the simplex category. One can deduce formally
from (1.2) that proper base change and the projection formula hold for
$Rf_{!}$. For instance, to establish the projection formula:

$\displaystyle Rf_{!}(A\otimes f^{*}B)\cong$
$\displaystyle\operatorname{colim}_{n\in\boldsymbol{\Delta}}R(f\circ
g_{n})_{!}Rg_{n}^{!}(A\otimes f^{*}B)$ $\displaystyle\cong$
$\displaystyle\operatorname{colim}_{n\in\boldsymbol{\Delta}}R(f\circ
g_{n})_{!}(Rg_{n}^{!}A\otimes Rg_{n}^{*}f^{*}B)$ (smoothness of $g_{n}$)
$\displaystyle\cong$
$\displaystyle\operatorname{colim}_{n\in\boldsymbol{\Delta}}R(f\circ
g_{n})_{!}Rg_{n}^{!}A\otimes B$ (projection formula) $\displaystyle\cong$
$\displaystyle Rf_{!}A\otimes B$ ($\otimes$ is closed)

The heuristic (1.2) follows formally from the expectation that the map

$\mathrm{colim}_{n\in\boldsymbol{\Delta}}Rg_{n}{}_{!}Rg_{n}^{!}A\to A$ (1.3)

should be an isomorphism, i.e. that “cohomological codescent for surjective
smooth maps” should hold. To see the implication, apply $Rf_{!}$ to the
equivalence in (1.3) to obtain

$\displaystyle Rf_{!}A$ $\displaystyle\simeq
Rf_{!}\mathrm{colim}_{n\in\boldsymbol{\Delta}}Rg_{n}{}_{!}Rg_{n}^{!}A$
$\displaystyle\simeq\mathrm{colim}_{n\in\boldsymbol{\Delta}}Rf_{!}Rg_{n}{}_{!}Rg_{n}^{!}A$
$\displaystyle\simeq\mathrm{colim}_{n\in\boldsymbol{\Delta}}R(f\circ
g_{n})_{!}Rg_{n}^{!}A,$

using the fact that $Rf_{!}$ is a left adjoint to pass it across the colimit.

There are two main difficulties with making sense of this heuristic. First and
foremost, the colimit in (1.2) is not guaranteed to exist, let alone behave
naturally as in Theorem 1.4(2), since colimits in triangulated categories are
generally ill-behaved.

The other main difficulty is that even if $g$ as above was chosen to be
separated (which is always possible), $g_{n}$ and $f\circ g_{n}$ will only be
locally separated in general, so the formalism in [11] does not apply to
construct $R(f\circ g_{n})_{!}$ and $Rg_{n}^{!}$.

The remedy for the first difficulty is to upgrade the functors $R(f\circ
g_{n})_{!}$ and $Rg_{n}^{!}$ to an $\infty$-categorical setting where colimits
are well-behaved. This will be accomplished using the machinery of enhanced
operation maps developed in [8, 9] in the context of schemes. We will review
this machinery in a more detail in 2.1, but for the moment we present a brief
outline of how enhanced operations were used to produce the six functor
formalism for Artin stacks.

Let $\mathcal{C}$ be a category of geometric objects (e.g., schemes or Artin
stacks) equipped with some marked classes of morphisms
$\mathcal{E}_{1},\dots,\mathcal{E}_{n}$, and a subset
$I\subset\left\\{1,\dots,n\right\\}$. An enhanced operation map for
$(\mathcal{C},\mathcal{E}_{1},\dots,\mathcal{E}_{n},I)$ associates to each
object $X\in\mathcal{C}$ a symmetric monoidal $\infty$-category
$\mathcal{D}(X)$, and it also associates to each morphism $X\to Y$ in
$\mathcal{E}_{i}$ a functor $\mathcal{D}(X)\to\mathcal{D}(Y)$ (if $i\not\in
I$) or a functor $\mathcal{D}(Y)\to\mathcal{D}(X)$ (if $i\in I$). This must be
done in such a way that is (a) compatible with cartesian diagrams built out of
morphisms in the $\mathcal{E}_{i}$ and (b) compatible with the symmetric
monoidal structure on $\mathcal{D}(X)$.

The main theorem of [8] asserts the existence of an enhanced operation map for
(Artin stacks, locally finite type morphisms, all morphisms,
$\left\\{2\right\\}$), which encodes the operations $f^{*}$ and $Rf_{!}$. The
compatibilities inherent in the enhanced operation map imply both (a) proper
base change and (b) the projection formula for these functors, in a homotopy
coherent way. The required enhanced operation map is built in stages.

  1. (1)

In the first stage it is observed that an enhanced operation map (EOM) exists
for (schemes, all morphisms, $\left\\{1\right\\}$). This is essentially just
the statement that pullbacks $f^{*}\colon\mathcal{D}(Y)\to\mathcal{D}(X)$
preserve symmetric monoidal structure. It is a special case of a general
theorem on enhanced operations for ringed topoi [8, §2.2]. We restrict this to
an EOM for (qc separated schemes, locally finite type morphisms,
$\left\\{1\right\\}$)

  2. (2)

In the next stage, we apply the fact that every morphism of qc separated
schemes $f\colon X\to Y$ which is locally of finite type can be factored as
$p\circ j$, where $j$ is an open immersion and $p$ is proper. A gluing
technique [9, Theorem 5.4] allows us to extend the EOM for (schemes, locally
finite type morphisms, $\left\\{1\right\\}$) to (qc separated schemes, proper
morphisms, local isomorphisms,$\left\\{1,2\right\\}$) and even to (qc
separated schemes, proper morphisms, local isomorphisms, all morphisms,
$\left\\{1,2,3\right\\}$). So far we are still only encoding structures
associated to the pullback functor $f^{*}$, but in a way that “remembers” all
possible factorizations of $f$ as $p\circ j$.

  3. (3)

Heuristically, if $f=p\circ j$ as above then $Rf_{!}$ should be defined as
$Rp_{*}\circ Rj_{!}$, where $Rp_{*}$ is right adjoint to $p^{*}$ and $Rj_{!}$
is left adjoint to $j^{*}$. [9, Proposition 1.4.4] is the abstract input
required to pass from an EOM on (qc separated schemes, proper morphisms, local
isomorphisms, all, $\left\\{1,2,3\right\\}$) to an EOM on (qc separated
schemes, proper morphisms, local isomorphisms, all, $\left\\{3\right\\}$),
i.e., the arrows have been reversed for the first two classes of morphisms.

  4. (4)

The same gluing technique as in (2) applied in reverse allows us to transfer
the EOM to (qc separated schemes, locally finite type morphisms, all,
$\left\\{2\right\\}$). By now, we have an $\infty$-categorical enhancement of
$Rf_{!}$ for $f$ a morphism of coproducts of qcqs schemes which is locally of
finite type.

  5. (5)

The “DESCENT” program developed in [8] is a means of extending an EOM from one
marked category $\mathcal{C}$ to a marked overcategory $\tilde{\mathcal{C}}$.
The input requires that for every object $X\in\tilde{\mathcal{C}}$, there
exists a marked morphism $Y\to X$ with $Y$ in $\mathcal{C}$. It is also
required that the marked morphisms be of “universal descent” with respect to
the EOM.

Repeated calls to DESCENT allow us to extend our EOM to the following domains:

    1. (a)

From qc separated schemes to qs schemes,

    2. (b)

From qs schemes to algebraic spaces,

    3. (c)

From algebraic spaces to Artin stacks.

The final output is the desired EOM on (Artin stacks, locally finite type
morphisms, all morphisms, $\left\\{2\right\\}$).

Our main result follows a similar strategy. We highlight the major
differences:

  1. (1)

In the first stage we have an EOM for (small coproducts of qcqs v-sheaves, all
morphisms, $\left\\{1\right\\}$).

  2. (2)

In the second stage, we start with the observation that if $f\colon X\to Y$ is
a morphism of qcqs v-sheaves which is representable in locally spatial
diamonds and compactifiable of locally finite dim.trg, then there is a
factorization $f=p\circ j$, where $j$ is an open immersion and $p$ is proper.
In fact there is a canonical compactification $j\colon X\to\overline{X}^{/Y}$
[11, Proposition 18.6]. We encounter a substantial problem here: we have
little control over $\overline{X}^{/Y}$, and in fact $p$ is not necessarily
representable in locally spatial diamonds. We resolve this issue by
introducing a larger auxiliary class of prespatial diamonds, which is
preserved under passing to the canonical compactification rather by design.
The result is an EOM on (small coproducts of qcqs v-sheaves, proper morphisms
representable in prespatial diamonds, separated local isomorphisms,
$\left\\{1,2\right\\}$). Let us emphasize that in this step, we make heavy use
of the main cohomological results in [11].

  3. (3)-(4)

These steps are similar to those in the scheme setting. The result is an EOM
on (small coproducts of qcqs v-sheaves, morphisms representable in locally
spatial diamonds and compactifiable of locally finite dim.trg,
$\left\\{2\right\\}$).

  4. (5)

The DESCENT program is applied repeatedly to extend the EOM to the following
domains:

    1. (a)

From small coproducts of separated spatial diamonds to quasiseparated locally
separated locally spatial diamonds,

    2. (b)

From quasiseparated locally separated locally spatial diamonds to locally
separated locally spatial diamonds.

    3. (c)

From locally separated locally spatial diamonds to decent v-stacks (with fine
morphisms).

The final output is the EOM which encodes the operation $Rf_{!}$ for fine
morphisms between v-stacks; this is what is necessary to prove Theorem 1.4.

###  1.3 Comments and conventions

Most readers of this article should simply take Theorem 1.4 as a black box.
However, for the scrupulous reader, we recommend having [8] and [9] close at
hand. Not only will we heavily use the machinery introduced there, but we will
borrow much notation from these papers, sometimes without comment.

We need to heavily use $\infty$-categorical techniques. As in [8, 9], we use
Lurie’s model: an $\infty$-category is a simplicial set satisfying the weak
Kan condition. We often conflate ordinary categories with $\infty$-categories
by identifying a category $\mathcal{C}$ with its nerve
$\mathrm{N}(\mathcal{C})$. Likewise, we often conflate (2,1)-categories with
$\infty$-categories by identifying a $(2,1)$-category $\mathcal{C}$ with its
Duskin nerve, which we again denote $\mathrm{N}(\mathcal{C})$ (see [7, Tag
009P] for some discussion of this notion). We will usually omit the nerve
functor from our notation (in a departure from the convention in [8]).

###  1.4 Acknowledgments

DH would like to thank Johan de Jong, Yifeng Liu, Jacob Lurie, Lucas Mann,
Peter Scholze, and Weizhe Zheng for some extremely helpful conversations
related to this material. DH is also very grateful to his wife for her
exceptional patience during several unusually intense periods of work on this
project.

##  2 Statement of main result

###  2.1 The notion of an enhanced operation map

A marked $\infty$-category is a pair $(\mathcal{C},\mathcal{F})$, where
$\mathcal{C}$ is a geometric $\infty$-category [8, Definition 4.1.3], and
$\mathcal{F}$ is a set of morphisms of $\mathcal{C}$ stable under composition,
arbitrary pullback, and small coproducts. The reader should imagine that
$\mathcal{C}$ is (the nerve of) some ordinary category of geometric
significance: the category of small coproducts of quasicompact separated
schemes, the category of locally spatial diamonds, etc. Consider the following
scenario, which is typical of what one sees in a six-functor formalism.

######  Scenario 2.1.

Let $(\mathcal{C},\mathcal{F})$ be a marked $\infty$-category.

  1. (1)

For all objects $X\in\mathcal{C}$, we have an associated closed symmetric
monoidal stable $\infty$-category $\mathcal{D}(X)$, which the reader should
imagine as the derived ($\infty$-)category of sheaves on some ringed site
associated with $X$. Moreover, we have an internal hom bifunctor
$R\mathscr{H}\mathrm{om}(-,-)$ such that $R\mathscr{H}\mathrm{om}(B,-)$ is
right-adjoint to $-\otimes B$ for all $B\in\mathcal{D}(X)$.

  2. (2)

For any morphism $f:X\to Y$ we have a symmetric monoidal pullback functor
$f^{\ast}:\mathcal{D}(Y)\to\mathcal{D}(X)$.

  3. (3)

For any morphism $p:X\to Y$ with $p\in\mathcal{F}$ we have an exceptional
pushforward functor $p_{!}:\mathcal{D}(X)\to\mathcal{D}(Y)$.

  4. (4)

The functors $f^{\ast}$ and $p_{!}$ commute with all direct sums, and
therefore admit right adjoints $f_{\ast}$ resp. $p^{!}$.

  5. (5)

For any cartesian square

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{q}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$
(2.1)

with $p\in\mathcal{F}$, there is a proper base change isomorphism
$f^{\ast}p_{!}\simeq q_{!}g^{\ast}$.

  6. (6)

There is a projection formula isomorphism $p_{!}A\otimes B\simeq
p_{!}(A\otimes p^{\ast}B)$ for $A\in\mathcal{D}(X)$, $B\in\mathcal{D}(Y)$.

We now consider the following problem:

######  Problem 2.2.

Give a sensible way of cleanly encoding all of the structures in Scenario
2.1,_together_ with all of their expected higher coherences with respect to
composition.

In this section we spell out one solution to this problem, closely following
the ideas of [8], namely the notion of an _enhanced operation map._ Before
making a formal definition, we recall two $\infty$-categorical constructions
from [6] and [9]. The first involves symmetric monoidal structures on general
$\infty$-categories, and the second has to do with the way we will encode base
change isomorphisms on the $\infty$-categorical level.

######  Construction 2.3 (Symmetric monoidal $\infty$-categories and
commutative algebra objects).

We review the definition of symmetric monoidal $\infty$-category [6,
Definition 2.0.0.7]. Let $\mathcal{F}\mathrm{in}_{\ast}$ be the category of
pointed finite sets, with objects
$\left<n\right>=\left\\{\ast,1,\dots,n\right\\}$, and where the morphisms
$\left<m\right>\to\left<n\right>$ are those functions which preserve $\ast$.
(Equivalently, $\mathcal{F}\mathrm{in}_{\ast}$ is the category of finite sets
where the morphisms are partially defined functions.)

A symmetric monoidal $\infty$-category is a coCartesian fibration of
simplicial sets
$p\colon\mathcal{C}^{\otimes}\to\operatorname{N}(\mathcal{F}\mathrm{in}_{\ast})$,
satisfying a certain condition ($\ast$). For $i=1,\dots,n$, let
$\rho^{i}\colon\left<n\right>\to\left<1\right>$ be the unique morphism with
$(\rho^{i})^{-1}(1)=\left\\{i\right\\}$. Since $p$ is a coCartesian fibration,
$\rho^{i}$ induces a functor
$\rho^{i}_{!}\colon\mathcal{C}^{\otimes}_{\left<n\right>}\to\mathcal{C}^{\otimes}_{\left<1\right>}$.
The condition ($\ast)$ is that the product of the $\rho^{i}_{!}$ is an
equivalence
$\mathcal{C}^{\otimes}_{\left<n\right>}\to(\mathcal{C}^{\otimes}_{\left<1\right>})^{n}$
for all $n$. We write $\mathcal{C}=\mathcal{C}^{\otimes}_{\left<1\right>}$ and
call it the underlying $\infty$-category of $\mathcal{C}^{\otimes}$. We may
sometimes start with an $\infty$-category $\mathcal{C}$, and say that
$\mathcal{C}^{\otimes}$ constitutes a symmetric monoidal structure on
$\mathcal{C}$, the map
$p\colon\mathcal{C}^{\otimes}\to\operatorname{N}(\mathcal{F}\mathrm{in}_{\ast})$
being understood.

In this situation we have a functor
$\underline{\;\;}\otimes\underline{\;\;}\colon\mathcal{C}\times\mathcal{C}\cong\mathcal{C}^{\otimes}_{\left<2\right>}\to\mathcal{C}$,
where the last map is induced from the unique active map
$\alpha\colon\left<2\right>\to\left<1\right>$; this is the composition law for
$\mathcal{C}$. The rest of the structure of $\mathcal{C}^{\otimes}$ furnishes
isomorphisms witnessing the fact that $\otimes$ is unital, commutative, and
associative, and it also encodes all higher coherences among those
isomorphisms.

A commutative algebra object of a symmetric monoidal $\infty$-category
$\mathcal{C}^{\otimes}$ is a section
$s\colon\operatorname{N}(\mathcal{F}\mathrm{in}_{\ast})\to\mathcal{C}^{\otimes}$
to the functor $p$. Let $X=s(\left<1\right>)$; then the unique active map
$\alpha\colon\left<2\right>\to\left<1\right>$ induces a composition law
$m\colon X\otimes X\to X$. Once again, the rest of the structure of $s$
furnishes isomorphisms encoding the fact that $m$ is unital, commutative, and
associative. The commutative algebra objects of $\mathcal{C}^{\otimes}$ form
an $\infty$-category $\operatorname{CAlg}(\mathcal{C})$ [6, Definition
2.1.3.1].

If $\mathcal{C}$ is any $\infty$-category admitting finite products (resp.,
coproducts), then $\mathcal{C}$ admits a canonical symmetric monoidal
structure $\mathcal{C}^{\times}$ (resp., $\mathcal{C}^{\amalg}$) for which the
composition law is the product (resp., the coproduct), see [6, Construction
2.4.1.4] (resp. [6, Construction 2.4.3.1]). For instance, the objects of
$\mathcal{C}^{\amalg}_{\left<n\right>}$ are $n$-tuples $(X_{1},\dots,X_{n})$
of objects of $\mathcal{C}$, and a 1-morphism
$(X_{1},\dots,X_{m})\to(Y_{1},\dots,Y_{n})$ in $\mathcal{C}^{\amalg}$ consists
of a morphism $\alpha\colon\left<m\right>\to\left<n\right>$ together with a
1-morphism $X_{i}\to Y_{\alpha(i)}$ for each $1\leq i\leq m$ with
$\alpha(i)\neq\ast$. In particular this forces $X_{1}\otimes X_{2}\cong
X_{1}\coprod X_{2}$.

If $\mathcal{C}$ is an $\infty$-category admitting finite products, we write
$\operatorname{CAlg}(\mathcal{C})=\operatorname{CAlg}(\mathcal{C}^{\times})$.
In particular $\operatorname{CAlg}(\mathcal{C}\mathrm{at}_{\infty})$ is the
$\infty$-category of symmetric monoidal $\infty$-categories.

There is a canonical functor
$\mathcal{C}\to\operatorname{CAlg}(\mathcal{C}^{\amalg})$ which assigns to an
object $A$ the commutative algebra structure for which the composition law is
the obvious map $A\coprod A\to A$. In fact $\mathcal{C}^{\amalg}$ is the
universal symmetric monoidal $\infty$-category $\mathcal{D}$ admitting a
functor $\mathcal{C}\to\operatorname{CAlg}(\mathcal{D})$, see [6, Theorem
2.4.3.18]. That is, for any such $\mathcal{D}$, there is an equivalence betwen
monoidal functors $T\colon\mathcal{C}^{\coprod}\to\mathcal{D}$ and functors
$T^{\otimes}\colon\mathcal{C}\to\operatorname{CAlg}(\mathcal{D})$.

In our desired application, $\mathcal{C}$ is (the nerve of) some category of
geometric objects, and we wish to assign to each object $X$ of $\mathcal{C}$ a
symmetric monoidal category $\infty$-category $\mathcal{D}(X)$ in a coherent
manner, such that morphisms $X\to Y$ induce functors
$\mathcal{D}(Y)\to\mathcal{D}(X)$. This may be accomplished by constructing a
functor
$\mathcal{C}^{\operatorname{op}}\to\operatorname{CAlg}(\mathcal{C}\mathrm{at}_{\infty})$,
which is equivalent to constructing a monoidal functor
$\mathcal{C}^{\operatorname{op},\amalg}\to\mathcal{C}\mathrm{at}_{\infty}$.

######  Construction 2.4.

Let $\mathcal{C}$ be a category, let $k\geq 2$ be an integer, let
$I\subset\left\\{1,\dots,k\right\\}$ be a subset, and let
$\mathcal{E}_{1},\dots,\mathcal{E}_{k}$ be sets of morphisms of $\mathcal{C}$,
each containing every identity morphism in $\mathcal{C}$. The restricted
multisimplicial nerve
$\delta_{k,I}^{\ast}\operatorname{N}(\mathcal{C})^{\operatorname{cart}}_{\mathcal{E}_{1},\dots,\mathcal{E}_{k}}$
is a simplicial set constructed in [8]. For the moment we will only need it in
the case $k=2$, $I=\left\\{2\right\\}$. The 0-simplices of
$\delta_{2,\left\\{2\right\\}}^{\ast}\operatorname{N}(\mathcal{C})_{\mathcal{E}_{1},\mathcal{E}_{2}}$
are simply the objects of $\mathcal{C}$. The 1-simplices are cartesian squares

$\textstyle{c_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{11}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{10}}$
(2.2)

where the vertical (resp., horizontal) arrows lie in $\mathcal{E}_{1}$ (resp.,
$\mathcal{E}_{2}$); the vertices of this edge are $c_{00}$ and $c_{11}$.333The
subscript $\left\\{2\right\\}$ in $\delta_{2,\left\\{2\right\\}}^{\ast}$
controls the shape of the diagram in (2.2), which resembles the diagram in
(2.1). If instead $I=\emptyset$ the columns of (2.2) would be transposed. The
2-simplices of
$\delta_{2,\left\\{2\right\\}}^{\ast}\operatorname{N}(\mathcal{C})_{\mathcal{E}_{1},\mathcal{E}_{2}}$
are diagrams

$\textstyle{c_{02}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{12}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{11}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{10}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{22}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{21}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{20}}$
(2.3)

where each square is cartesian, and where once again the vertical (resp.,
horizontal) arrows lie in $\mathcal{E}_{1}$ (resp., $\mathcal{E}_{2}$). The
three edges of this 2-simplex are given by the upper-right square, the lower-
left square, and the outer square in (2.3), respectively. Given a 1-simplex
corresponding to the diagram in (2.2), its degenerate 2-simplices are obtained
by placing it in the upper-right or lower-left of a diagram like in (2.3), and
“filling in” the rest of the diagram using identity morphisms. Higher
simplices, face maps and degeneracy maps are definted similarly.

We can now describe the machinery of enhanced operation maps built in [9].
Suppose given a marked $\infty$-category $(\mathcal{C},\mathcal{F})$ as in the
beginning of 2.1. We then have an $\infty$-category
$\mathcal{C}_{\amalg}=(\mathcal{C}^{\mathrm{op},\amalg})^{\mathrm{op}}$
equipped with a map of simplicial sets
$\mathcal{C}_{\amalg}\to\mathrm{N}(\mathcal{F}\mathrm{in}_{\ast})^{\mathrm{op}}$
[6, p. 297]. An object of $\mathcal{C}_{\amalg}$ is a pair $\left\langle
n\right\rangle\in\mathcal{F}\mathrm{in}_{\ast}$ together with a sequence
$(X_{1},\dots,X_{n})$ of objects in $\mathcal{C}$. A morphism $f$ in
$\mathcal{C}_{\amalg}$ from $(X_{1},\dots,X_{n})$ to $(Y_{1},\dots,Y_{m})$
consists of a map of pointed sets $\alpha:\left\langle
m\right\rangle\to\left\langle n\right\rangle$ together with a sequence of
morphisms $\left\\{X_{\alpha(i)}\to
Y_{i}\right\\}_{i\in\alpha^{-1}\left\langle n\right\rangle^{\circ}}$ in
$\mathcal{C}$. Note that $\mathcal{F}$ induces a marking on
$\mathcal{C}_{\amalg}$ by taking those morphisms for which
$\alpha:\left\langle m\right\rangle\to\left\langle m\right\rangle$ is the
identity map and all the associated morphisms $\\{X_{i}\to Y_{i}\\}_{1\leq
i\leq m}$ lie in $\mathcal{F}$. In the terminology of [8], these are the edges
of $\mathcal{C}_{\amalg}$ which _statically belong to $\mathcal{F}$_. By abuse
of notation, we will also denote this marking by $\mathcal{F}$. Note that the
fiber of $\mathcal{C}_{\amalg}$ over $\left\langle 1\right\rangle$ is just
$\mathcal{C}$, on which the marking of $\mathcal{C}_{\amalg}$ restricts to the
original marking $\mathcal{F}$, so this abuse should cause no confusion.

Now, as in [9] we can form the simplicial set
$\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}$.
A 0-simplex is just an object $(X_{1},\dots,X_{n})$ of $\mathcal{C}_{\amalg}$.
_If_ $\mathcal{C}$ admits finite products, then a 1-simplex consists of a map
of pointed sets $\alpha:\left\langle m\right\rangle\to\left\langle
n\right\rangle$ together with a diagram

$\textstyle{(X_{1}^{\prime},\dots,X_{n}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y_{1}^{\prime},\dots,Y_{m}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X_{1},\dots,X_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y_{1},\dots,Y_{m})}$

where the vertical edges statically belong to $\mathcal{F}$, the horizontal
edges are morphisms in $\mathcal{C}_{\amalg}$ lying over $\alpha$, and for all
$j\in\left\langle n\right\rangle^{\circ}$ the induced diagrams

$\textstyle{X_{j}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i\in\alpha^{-1}(j)}Y_{i}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{i\in\alpha^{-1}(j)}Y_{i}}$

are cartesian.

Given this setup, an enhanced operation map is a functor

${}_{\mathcal{C}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

satisfying various properties. To state these properties, we introduce some
notation attached to such a functor.

  * •

Any object $X\in\mathcal{C}$ defines a 0-simplex of
$\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}$
lying over $\left\langle 1\right\rangle\in\mathcal{F}\mathrm{in}_{\ast}$, and
we set $\mathcal{D}(X):=\mathrm{{}_{\mathcal{C}}EO}(X)$.

  * •

Restriction to the “all” direction defines a functor

${}_{\mathcal{C}}\mathrm{EO}^{\mathrm{I}}:\mathcal{C}^{\mathrm{op},\amalg}\to\mathcal{C}\mathrm{at}_{\infty},$

and further restriction to the fiber over $\left\langle 1\right\rangle$
defines a functor

${}_{\mathcal{C}}\mathrm{EO}^{\ast}:\mathcal{C}^{\mathrm{op}}\to\mathcal{C}\mathrm{at}_{\infty}.$

Given any morphism $f:X\to Y$ in $\mathcal{C}$, we write
$f^{\ast}:\mathcal{D}(Y)\to\mathcal{D}(X)$ for the functor given by the image
of $f$ under ${}_{\mathcal{C}}\mathrm{EO}^{\ast}$.

  * •

Restriction of ${}_{\mathcal{C}}\mathrm{EO}$ to the fiber over $\left\langle
1\right\rangle$ defines a functor

${}_{\mathcal{C}}\mathrm{EO}_{!}^{\ast}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}{}_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}})\to\mathcal{C}\mathrm{at}_{\infty},$

and further restriction to the $\mathcal{F}$ direction defines a functor

${}_{\mathcal{C}}\mathrm{EO}_{!}:\mathcal{C}_{\mathcal{F}}\to\mathcal{C}\mathrm{at}_{\infty}.$

Given any morphism $p:X\to Y$ in $\mathcal{C}$ with $p\in\mathcal{F}$, we
write $p_{!}:\mathcal{D}(X)\to\mathcal{D}(Y)$ for the functor given by the
image of $p$ under ${}_{\mathcal{C}}\mathrm{EO}_{!}$.

######  Definition 2.5.

Notation as above, the functor

${}_{\mathcal{C}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

is an _enhanced operation map_ if the following two conditions are satisfied.

  1. 1.

The functor ${}_{\mathcal{C}}\mathrm{EO}^{\mathrm{I}}$ is a weak Cartesian
structure [6, Definition 2.4.1.1], and the induced functor (see Construction
2.3)444See [8, Remark 1.5.6] for this notation.

$\left({}_{\mathcal{C}}\mathrm{EO}^{\mathrm{I}}\right)^{\otimes}:\mathcal{C}^{\mathrm{op}}\to\mathrm{CAlg}(\mathcal{C}\mathrm{at}_{\infty})$

factors through
$\mathrm{CAlg}(\mathcal{C}\mathrm{at}_{\infty})_{\mathrm{pr,st,cl}}^{\mathrm{L}}$
and sends small products in $\mathcal{C}^{\mathrm{op}}$ (i.e. small coproducts
in $\mathcal{C}$) to products. Here
$\operatorname{CAlg}(\mathcal{C}\mathrm{at}_{\infty})^{L}_{\mathrm{pr,st,cl}}$
is the category appearing in [8, Definition 1.5.2]. It is the subcategory of
$\operatorname{CAlg}(\mathcal{C}\mathrm{at}_{\infty})$ whose objects are
symmetric monoidal $\infty$-categories which are:

     * •

presentable [5, Definition 5.5.0.1],

     * •

stable [6, Definition 1.1.1.9], and

     * •

closed [6, Definition 4.1.1.17],

and for which the morphisms $\mathcal{C}^{\otimes}\to\mathcal{D}^{\otimes}$
are left adjoints.

  2. 2.

The functor
${}_{\mathcal{C}}\mathrm{EO}_{!}^{\ast}\colon\delta_{2,\left\\{2\right\\}}^{\ast}(\mathcal{C}_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}})\to\mathcal{C}\mathrm{at}_{\infty}$
factors through $\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}}$. Here,
$\mathcal{P}\mathrm{r}_{\operatorname{st}}^{\operatorname{L}}\subset\mathcal{C}\mathrm{at}_{\infty}$
is the subcategory whose objects are presentable stable $\infty$-categories,
and whose morphisms are left adjoint functors.

An enhanced operation map is an extremely dense piece of structure, and the
full content of this structure is probably opaque at first glance. However, we
claim this notion gives a reasonable solution to Problem 2.1. This claim is
justified as follows:

  * •

Condition (1) in Definition 2.5 has the following consequences. For all
$X\in\mathcal{C}$ we have a presentable stable closed symmetric monoidal
$\infty$-category
$\mathcal{D}(X):=(_{\mathcal{C}}\operatorname{EO}^{I})^{\otimes}(X)$. Let us
note here that
${}_{\mathcal{C}}\operatorname{EO}(X,X)=\mathcal{D}(X)\times\mathcal{D}(X)$,
and that evaluating ${}_{\mathcal{C}}\operatorname{EO}$ on the 1-simplex

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,X)}$

defines a functor $\mathcal{D}(X)\times\mathcal{D}(X)\to\mathcal{D}(X)$, which
is none other than the symmetric monoidal structure on $\mathcal{D}(X)$. Since
the symmetric monoidal structure on $\mathcal{D}(X)$ is closed, the functor
$A\mapsto A\otimes B$ commutes with all colimits and therefore admits a right
adjoint, giving the desired internal hom. This verifies item (1) of Scenario
2.1. The construction of $f^{*}$ and $p_{!}$ we have given above verifies
items (2) and (3).

  * •

Condition (2) in Definition 2.5 implies that $f^{\ast}$ (for arbitrary
morphisms $f$) and $p_{!}$ (for morphisms $p\in\mathcal{F}$) are morphisms in
$\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}}$, and therefore admit right
adjoints. This verifies item (4) of Scenario 2.1.

  * •

Applying ${}_{\mathcal{C}}\mathrm{EO}_{!}^{\ast}$ to a 1-simplex

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{q}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

of
$\delta_{2,\\{2\\}}^{\ast}\mathcal{C}{}_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}$
defines a functor $F\colon\mathcal{D}(X)\to\mathcal{D}(Y^{\prime})$. Now the
2-simplices

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$
and
$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

witness isomorphisms $F\cong f^{*}p_{!}$ and $F\cong q_{!}g^{*}$,
respectively. In particular, we get a proper base change equivalence
$f^{\ast}p_{!}\simeq q_{!}g^{\ast}$, verifying item (5) of Scenario 2.1.

  * •

Suppose $\mathcal{C}$ admits finite products. Then for any morphism $p\colon
X\to Y$ in $\mathcal{F}$, there is a 1-simplex

$\textstyle{(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(p,\mathrm{id})}$$\textstyle{(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y,Y)}$

of
$\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}$
lying over the unique active map $\alpha:\left\langle
2\right\rangle\to\left\langle 1\right\rangle$, and applying
${}_{\mathcal{C}}\mathrm{EO}$ to this 1-simplex defines a functor
$G\colon\mathcal{D}(X)\times\mathcal{D}(Y)\to\mathcal{D}(Y)$. The 2-simplices

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y,Y)}$
and
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y,Y)}$

witness isomorphisms $G(A,B)\cong p_{!}A\otimes B$ and $G(A,B)\cong
p_{!}(A\otimes p^{*}B)$, respectively. In particular, we get the projection
formula, verifying item (6) of Scenario 2.1.

###  2.2 The main theorem

Using the language and notation of the previous section, we can now state the
main technical theorem proved in this paper.

######  Theorem 2.6.

Consider the marked $\infty$-category
$(\mathcal{V}\mathrm{stk}^{\mathrm{dct}},\mathcal{F})$ where
$\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$ denotes the category of decent
v-stacks, and $\mathcal{F}$ is the class of fine morphisms. Fix a ring
$\Lambda$ killed by some integer prime to $p$. Then there is an enhanced
operation map

${}_{\mathcal{V}\mathrm{stk}^{\mathrm{dct}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{stk}_{\amalg}^{\mathrm{dct}})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

such that:

  1. 1.

$\mathcal{D}(X)=\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$ with its natural
symmetric monoidal structure for all
$X\in\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$.

  2. 2.

$f^{\ast}$ coincides with the pullback functor on
$\mathcal{D}_{\mathrm{\acute{e}t}}$ constructed in [11] for all morphisms $f$.

  3. 3.

For morphisms $f\in\mathcal{F}$ which are separated and representable in
locally spatial diamonds, $f_{!}$ coincides with the functor $Rf_{!}$
constructed in [11].

Theorem 2.6 implies Theorem 1.4. The functors $Rf_{!}$ required by Theorem 1.4
are obtained from the functors $f_{!}$ coming from the enhanced operation
functor by passing from $\mathcal{D}_{\text{\rm\'{e}t}}(X,\Lambda)$ to its
homotopy category $D_{\text{\rm\'{e}t}}(X,\Lambda)$. In light of the
discussion in the previous section, $Rf_{!}$ satisfies the projection formula
and proper base change. Since ${}_{\mathcal{C}}\operatorname{EO}^{*}_{!}$
takes values in $\Pr_{\operatorname{st}}^{\operatorname{L}}$, the functor
$Rf_{!}$ is a left adjoint.

##  3 Enhanced operations for qcqs v-sheaves

###  3.1 Prespatial diamonds

In the theory of [11], (locally) spatial diamonds play a central role. There
are several justifications for this centrality: they capture most diamonds of
practical interest, they have excellent categorical properties, and their
étale cohomology admits some a priori control in terms of simple “dimensional”
invariants. However, from the perspective of this paper, they suffer one
serious defect: spatial diamonds are not known to be stable under the
formation of _canonical compactifications_. In particular, if $f:X\to Y$ is a
map which is compactifiable and representable in spatial diamonds, it is not
known whether $f$ admits a factorization
$X\overset{j}{\to}\overline{X}\overset{p}{\to}Y$ where $j$ is an open
immersion and $p$ is proper and representable in spatial diamonds. This lacuna
in our knowledge is a serious obstacle to implementing the categorical gluing
arguments from [9].

In this section, we overcome this difficulty by slightly enlarging the
category of (locally) spatial diamonds. More precisely, we introduce a notion
of (locally) _prespatial_ diamonds. On one hand, we will see that these
gadgets enjoy all the same good properties as (locally) spatial diamonds,
including the same crucial a priori control on étale cohomology. On the other
hand, they are stable under passing to canonical compactifications,
essentially by design.555Finding the “right” generalization of locally spatial
diamonds with all of these properties involved several years of trial and
error, and turned out to be the main bottleneck in the completion of this
project. In the end, we arrived at the definition presented here through
extended meditation on the proof of [11, Theorem 22.5]. We will revisit that
proof in the argument for Proposition 3.7 below.

We now turn to the key definitions.

######  Definition 3.1.

A qcqs diamond $X$ is _prespatial_ if there exists a spatial subdiamond
$X_{0}\subset X$ such that $X_{0}(K,\mathcal{O}_{K})=X(K,\mathcal{O}_{K})$ for
all perfectoid fields $K$.

A diamond $X$ is _locally prespatial_ if there exists a locally spatial
subdiamond $X_{0}\subset X$ such that the inclusion map $X_{0}\to X$ is
quasicompact and $X_{0}(K,\mathcal{O}_{K})=X(K,\mathcal{O}_{K})$ for all
perfectoid fields $K$.

A pair $(X_{0},X)$ satisfying the above conditions will be called a _(locally)
prespatial pair_.

Note that in the prespatial case, the inclusion $X_{0}\to X$ is automatically
qcqs. However, we do not impose any further conditions on the map $X_{0}\to
X$. Note also that $X_{0}$ is far from unique in general: if
$X_{0}^{\prime}\subset X$ and $X_{0}^{\prime\prime}\subset X$ both verify the
definition of a (locally) prespatial diamond, then so does
$X_{0}^{\prime}\times_{X}X_{0}^{\prime\prime}$. More generally, the following
lemma holds.

######  Lemma 3.2.

Let $(X_{0},X)$ be a (locally) prespatial pair, let $(Y_{0},Y)$ be a locally
prespatial pair, and let $f\colon X\to Y$ be a morphism of diamonds. Then
$(X_{0}\times_{Y}Y_{0},X)$ is also a (locally) prespatial pair.

###### Proof.

Since $Y_{0}\to Y$ is a qc injection, its base change $X_{0}\times_{Y}Y_{0}\to
X_{0}$ is also a qc injection, so $X_{0}\times_{Y}Y_{0}$ is locally spatial by
[11, Proposition 11.20]. Moreover, $X_{0}\times_{Y}Y_{0}\to X$ is qc since it
factors as $X_{0}\times_{Y}Y_{0}\to X_{0}\to X$ where both maps are qc. If
moreover $X_{0}$ is spatial, then $X_{0}\times_{Y}Y_{0}$ is qcqs, hence
spatial. It is clear that for any perfectoid field $K$,
$(X_{0}\times_{Y}Y_{0})(K,\mathcal{O}_{K})=X_{0}(K,\mathcal{O}_{K})=X(K,\mathcal{O}_{K})$.
This gives the result. ∎

######  Definition 3.3.

Let $f\colon X\to Y$ be a map of v-stacks. Say $f$ is _representable in
(locally) prespatial diamonds_ if $f$ is 0-truncated and for all maps $W\to Y$
with $W$ a (locally) prespatial diamond, $X\times_{Y}W$ is a (locally)
prespatial diamond.

######  Lemma 3.4 (Sanity Checks).

  1. (i)

A diamond $X$ is prespatial iff it is locally prespatial and qcqs.

  2. (ii)

Any open subdiamond of a locally prespatial diamond is locally prespatial.

  3. (iii)

Any (locally) spatial diamond is (locally) prespatial.

  4. (iv)

If $f$ is representable in (locally) spatial diamonds, it is representable in
(locally) prespatial diamonds.

  5. (v)

Any fiber product of (locally) prespatial diamonds is a (locally) prespatial
diamond.

  6. (vi)

Morphisms which are representable in (locally) prespatial diamonds are stable
under composition and base change.

###### Proof.

  1. (i)

Suppose that $(X_{0},X)$ is a prespatial pair. Then $X_{0}\to X$ is qc since
$X_{0}$ is qc and $X$ is qs. So $X$ is also locally prespatial.

Conversely, suppose that $(X_{0},X)$ is a locally prespatial pair and $X$ is
qcqs. Then $X_{0}$ is qcqs since $X$ is qcqs and $X_{0}\hookrightarrow X$ is a
qc injection. So $X_{0}$ is spatial.

  2. (ii)

If $(X_{0},X)$ is a locally prespatial pair and $U$ is an open subdiamond of
$X$, then $U\times_{X}X_{0}$ is locally spatial by [11, Proposition 11.19(ii)]
(or Lemma 3.2). Since base change preserves the property of being a qc
injection, $(U\times_{X}X_{0},U)$ is a locally prespatial pair.

  3. (iii)

If $X$ is (locally) spatial, then $(X,X)$ is a (locally) prespatial pair.

  4. (iv)

Let $f\colon X\to Y$ be a morphism that is representable in locally spatial
diamonds, let $(Z_{0},Z)$ be a locally prespatial pair, and suppose that we
are given a map $Z\to Y$. Then $(Z_{0}\times_{Y}X,Z\times_{Y}X)$ is a
prespatial pair. Suppose $f$ is representable in spatial diamonds and
$(Z_{0},Z)$ is a prespatial pair. Then $f$, $Z_{0}$, and $Z$ are qcqs, so
$Z_{0}\times_{Y}X$ is qcqs, hence spatial, and $Z\times_{Y}X$ is also qcqs.

  5. (v)

Suppose we have (locally) prespatial pairs $(X_{0},X)$, $(Y_{0},Y)$,
$(Z_{0},Z)$, and a fiber product $X\times_{Z}Y$. Then $X_{0}\times_{Z}Z_{0}$
and $Y_{0}\times_{Z}Z_{0}$ are (locally) spatial by Lemma 3.2. Hence
$(X_{0}\times_{Z}Z_{0})\times_{Z_{0}}(Y_{0}\times_{Z}Z_{0})=X_{0}\times_{Z}Y_{0}\times_{Z}Z_{0}$
is (locally) spatial, and $(X_{0}\times_{Z}Y_{0}\times_{Z}Z_{0},X\times_{Z}Y)$
is a (locally) spatial pair.

  6. (vi)

This is clear from the definition.

∎

######  Warning 3.5.

If $X$ is a diamond, it is not clear whether the property of being locally
prespatial can be checked locally on an open cover of $X$. In light of this,
the name is perhaps slightly misleading.

######  Proposition 3.6.

If $f:X\to Y$ is separated and representable in prespatial diamonds, then
$\overline{f}^{/Y}:\overline{X}^{/Y}\to Y$ is proper and representable in
prespatial diamonds.

###### Proof.

We can assume that $X$ and $Y$ are prespatial diamonds. Then
$\overline{X}^{/Y}\to Y$ is proper by [11, Prop. 18.7.(vii)]. In particular,
$\overline{X}^{/Y}$ admits a separated map to a quasiseparated target and thus
is quasiseparated. Quasicompacity of $\overline{X}^{/Y}$ is clear. Finally,
taking any $X_{0}\subset X$ as in the definition of a prespatial diamond, we
have
$\overline{X}^{/Y}(K,\mathcal{O}_{K})=X(K,\mathcal{O}_{K})=X_{0}(K,\mathcal{O}_{K})$
by the definition of $\overline{X}^{/Y}$, so $\overline{X}^{/Y}$ is
prespatial. ∎

######  Proposition 3.7.

If $f:X\to Y$ is proper and representable in prespatial diamonds, then
$Rf_{\ast}:D_{\mathrm{\acute{e}t}}(X,\Lambda)\to
D_{\mathrm{\acute{e}t}}(Y,\Lambda)$ has cohomological amplitude $\leq
3\mathrm{dim.trg}f$.

###### Proof.

We follow the proof of [11, Theorem 22.5] closely. Arguing as in that proof,
we can assume that $Y=\mathrm{Spd}(C,C^{+})$, $X$ is a separated prespatial
diamond proper over $Y$, and $A\in D_{\mathrm{\acute{e}t}}(X,\Lambda)$ is
concentrated in degree zero with $j^{\ast}A=0$. Here $j:f^{-1}(U)\to X$ is the
natural open immersion, where $U\subset Y$ is the complement of the unique
closed point.

Let $|X|^{h}$ be the maximal Hausdorff quotient of $|X|$, or equivalently the
maximal Hausdorff quotient of
$|X\times_{\mathrm{Spd}(C,C^{+})}\mathrm{Spd}(C,\mathcal{O}_{C})|$. As in
_loc. cit.,_ we factor $f$ as $g:X\to\underline{|X|^{h}}\times Y$ followed by
$h:\underline{|X|^{h}}\times Y\to Y$. As in _loc. cit.,_ $g$ is proper and
representable in spatial diamonds, and thus $Rg_{\ast}$ has cohomological
amplitude $\leq 2\mathrm{dim.trg}f$.

It remains to check that if $B\in
D_{\mathrm{\acute{e}t}}(\underline{|X|^{h}}\times Y,\Lambda)$ is concentrated
in degree zero and trivial on $\underline{|X|^{h}}\times U$, then
$R^{i}h_{\ast}B=0$ for all $i>\mathrm{dim.trg}f$. As in _loc. cit.,_ we can
identify

$R^{i}h_{\ast}B\cong H^{i}(\underline{|X|^{h}}\times Y,B)\cong
H^{i}(|X|^{h},B|_{|X|^{h}\times{s}})$

(where we implicitly appeal to [11, Proposition 22.7]). Thus it remains to
control the cohomology of abelian sheaves on $|X|^{h}$. The key observation
now is that for any choice of $X_{0}\subset X$ as in the definition of a
prespatial diamond, we get an induced homeomorphism

$|X_{0}\times_{\mathrm{Spd}(C,C^{+})}\mathrm{Spd}(C,\mathcal{O}_{C})|^{h}\cong|X|^{h}.$

This follows from the general observation that if $i:U\to V$ is any injection
of qcqs diamonds which induces a bijection on $(K,\mathcal{O}_{K})$-points for
all perfectoid fields $K$, then $i$ induces a homeomorphism
$|U|^{h}\cong|V|^{h}$.666To verify this, observe that $|U|^{h}\to|V|^{h}$ is a
continuous bijection by the assumption on $(K,\mathcal{O}_{K})$-points. But
any continuous bijection from a compact space to a Hausdorff space is a
homeomorphism. Since $X_{0}$ is spatial, the Krull dimension of
$|X_{0}\times_{\mathrm{Spd}(C,C^{+})}\mathrm{Spd}(C,\mathcal{O}_{C})|$ is
bounded above by

$\displaystyle\mathrm{dim.trg}X_{0}\times_{\mathrm{Spd}(C,C^{+})}\mathrm{Spd}(C,\mathcal{O}_{C})/\mathrm{Spd}(C,\mathcal{O}_{C})$
$\displaystyle\leq\mathrm{dim.trg}X_{0}/\mathrm{Spd}(C,C^{+})$
$\displaystyle\leq\mathrm{dim.trg}f.$

The desired bound on the cohomological dimension of $|X|^{h}$ now follows from
the next lemma. ∎

######  Lemma 3.8.

Let $X$ be a separated spatial diamond, and let $d$ be the Krull dimension of
the spectral space $|X|$. Then the cohomological dimension of the compact
Hausdorff space $|X|^{h}$ is $\leq d$.

###### Proof.

Note that the set of generizations of any point in $|X|$ forms a totally
ordered chain, as this is true for any locally spatial diamond. Now, let $S$
be any spectral space in which the generizations of any point form a chain,
and let $q:S\to S^{h}$ be the natural map to the maximal Hausdorff quotient.
We claim that in fact $R\Gamma(S^{h},\mathcal{F})\cong
R\Gamma(S,q^{\ast}\mathcal{F})$ for any abelian sheaf $\mathcal{F}$ on
$S^{h}$. In the case of interest to us, $S=|X|$ has cohomological dimension
$\leq d$ by Scheiderer’s theorem [10], so this implies the desired result.

It’s clearly enough to prove that $\mathcal{F}\cong
Rq_{\ast}q^{\ast}\mathcal{F}$. Let $x\in S^{h}$ be any point, and let
$\tilde{x}\in S$ be the unique maximal point in the fiber $q^{-1}(x)$, so
$q^{-1}(x)=\overline{\\{\tilde{x}\\}}$. Let $\mathcal{P}$ be the cofiltered
set of all open neighborhoods of $x$ in $S^{h}$, and let $\mathcal{N}$ be the
cofiltered set of all quasicompact open neighborhoods of $q^{-1}(x)$ in $S$.
By [4, Lemma 8.1.5], each of the collections $\\{V\subset
S\\}_{V\in\mathcal{N}}$ and $\\{q^{-1}(U)\subset S\\}_{U\in\mathcal{P}}$ is a
fundamental system of neighborhoods of $q^{-1}(x)$ in $S$, and moreover

$q^{-1}(x)=\cap_{V\in\mathcal{N}}V=\cap_{U\in\mathcal{P}}q^{-1}(U).$

Then

$\displaystyle(R^{i}q_{\ast}q^{\ast}\mathcal{F})_{x}\cong$
$\displaystyle\mathrm{colim}_{U\in\mathcal{P}}H^{i}(q^{-1}(U),q^{\ast}\mathcal{F})$
$\displaystyle\cong$
$\displaystyle\mathrm{colim}_{V\in\mathcal{N}}H^{i}(V,q^{\ast}\mathcal{F})$
$\displaystyle\cong$ $\displaystyle H^{i}(q^{-1}(x),q^{\ast}\mathcal{F}).$

Since $q^{\ast}\mathcal{F}$ is constant on the fiber $q^{-1}(x)$, we’re
reduced to showing that $R\Gamma(q^{-1}(x),A)\cong A$ for any constant sheaf
of abelian groups $A$ on $q^{-1}(x)$. This is an easy exercise, using the fact
that $q^{-1}(x)$ is a spectral space with a unique maximal point in which the
generalizations of any point form a chain. (Precisely: If $T$ is a such a
spectral space, and $j:\eta\to T$ is the inclusion of the unique maximal
point, then $A\overset{\sim}{\to}Rj_{\ast}A$ for any constant sheaf of abelian
groups $A$.) ∎

######  Corollary 3.9.

If $f:X\to Y$ is separated and representable in prespatial diamonds with
$\mathrm{dim.trg}f<\infty$, then $Rf_{\ast}$ has cohomological amplitude $\leq
3\mathrm{dim.trg}f$. If moreover $f$ is compactifiable, then $Rf_{!}$ has
cohomological amplitude $\leq 3\mathrm{dim.trg}f$.

###### Proof.

Factor $f$ as $\overline{f}^{/Y}\circ j$, where $j:X\to\overline{X}^{/Y}$ is
the natural quasicompact injection. By Proposition 3.6, $\overline{f}^{/Y}$ is
proper and representable in prespatial diamonds with
$\mathrm{dim.trg}\overline{f}^{/Y}=\mathrm{dim.trg}f$, so
$R\overline{f}_{\ast}^{/Y}$ has cohomological amplitude $\leq
3\mathrm{dim.trg}f$ by Proposition 3.7. Since
$Rf_{!}=R\overline{f}_{\ast}^{/Y}\circ j_{!}$, this implies the claim for
$Rf_{!}$. Similarly, writing $Rf_{\ast}=R\overline{f}_{\ast}^{/Y}\circ
Rj_{\ast}$, the desired bound for $Rf_{\ast}$ follows from the observation
that $Rj_{\ast}$ has cohomological amplitude zero, which is a special case of
the next lemma. ∎

######  Lemma 3.10.

If $j:X\to Y$ is a quasicompact injection of small v-stacks, then $Rj_{\ast}$
has cohomological amplitude zero.

###### Proof.

By the first half of [11, Prop. 17.6], we can assume that $Y$ is a spatial
diamond, in which case $X$ is also a spatial diamond. This reduces us to [11,
Lemma 21.13]. ∎

With these results on the books, we now make another definition.

######  Definition 3.11.

Let $f:X\to Y$ be a morphism of small v-stacks. We say $f$ is _strongly
compactifiable_ if it is compactifiable, representable in prespatial diamonds,
and locally of finite dim.trg. We say $f$ is _weakly compactifiable_ if it is
compactifiable, representable in locally prespatial diamonds, and $X$ admits
an open cover $X=\cup X_{i}$ with each $X_{i}\to Y$ strongly compactifiable.

Recall that by definition, a compactifiable morphism is necessarily
0-truncated and separated.

######  Lemma 3.12.

  1. (i)

The property of being strongly resp. weakly compactifiable is stable under
composition and base change.

  2. (ii)

A morphism $f$ is strongly compactifiable iff it is weakly compactifiable and
quasicompact.

  3. (iii)

A morphism is proper and weakly compactifiable iff it is proper and strongly
compactifiable.

  4. (iv)

Any strongly compactifiable morphism $f$ can be factored as $\overline{f}\circ
j$ where $j$ is a quasicompact open immersion and $\overline{f}$ is proper and
strongly compactifiable.

  5. (v)

If $f:X\to Y$ is strongly compactifiable, then $Rf_{\ast}$ and
$R\overline{f}_{\ast}^{/Y}$ satisfy base change on unbounded complexes and
commute with all colimits. Moreover, $Rf_{!}=R\overline{f}_{\ast}^{/Y}\circ
j_{!}$ satisfies composability, base change, and the projection formula, and
commutes with all colimits.

###### Proof.

Parts (i)-(iv) are easy and left to the reader. Part (v) follows from the same
arguments used in [11] for the case where $f$ is representable in spatial
diamonds, using the cohomological dimension bounds from Corollary 3.9. ∎

Note that in [11], $Rf_{!}$ and $Rf^{!}$ are constructed exactly for morphisms
$f$ which are weakly compactifiable _and_ representable in locally spatial
diamonds. We now have the following basic claim.

######  Scholium 3.13.

All constructions and results involving $Rf_{!}$ and $Rf^{!}$ established in
[11, §22-24] extend to the setting of weakly compactifiable morphisms.

One simply repeats all arguments in [11] with extremely minor changes; we will
not need the full generality of this claim, so we omit the details. However,
let us analyze one particular case of this more general $Rf_{!}$ construction,
which will be encoded in the enhanced operation map constructed in the next
section. Suppose $f:X\to Y$ is a weakly compactifiable map, and that $X$ and
$Y$ are coproducts of qcqs v-sheaves, say with $X=\coprod_{i\in I}X_{i}$ where
all $X_{i}$ are qcqs. There is a naturally associated diagram

$\textstyle{X=\coprod_{i\in
I}X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{f}$$\textstyle{\coprod_{i\in
I}\overline{X_{i}}^{/Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y}$$\textstyle{\coprod_{i\in
I}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$

where $p$ is proper and strongly compactifiable, $j$ is an open immersion, and
$h$ is a local isomorphism. In this notation, we have a natural identification
$Rf_{!}\cong h_{!}\circ Rp_{\ast}\circ j_{!}$.

###  3.2 Enhanced operations for qcqs v-sheaves

The main goal of this section is the following theorem.

######  Theorem 3.14.

Let $\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ denote the category spanned by
small coproducts of qcqs v-sheaves, and let $W$ denote the class of weakly
compactifiable maps. Fix a ring $\Lambda$ killed by some integer prime to $p$.
Then there is an enhanced operations map

${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{W,\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

with the following properties.

_i._ $\mathcal{D}(X)=\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$ with its
natural symmetric monoidal structure for all
$X\in\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$.

_ii._ $f^{\ast}$ coincides with the pullback functor on
$\mathcal{D}_{\mathrm{\acute{e}t}}$ constructed in [11] for all morphisms $f$.

_iii._ For morphisms $g\in W$, $g_{!}$ coincides with the functor $Rg_{!}$
constructed in Scholium 3.13 and the discussion immediately afterwards.

Our construction of ${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}$
closely follows the arguments in [8, pp. 37-39], where an analogous map is
constructed in the setting of quasicompact separated schemes. The key input is
the following “categorical gluing” result, which is the analogue of [9,
Corollary 0.4] in our setup.

######  Proposition 3.15.

Let $\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ denote the category of small
coproducts of qcqs v-sheaves. Let $S\subset W$ be the classes of strongly
compactifiable and weakly compactifiable maps, respectively. Let $P$ be the
class of proper weakly compactifiable maps, let $I$ be the class of separated
local isomorphisms, and let $I_{\mathrm{qc}}\subset I$ be the class of
quasicompact separated local isomorphisms. Then $P$ and $I$ are _admissible
subsets_ , in the terminology of [9], and every strongly compactifiable
morphism $f$ in $\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ can be factored as
$f=p\circ q$ for some $p\in P$ and $q\in I_{\mathrm{qc}}$.

In this notation, the natural map

$\delta_{2}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{P,I}^{\mathrm{cart}}\to\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{W}$

is a categorical equivalence. Similarly, the natural map

$\delta_{3,\\{3\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{P,I,\mathrm{all}}^{\mathrm{cart}}\to\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{W,\mathrm{all}}^{\mathrm{cart}}$

is a categorical equivalence. Finally, analogous claims hold with
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ replaced by
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg}$.

###### Proof.

All claims in the first paragraph follow immediately from the definitions. For
the admissibility claims in particular, use the criterion in [9, Remark 3.19].
For the first categorical equivalence, argue exactly as in the proof of
Corollary 0.4 in [9], with the square

$\textstyle{\delta_{3}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{P,I_{\mathrm{qc}},I}^{\mathrm{cart}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\delta_{2}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{S,I}^{\mathrm{cart}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\delta_{2}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{P,I}^{\mathrm{cart}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{W}}$

playing the role of the square in loc. cit. The argument for the second
categorical equivalence is completely analogous. For the final claim, note
that the passage from $\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ to
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg}$ poses no additional
complications, since by convention the various markings $?$ on
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ induce markings on
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg}$ by consideration of those
edges which statically belong to ?. ∎

Returning to the problem at hand, consider the composite map

$\delta_{3,\\{1,2,3\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{P,I,\mathrm{all}}^{\mathrm{cart}}\to(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})^{\mathrm{op},\amalg}\overset{\mathrm{EO}^{\mathrm{I}}}{\longrightarrow}\mathcal{C}\mathrm{at}_{\infty}$
(3.1)

where the first arrow is the evident diagonal map, and
$\mathrm{EO}^{\mathrm{I}}$ is the map encoding the association
$X\mapsto\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$ together with its
symmetric monoidal $\ast$-pullback functoriality.777This map exists by general
nonsense, as in [8, Section 2] and the discussion on [11, p. 133].

To construct the desired enhanced operation map, we will encode
$!$-pushfowards by arguing as in [8, pp. 37-39]: in the source of the map ??,
we will pass to _right_ adjoints in direction 1, and _left_ adjoints in
direction 2.

To pass to right adjoints in direction 1, we apply the dual of [8, Proposition
1.4.4]. To apply this proposition, we need to check the relevant
adjointability in directions (1,2) and (1,3). The former is a special case of
the latter. For the latter, adjointability follows from proper base change and
the projection formula as in [8, Lemma 3.2.5], both of which hold in the
present situation by Lemma 3.12.(v). Therefore, passing to right adjoints in
direction 1, we get a map

$\delta_{3,\\{2,3\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{P,I,\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}.$

By another application of [8, Proposition 1.4.4], we now pass to left adjoints
in direction 2. Here the necessary adjointability in direction (2,1) follows
from the adjunction $f_{!}\vdash f^{\ast}$ for $f\in I$, and the
adjointability in direction (2,3) follows from étale base change and the
projection formula for $j_{!}$ with $j\in I$, which is trivial. Passing to
left adjoints in direction 2, we get a map

$\mathrm{EO}^{\prime}:\delta_{3,\\{3\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{P,I,\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}.$

Finally, by the final claim in Proposition 3.15, the map

$\delta_{3,\\{3\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{P,I,\mathrm{all}}^{\mathrm{cart}}\to\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{W,\mathrm{all}}^{\mathrm{cart}}$

is a categorical equivalence of simplicial sets, so the induced functor

$f:\mathrm{Fun}\left(\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{W,\mathrm{all}}^{\mathrm{cart}},\mathcal{C}\mathrm{at}_{\infty}\right)\to\mathrm{Fun}\left(\delta_{3,\\{3\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{P,I,\mathrm{all}}^{\mathrm{cart}},\mathcal{C}\mathrm{at}_{\infty}\right)$

is a categorical equivalence of $\infty$-categories by [5, Proposition
1.2.7.3]. Therefore, choosing any $x$ in the source of $f$ such that
$f(x)\simeq\mathrm{EO}^{\prime}$, we obtain the desired map

${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{W,\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}.$

This completes the proof of Proposition 3.14.

###  3.3 Descent and codescent

In this section we prove some descent and codescent properties satisfied by
the enhanced operation map constructed in the previous section. These will be
crucial inputs for the first iteration of the DESCENT algorithm.

Consider the enhanced operation map

${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}_{\amalg})_{W,\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

constructed in the previous section. By restriction and passing to suitable
adjoints, we obtain from this the following functors (with notation as in
Section 2.1):

1\. A functor

$(_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO^{I}})^{\otimes}:\left(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}\right)^{\mathrm{op}}\to\mathrm{CAlg}(\mathcal{C}\mathrm{at}_{\infty})_{\mathrm{pr,st,cl}}^{\mathrm{L}}$

encoding the assignment $X\mapsto\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$
and all $\ast$-pullbacks together with their symmetric monoidal structures,
with all higher coherences.

2\. A functor

${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}_{!}:(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{W}\to\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}}$

encoding the assignment $X\mapsto\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$
together with all $!$-pushforwards for weakly compactifiable maps, with all
higher coherences.

2’. A functor

${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}^{!}:(\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}})_{W}^{\mathrm{op}}\to\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{R}}$

encoding the assignment $X\mapsto\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$
together with all $!$-pullbacks for weakly compactifiable maps, with all
higher coherences.

######  Proposition 3.16.

Let $f:X\to Y$ be any surjective map in
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$.

  1. 1.

$f$ is of universal
$(_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO^{I}})^{\otimes}$-descent.

  2. 2.

If $f$ is weakly compactifiable, representable in locally spatial diamonds,
and cohomologically smooth, then $f$ is of universal
${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}_{!}$-codescent.

Here the terminology follows [8, Definition 3.1.1].

###### Proof.

Part 1. amounts to the claim that
$\mathcal{D}_{\mathrm{\acute{e}t}}(-,\Lambda)$ is a v-sheaf of closed
symmetric monoidal stable $\infty$-categories, which is clear from its
construction.

For 2., arguing as in Lemma 1.3.3 of [2], it is equivalent to prove that $f$
is of universal
${}_{\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}}\mathrm{EO}^{!}$-descent. In other
words, if $f_{0}:X=X_{0}\to Y=X_{-1}$ is any map as in 2., with Cech nerve
$f_{\bullet}:X_{\bullet}\to Y$, we need to prove that

$\mathcal{D}_{\mathrm{\acute{e}t}}(Y,\Lambda)\simeq\lim_{n\in\boldsymbol{\Delta}}\mathcal{D}_{\mathrm{\acute{e}t}}(X_{n},\Lambda)$

where the transition maps are given by $!$-pullback.

Quite generally, when computing the limit of a cosimplicial $\infty$-category,
it is equivalent to compute the limit of the associated semi-cosimplicial
$\infty$-category, by (the dual of) Lemma 6.5.3.7 and the subsequent remarks
in [5]. Thus, let
$\mathcal{C}^{\bullet}:N(\boldsymbol{\Delta}_{s})\to\mathcal{C}\mathrm{at}_{\infty}$
be the (augmented) semi-cosimplicial $\infty$-category with
$\mathcal{C}^{n}=\mathcal{D}_{\mathrm{\acute{e}t}}(X_{n},\Lambda)$ and with
the transition maps given by $!$-pullback. We need to prove that

$\mathcal{D}_{\mathrm{\acute{e}t}}(Y,\Lambda)\simeq\lim_{n\in\boldsymbol{\Delta}_{s}}\mathcal{C}^{n}.$

Let
$\mathcal{D}^{\bullet}:N(\boldsymbol{\Delta}_{s})\to\mathcal{C}\mathrm{at}_{\infty}$
be the (augmented) semi-cosimplicial $\infty$-category with
$\mathcal{D}^{n}=\mathcal{D}_{\mathrm{\acute{e}t}}(X_{n},\Lambda)$ and with
the transition maps given by $\ast$-pullback. Then we have a natural
equivalence
$\tau:\mathcal{D}^{\bullet}\overset{\sim}{\to}\mathcal{C}^{\bullet}$ sending
$A_{n}\in\mathcal{D}_{\mathrm{\acute{e}t}}(X_{n},\Lambda)$ to $A_{n}\otimes
Rf_{n}^{!}\Lambda$, and this equivalence is compatible with the augmentations.
Thus

$\mathcal{D}_{\mathrm{\acute{e}t}}(Y,\Lambda)\simeq\lim_{n\in\boldsymbol{\Delta}_{s}}\mathcal{D}^{n}\overset{\sim}{\to}\lim_{n\in\boldsymbol{\Delta}_{s}}\mathcal{C}^{n},$

where the first isomorphism follows from part 1 and the second isomorphism is
induced by $\tau$. This gives the desired result. ∎

##  4 Running DESCENT

###  4.1 Decent v-stacks and fine morphisms

Recall that in the introduction, we have defined decent v-stacks and fine
morphisms between them. In this section we study this notion in detail.

To streamline the discussion, it will be convenient to make the following
definition.

######  Definition 4.1.

A map of small v-stacks $f:X\to Y$ is _representable in locally separated
locally spatial diamonds_ if for all locally separated locally spatial
diamonds $T$ with a map $T\to Y$, the fiber product $X\times_{Y}T$ is a
locally separated locally spatial diamond.

Note that it is enough to quantify over all separated spatial $T$. In this
language, condition 1. in Definition 1.1 is exactly the condition that the
diagonal be representable in locally separated locally spatial diamonds.

######  Warning 4.2.

One must be slightly careful when using this definition. In particular, it is
not clear to us whether $f:X\to Y$ being representable in locally separated
locally spatial diamonds implies that $f$ is representable in locally spatial
diamonds, although see Proposition 4.3.i below for a partial result. Even if
this were true, it is still not clear whether $f:X\to Y$ being representable
in locally separated locally spatial diamonds is equivalent to $f$ being
locally separated and representable in locally spatial diamonds. It seems
plausible that such an equivalence is actually false, since the condition of
$f$ being locally separated can be phrased as a quantification over open
subsets of $|X|$, and in general $|X|$ can have very few open subsets.

######  Proposition 4.3.

_i._ Suppose $f:X\to Y$ is representable in locally separated locally spatial
diamonds. Then $f$ is representable in diamonds. If also $f$ is
quasiseparated, then $f$ is representable in locally spatial diamonds.

_ii._ The property of being representable in locally separated locally spatial
diamonds is stable under composition and base change.

_iii._ Quasicompact injections are representable in locally separated locally
spatial diamonds.

_iv._ If $f:X\to Y$ and $g:Y\to Z$ are maps of small v-stacks such that
$g\circ f$ is representable in locally separated locally spatial diamonds, and
$g$ is 0-truncated and quasiseparated, then $f$ is representable in locally
separated locally spatial diamonds.

###### Proof.

We show the first claim in i. Quite generally, to show that a map $f:X\to Y$
is representable in diamonds, it suffices to show that for any map $T\to Y$
with $T$ affinoid perfectoid, the fiber product $X\times_{Y}T$ is a diamond.
Since affinoid perfectoids are separated spatial diamonds, the hypothesis in
i. implies that $X\times_{Y}T$ is a (locally separated locally spatial)
diamond, giving the first claim. The second claim in i. is left as a slightly
tricky exercise for the reader (since we make no use of it in this paper).

Part ii. is clear, and iii. follows from [11, Proposition 11.20]. For iv., let
$T$ be any locally separated locally spatial diamond with a map $T\to Y$. Then
we have a cartesian diagram

$\textstyle{X\times_{Y}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times_{Z}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\times_{Z}Y}$

of small v-stacks, and we need to see that $X\times_{Y}T$ is a locally
separated locally spatial diamond. Since $g\circ f$ is representable in
locally separated locally spatial diamonds, $X\times_{Z}T$ is a locally
separated locally spatial diamond. Then since $g$ is 0-truncated and
quasiseparated, $Y\to Y\times_{Z}Y$ is a quasicompact injection, so
$X\times_{Y}T\to X\times_{Z}T$ is a quasicompact injection. The claim now
follows from iii. ∎

######  Proposition 4.4.

Decent v-stacks are Artin v-stacks in the sense of [1, Definition IV.1.1]. In
particular, the diagonal of any decent v-stack is quasiseparated and
representable in locally spatial diamonds.

###### Proof.

Suppose $X$ is decent, so $\Delta:X\to X\times X$ is representable in locally
separated locally spatial diamonds. First we show that $\Delta$ is
quasiseparated and representable in locally spatial diamonds. Pick any chart
$f:U\to X$. Then [1, Remark IV.1.4.ii] implies that $\Delta:X\to X\times X$ is
quasiseparated, and reduces us to checking that $\Delta$ is representable in
diamonds. Since $X$ is decent, this follows from the first part of Proposition
4.3.i.

Refining any given chart for $X$ as in Remark 1.2.ii, we get a surjective
separated cohomologically smooth map $U^{\prime}\to X$ from a locally spatial
diamond. Hence $X$ is Artin. ∎

######  Proposition 4.5.

If $X_{2}\to X_{1}\leftarrow X_{3}$ is any diagram of decent v-stacks, then
$X_{2}\times_{X_{1}}X_{3}$ is a decent v-stack.

###### Proof.

Pick charts $f_{i}:U_{i}\to X_{i}.$ Then $U_{2}\times_{X_{1}}U_{3}$ is a
locally separated locally spatial diamond, by Remark 1.2.i applied to $X_{1}$.
One now verifies that $U_{2}\times_{X_{1}}U_{3}\to X_{2}\times_{X_{1}}X_{3}$
is a chart, by factoring it as the composition of maps

$U_{2}\times_{X_{1}}U_{3}\overset{g_{2}}{\to}X_{2}\times_{X_{1}}U_{3}\overset{g_{3}}{\to}X_{2}\times_{X_{1}}X_{3}$

where $g_{i}$ is a base change of $f_{i}$.

For the condition on the diagonal, let $u:X_{2}\times_{X_{1}}X_{3}\to
X_{2}\times X_{3}$ be the pullback of $\Delta_{X_{1}}:X_{1}\to X_{1}\times
X_{1}$ along $X_{2}\times X_{3}\to X_{1}\times X_{1}$, so $u$ is representable
in locally separated locally spatial diamonds by the decency of $X_{1}$ and
quasiseparated by Proposition 4.4. Now consider the commutative diagram

$\textstyle{X_{2}\times_{X_{1}}X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{\Delta_{X_{2}\times_{X_{1}}X_{3}}\;\;\;\;\;\;\;\;\;\;}$$\textstyle{(X_{2}\times_{X_{1}}X_{3})\times(X_{2}\times_{X_{1}}X_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u\times
u}$$\textstyle{X_{2}\times
X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{X_{2}\times
X_{3}}\;\;\;\;\;\;}$$\textstyle{(X_{2}\times X_{3})\times(X_{2}\times X_{3})}$

of small v-stacks. Using the decency of $X_{2}$ and $X_{3}$, one sees by
repeated applications of Proposition 4.3.ii that $\Delta_{X_{2}\times X_{3}}$
and then also $\Delta_{X_{2}\times X_{3}}\circ u$ are both representable in
locally separated locally spatial diamonds. Now going around the diagram via
the upper right, we also get that

$(u\times u)\circ\Delta_{X_{2}\times_{X_{1}}X_{3}}\simeq\Delta_{X_{2}\times
X_{3}}\circ u$

is representable in locally separated locally spatial diamonds. Since $u$ is
0-truncated and quasiseparated, it is formal that $u\times u$ is 0-truncated
and quasiseparated, by factoring it as a composition of pullbacks of $u$ and
using Proposition 4.3.ii again. Therefore, applying Proposition 4.3.iv with
$f=\Delta_{X_{2}\times_{X_{1}}X_{3}}$ and $g=u\times u$, we deduce that
$\Delta_{X_{2}\times_{X_{1}}X_{3}}$ is representable in locally separated
locally spatial diamonds, as desired. ∎

We now turn to the study of fine morphisms. We begin with some remarks on the
definition.

######  Remark 4.6.

Let $f$ be a fine morphism, with

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

a commutative diagram witnessing the fineness of $f$. Successively refining
$V$ and $W$ as in Remark 1.2.ii, we can extend this to a commutative diagram

$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{g^{\prime}}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

where $V^{\prime}$ and $W^{\prime}$ are separated locally spatial diamonds,
$\alpha$ and $\beta$ are surjective separated local isomorphisms, and
$a\circ\alpha$ and $b\circ\beta$ are clean charts. Using [11, Proposition
22.3], the condition that $g$ is locally on $W$ compactifiable of finite
dim.trg is then _equivalent_ to the analogous condition for $g^{\prime}$.
However, $g^{\prime}$ is a map between separated v-sheaves, thus is separated
itself, so using [11, Proposition 22.3] again, one sees that the condition on
$g^{\prime}$ boils down to the condition that $g^{\prime}$ is compactifiable
of locally finite dim.trg. Summarizing this discussion, we conclude that a
given morphism $f$ is fine if and only if there is a commutative diagram

$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{g^{\prime}}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

where $a^{\prime}$ and $b^{\prime}$ are clean charts and $g^{\prime}$ is
compactifiable of locally finite dim.trg.

Similar arguments show that if $f:X\to Y$ is any map of decent v-stacks, then
$f$ is $\ell$-cohomologically smooth in the sense of Definition 1.3.ii if and
only if it is $\ell$-cohomologically smooth in the sense of [1, Definition
IV.1.11].

######  Remark 4.7.

In the definition of a fine morphism, one might instead ask for an a priori
stronger condition, namely the existence of a commutative diagram

$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{g^{\prime}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{a}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

where $a$ is a chart for $Y$, the square is cartesian, $b^{\prime}$ is a chart
for $U$, and $g^{\prime}$ is locally on $W^{\prime}$ compactifiable of finite
dim.trg. The existence of such a diagram certainly implies that $f$ is fine,
since $\tilde{a}\circ b^{\prime}:W^{\prime}\to X$ is a chart for $X$. In fact,
these two conditions are equivalent.

To see this, pick a commutative square as in the definition of a fine
morphism. Set $U=X\times_{Y}V$, so we get a commutative diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{b}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\tilde{a}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

where $a$ and $b$ are charts, the square is cartesian, and $g=h\circ i$. Then
setting $W^{\prime}=W\times_{b,X,\tilde{a}}U$, this diagram extends to a
diagram

$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\tilde{a}}}$$\scriptstyle{b^{\prime}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{b}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\tilde{a}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

where the trapezoid is cartesian. (Warning! The triangle spanned by
$\tilde{\tilde{a}},b^{\prime},i$ is typically not commutative.) Using the
alternative presentation $W^{\prime}=W\times_{Y}V$, decency of $Y$ implies by
Remark 1.2.i that $W^{\prime}$ is a locally separated locally spatial diamond.
One then sees (using that $b$ is a chart) that $b^{\prime}$ is a chart.

To conclude, it is enough to see that $g^{\prime}:=h\circ b^{\prime}$ is
locally on $W^{\prime}$ compactifiable of finite dim.trg. For this, observe
that by our assumptions on $g=h\circ i$ and $a$, $a\circ g=a\circ h\circ
i=f\circ b$ is representable in locally spatial diamonds and is locally on $W$
compactifiable of finite dim.trg. Indeed, the claimed properties are true for
$g$ and $a$ separately by assumption; to get it for the composition, refine
the chart $a$ as in Remark 1.2.ii if necessary. Then $h\circ b^{\prime}$ is
the base change of $f\circ b$ along $a$, which gives the desired conclusion.

######  Proposition 4.8.

_o._ If $f:X\to Y$ is fine, then for _any_ commutative diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

where the vertical maps are charts, the map $W\to V$ is locally on $W$
compactifiable of finite dim.trg.

_i._ Fine morphisms are stable under composition.

_ii._ If

$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

is any Cartesian diagram of decent v-stacks, and $f$ is fine, then also
$\tilde{f}$ is fine.

_iii._ If

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{f}$$\textstyle{Z}$

is any commutative diagram of decent v-stacks, and $h$ and $g$ are fine, then
so is $f$.

_iv._ If $f:X\to Y$ is a map of decent v-stacks which is separated and
representable in locally spatial diamonds, then $f$ is fine if and only if $f$
is compactifiable of locally finite dim.trg.

###### Proof.

o. By Remark 4.6, it suffices to show that if $f\colon X\to Y$ is a fine
morphism, there are commutative squares

$\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.41667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-18.41666pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.92726pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.41667pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.41667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.44444pt\raise-18.41666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 39.44444pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.63193pt\raise-30.72221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
32.43056pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.43056pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y}$}}}}}}}\ignorespaces}}}}\ignorespaces}\text{ and
}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.18666pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-18.80554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.98611pt\hbox{$\scriptstyle{b^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.77777pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.14726pt\raise 5.74306pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{g^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.18666pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.18666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.98444pt\raise-18.80554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.98444pt\raise-27.77777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.40193pt\raise-31.49998pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
33.97055pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.97055pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y}$}}}}}}}\ignorespaces}}}}\ignorespaces}\,,$

where the vertical maps are clean charts, and $g^{\prime}$ is compactifiable
of locally finite dim.trg, then $g$ is also compactifiable of locally finite
dim.trg.

We apply various parts of [11, Proposition 22.3] to show that the following
maps are compactifiable: $W\times_{X}W^{\prime}\to W^{\prime}$ by (ii),
$W\times_{X}W^{\prime}\to V\times_{Y}W^{\prime}$ by (viii),
$W\times_{X}W^{\prime}\to V$ by (iv), and finally, $g\colon W\to V$ by (vii).

Let $w\in W$, let $w^{\prime}\in W^{\prime}$ satisfy
$b^{\prime}(w^{\prime})=b(w)$, and choose open neighborhoods $U\owns w$,
$U^{\prime}\owns w^{\prime}$ so that $b|_{U}$ and $(a^{\prime}\circ
g^{\prime})|_{U^{\prime}}=(f\circ b^{\prime})|_{U^{\prime}}$ are of finite
dim.trg. Then, using a chain of maps similar to the one in the previous
paragraph, we see that $U\times_{X}U^{\prime}\to V$ is of finite dim.trg. By
[11, Proposition 23.11], $a^{\prime}$ is universally open, so
$a^{-1}(a^{\prime}(U^{\prime}))\subset W$ is open. Then the restriction of $g$
to $U\cap a^{-1}(a^{\prime}(U^{\prime}))$ has finite dim.trg. Since $w$ was
chosen arbitrarily, $g$ is of locally finite dim.trg.

i. Suppose $X_{1}\to X_{2}$ and $X_{2}\to X_{3}$ are fine maps of decent
v-stacks. Pick commutative squares

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}}$

and

$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}}$

witnessing the fineness of these maps. Set $T=V\times_{X_{2}}V^{\prime}$. Then
we get a commutative diagram

---
$\textstyle{U\times_{V}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}}$

and throwing away most of it we get a diagram

$\textstyle{U\times_{V}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}}$

witnessing the fineness of $X_{1}\to X_{3}$.

ii. Pick a diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

witnessing the fineness of $f$. Let $U\to\tilde{Y}$ be a chart for
$\tilde{Y}$. Then we get a commutative diagram

---
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{W}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{V}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{X}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where all vertical arrows are locally separated smooth etc. and the
parallelograms involving slanted arrows are all cartesian. Pick a chart
$T\to\tilde{X}_{U}$. Then we get a commutative square

$\textstyle{T\times_{\tilde{X}_{U}}\tilde{W}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{V}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{Y}}$

which I claim witnesses the fineness of $\tilde{X}\to\tilde{Y}$.

iii. We can find commutative diagrams

$\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.02777pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-27.90222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
31.02777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.02777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{U_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.38678pt\raise-27.90222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.89844pt\raise-32.54803pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
32.61595pt\raise-37.73553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.61595pt\raise-37.73553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z}$}}}}}}}\ignorespaces}}}}\ignorespaces}\text{ and
}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.41667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-27.90222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.41667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.41667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{U_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.77568pt\raise-27.90222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.37128pt\raise-32.30498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{h}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
34.00485pt\raise-37.73553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.00485pt\raise-37.73553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z}$}}}}}}}\ignorespaces}}}}\ignorespaces}\,,$

where the vertical arrows are clean charts and the horizontal arrows are
compactifiable of locally finite dim.trg. Then

$\textstyle{V\times_{Y}W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\times_{Z}U_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{1}\times_{Z}U_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Z}$

is also a commutative diagram where the vertical arrows are clean charts, and
$V\times_{Y}W\to U_{1}\times_{Z}U_{2}$ and $V\times_{Z}U_{2}\to
U_{1}\times_{Z}U_{2}$ are compactifiable of locally finite dim.trg. by o. Then
$V\times_{Y}W\to V\times_{Z}U_{2}$ is compactifiable by [11, Proposition
22.3(viii)], and it has locally finite dim.trg.

iv. Choose a commutative diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

such that the vertical arrows are clean charts. By o., $f$ is fine if and only
if $g$ is compactifiable of locally finite dim.trg. By [11, Proposition
22.3(iv,vii,viii)], $g$ is compactifiable iff $f$ is compactifable. By an
argument similar to that of the proof of o., $b$ is universally open, and $g$
is locally on $W$ of finite dim.trg. iff $f$ is locally on $X$ of finite
dim.trg. ∎

###  4.2 A fragment of DESCENT

In this section we describe the fragment of Liu-Zheng’s DESCENT algorithm we
will need. The strange numberings below are chosen to match the numbering in
[8].

Input 0. Suppose given a 4-marked $\infty$-category
$(\tilde{\mathcal{C}},\tilde{\mathcal{E}}_{s},\tilde{\mathcal{E}}^{\prime},\tilde{\mathcal{E}}^{\prime\prime},\tilde{\mathcal{F}})$
together with a full subcategory $\mathcal{C}\subset\tilde{\mathcal{C}}$.
Write $\mathcal{E}_{s}=\tilde{\mathcal{E}}_{s}\cap\mathcal{C}$, and likewise
$\mathcal{E}^{\prime}$, $\mathcal{E}^{\prime\prime}$, $\mathcal{F}$. They are
assumed to satisfy:

1\. $\tilde{\mathcal{C}}$ is geometric,
$\mathcal{C}\subseteq\tilde{\mathcal{C}}$ is stable under finite limits, and
for all small coproducts $X=\coprod_{i}X_{i}$ in $\tilde{\mathcal{C}}$, $X$
belongs to $\mathcal{C}$ if and only if all $X_{i}$ belong to $\mathcal{C}$.

3\.
$\tilde{\mathcal{E}}_{s},\tilde{\mathcal{E}}^{\prime},\tilde{\mathcal{E}}^{\prime\prime},\tilde{\mathcal{F}}$
are stable under composition, pullback and small coproducts, and
$\tilde{\mathcal{E}}^{\prime}\subseteq\tilde{\mathcal{E}}^{\prime\prime}\subseteq\tilde{\mathcal{F}}$.

4\. For every object $X$ of $\tilde{\mathcal{C}}$, there exists an edge
$f:Y\to X$ in $\tilde{\mathcal{E}}_{s}\cap\tilde{\mathcal{E}}^{\prime}$ with
$Y$ in $\mathcal{C}$. Such an edge is called an _atlas_ for $X$.

5\. For every pullback square

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$

with $X\in\mathcal{\tilde{C}}$, $Y\in\mathcal{C}$ and $Z\in\mathcal{C}$, and
$f$ an atlas, then also $W\in\mathcal{C}$. Intuitively, “atlas maps are
representable in $\mathcal{C}$”.

Input I. Suppose given an enhanced operation map

${}_{\mathcal{C}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{C}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

satisfying the following properties.

P0-P2.888This is of course redundant, since it is exactly condition 1. in the
definition of an enhanced operation map. We have written it out nonetheless to
aid the reader in comparing with [8, Section 4.1], where the notion of an
enhanced operation map is not explicitly formalized and so P0-P2 have actual
content. The functor

${}_{\mathcal{C}}\mathrm{EO}^{\mathrm{I}}:\mathcal{C}^{\mathrm{op},\amalg}\to\mathcal{C}\mathrm{at}_{\infty}$

induced by restriction to the “all” direction is a weak Cartesian structure,
and the induced functor
$({{}_{\mathcal{C}}\mathrm{EO}^{\mathrm{I}}})^{\otimes}$ factors through
$\mathrm{CAlg}(\mathcal{C}\mathrm{at}_{\infty})_{\mathrm{pr,st,cl}}^{\mathrm{L}}$
and sends small coproducts to products.

As in section 2.1, from ${}_{\mathcal{C}}\mathrm{EO}$ we obtain a map

${}_{\mathcal{C}}\mathrm{EO}_{!}^{\ast}:\delta_{2,\\{2\\}}^{\ast}\mathcal{C}{}_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}},$

which restricts further to maps

${}_{\mathcal{C}}\mathrm{EO}^{\ast}:\mathcal{C}^{\mathrm{op}}\to\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}}\;\;\;\;\;\mathrm{and}\;\;\;\;\;{}_{\mathcal{C}}\mathrm{EO}_{!}:\mathcal{C}_{\mathcal{F}}\to\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}}.$

As before we write $\mathcal{D}(X)$ for the image of a 0-cell
$X\in\mathcal{C}$ under either of these maps, and $f^{\ast}$ resp. $f_{!}$ for
the image of a 1-cell $f:Y\to X$ under the first resp. second map. Now we can
state the remaining properties we impose.

P3. If $f:Y\to X$ is an edge in $\mathcal{E}_{s}$, then
$f^{\ast}:\mathcal{D}(X)\to\mathcal{D}(Y)$ is conservative.

P4. If $f$ is an edge in $\mathcal{E}_{s}\cap\mathcal{E}^{\prime\prime}$, then
$f$ is of universal ${}_{\mathcal{C}}\mathrm{EO}^{\otimes}$-descent and of
universal ${}_{\mathcal{C}}\mathrm{EO}_{!}$-codescent.

P5. Let

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{q}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$

be a cartesian diagram in $\mathcal{C}$, with $f$ in $\mathcal{E}^{\prime}$.
Then:

1) The square

$\textstyle{\mathcal{D}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\ast}}$$\scriptstyle{g^{\ast}}$$\textstyle{\mathcal{D}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\ast}}$$\textstyle{\mathcal{D}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\ast}}$$\textstyle{\mathcal{D}(Y)}$

is right-adjointable, with a right adjoint a square in
$\mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{R}}$.

2) If $p$ is also in $\mathcal{E}^{\prime}$, then the square

$\textstyle{\mathcal{D}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{!}}$$\scriptstyle{p^{\ast}}$$\textstyle{\mathcal{D}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\ast}}$$\textstyle{\mathcal{D}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{!}}$$\textstyle{\mathcal{D}(W)}$

is right-adjointable.

P5bis. Same as P5 but with $\mathcal{E}^{\prime}$ replaced by
$\mathcal{E}^{\prime\prime}$.

Output I. This consists of an enhanced operation map

${}_{\mathcal{\tilde{C}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{\tilde{C}}_{\amalg})_{\tilde{\mathcal{F}},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

extending Input I, and satisfying the obvious analogues of P0-P5bis with
$\tilde{\mathcal{C}},\tilde{\mathcal{E}}_{s},\tilde{\mathcal{E}}^{\prime},\tilde{\mathcal{E}}^{\prime\prime},\tilde{\mathcal{F}}$
in place of
$\mathcal{C},\mathcal{E}_{s},\mathcal{E}^{\prime},\mathcal{E}^{\prime\prime},\mathcal{F}$.

We now have the following key theorem, which is the fragment of DESCENT we
will need.

######  Theorem 4.9.

Fix an Input 0. Then every Input I can be extended to an Output I in an
essentially unique way.

###### Proof.

This is a special case of (the proof of) [8, Theorem 4.1.8.(1)]. ∎

###  4.3 First iteration of DESCENT

We begin by running DESCENT with the following inputs.

For Input 0, we make the following choices.

  * •

$\tilde{\mathcal{C}}=\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$ where
$\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$ is the category of
quasiseparated locally separated locally spatial diamonds.

  * •

$\mathcal{C}=\mathcal{D}\mathrm{ia}^{\mathrm{sep.spat}}$ where
$\mathcal{D}\mathrm{ia}^{\mathrm{sep.spat}}\subset\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$
is the full subcategory spanned by small coproducts of separated spatial
diamonds.

  * •

$\tilde{\mathcal{E}}_{s}$ is the set of morphisms which are surjective as maps
of v-sheaves.

  * •

$\tilde{\mathcal{E}}^{\prime}$ is the set of (locally separated) étale
morphisms in $\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$.

  * •

$\tilde{\mathcal{E}}^{\prime\prime}$ is the set of (locally separated)
cohomologically smooth morphisms in
$\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$.

  * •

$\tilde{\mathcal{F}}$ is the set of fine morphisms.

It is easy to see that these choices satisfy the conditions required of an
Input 0. The only point which isn’t immediate is condition 5., which follows
from the next lemma.

######  Lemma 4.10.

Let

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Z}$

be a cartesian diagram of v-sheaves, where $X$ and $Y$ are separated spatial
diamonds, and $Z$ is a quasiseparated locally separated locally spatial
diamond. Then $W$ is a separated spatial diamond.

###### Proof.

Since $W=X\times_{Z}Y$ is a subfunctor of $X\times Y$, and $X\times Y$ is
separated, we immediately get that $W$ is separated. Local spatiality of $W$
is clear, so it remains to see that $W$ is qcqs. Since $X$ and $Y$ are
quasicompact and $Z$ is quasiseparated, the maps $f,g$ are quasicompact. In
particular, $|f|$ and $|g|$ have quasicompact images in $|Z|$. By local
spatiality of $Z$, we may pick some sufficiently large quasicompact open
subfunctor $U\subset Z$ such that
$|U|\supset(\mathrm{im}|f|\cup\mathrm{im}|g|)$. Then since $Z$ is
quasiseparated, $U$ is automatically qcqs, so $W=X\times_{Z}Y=X\times_{U}Y$ is
a fiber product of qcqs objects, and therefore is qcqs as desired. ∎

As Input I, we take the enhanced operation map constructed in section 3.2 for
a specific choice of coefficient ring $\Lambda$, restricted from
$\mathcal{V}\mathrm{sh}^{\mathrm{qcqs}}$ to
$\mathcal{D}\mathrm{ia}^{\mathrm{sep.spat}}$, noting that in
$\mathcal{D}\mathrm{ia}^{\mathrm{sep.spat}}$ weakly compactifiable maps
exactly coincide with fine maps. P0-P2 and P3 are clear. P4 follows from
Proposition 3.16. P5 and P5bis follow from a combination of smooth and proper
base change.

Theorem 4.9 now applies, yielding an enhanced operation map on
$\tilde{\mathcal{C}}$, and in particular a functor

${}_{\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

where $\mathcal{F}$ denotes the set of fine morphisms.

###  4.4 Second iteration of DESCENT

We run DESCENT with the following input.

For Input 0, we make the following choices.

  * •

$\tilde{\mathcal{C}}=\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$ where
$\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$ is the category of locally
separated locally spatial diamonds.

  * •

$\mathcal{C}=\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$ where
$\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}\subset\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$
is the category of quasiseparated locally separated locally spatial diamonds.

  * •

$\tilde{\mathcal{E}}_{s}$ is the set of surjective morphisms of v-sheaves.

  * •

$\tilde{\mathcal{E}}^{\prime}$ is the set of (locally separated) étale
morphisms in $\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$.

  * •

$\tilde{\mathcal{E}}^{\prime\prime}$ is the set of (locally separated)
cohomologically smooth morphisms in
$\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$.

  * •

$\tilde{\mathcal{F}}$ is the set of fine morphisms in
$\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$.

It is easy to see that these choices satisfy the conditions required of an
Input 0. The only point which isn’t immediate is condition 5., which follows
from the next lemma.

######  Lemma 4.11.

Let

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Z}$

be a cartesian diagram of small v-sheaves, where $X$ and $Y$ are
quasiseparated locally separated locally spatial diamonds, and $Z$ is a
locally separated locally spatial diamond. Then $W$ is a quasiseparated
locally separated locally spatial diamond.

###### Proof.

It is clear that $W$ is a locally separated locally spatial diamond. We need
to see that $W$ is quasiseparated. Since $W=X\times_{Z}Y\subset X\times Y$ is
a subfunctor of $X\times Y$, and quasiseparatedness passes to subfunctors,
it’s enough to see that $X\times Y$ is quasiseparated. But small v-sheaves
form an algebraic topos by [11, Proposition 8.3], so for any quasiseparated
objects $X$ and $Y$, also $X\times Y$ is quasiseparated by [12, VI,
Proposition 2.2.(ii)]. ∎

As Input I, we take the enhanced operation map constructed as Output I of the
first iteration. P0-P5bis are automatic, since they hold for any Output I.

Theorem 4.9 now applies again, yielding an enhanced operation map on
$\tilde{\mathcal{C}}$, and in particular a functor

${}_{\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}_{\amalg})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

where $\mathcal{F}$ denotes the set of fine morphisms.

###  4.5 Third iteration of DESCENT

We run DESCENT with the following input.

For Input 0, we make the following choices.

  * •

$\tilde{\mathcal{C}}=\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$ where
$\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$ is the category of decent v-stacks.

  * •

$\mathcal{C}=\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}$ where
$\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}}\subset\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$
is the category of locally separated locally spatial diamonds.

  * •

$\tilde{\mathcal{E}}_{s}$ is the set of surjective morphisms of v-stacks.

  * •

$\tilde{\mathcal{E}}^{\prime}=\tilde{\mathcal{E}}^{\prime\prime}$ is the set
of cohomologically smooth morphisms in
$\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$.

  * •

$\tilde{\mathcal{F}}$ is the set of fine morphisms in
$\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$.

It is easy to see that these choices satisfy the conditions required of an
Input 0. As Input I, we take the enhanced operation map constructed as Output
I of the second iteration. P0-P5bis are automatic, since they hold for any
Output I.

Theorem 4.9 now applies again, yielding an enhanced operation map on
$\tilde{\mathcal{C}}$, and in particular a functor

${}_{\mathcal{V}\mathrm{stk}^{\mathrm{dct}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{stk}_{\amalg}^{\mathrm{dct}})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$

where $\mathcal{F}$ denotes the set of fine morphisms.

###  4.6 Endgame

In this section we complete the proof of Theorem 2.6.

Let

${}_{\mathcal{V}\mathrm{stk}^{\mathrm{dct}}}\mathrm{EO}:\delta_{2,\\{2\\}}^{\ast}(\mathcal{V}\mathrm{stk}_{\amalg}^{\mathrm{dct}})_{\mathcal{F},\mathrm{all}}^{\mathrm{cart}}\to\mathcal{C}\mathrm{at}_{\infty}$
(4.1)

be the enhanced operation map obtained as the output of the third iteration of
DESCENT in the previous section. We freely use the notation from section 2.1
for various subordinate structures obtained from this map.

######  Proposition 4.12.

For any $X\in\mathcal{V}\mathrm{stk}^{\mathrm{dct}}$, we have
$\mathcal{D}(X)\cong\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$ compatibly
with the symmetric monoidal structures and with all $\ast$-pullbacks.

###### Proof.

For this we need to dig into the proof of Theorem 4.9. For the moment, let
$\mathcal{C}\subset\tilde{\mathcal{C}}$ be as in the general setup for
DESCENT. The proof of Theorem 4.9 extends the association
$X\rightsquigarrow\mathcal{D}(X)$ from $\mathcal{C}$ to $\tilde{\mathcal{C}}$
as follows. For any $X\in\tilde{\mathcal{C}}$, let $X_{0}\to X$ be a choice of
atlas, with Cech nerve $X_{\bullet}\to X$. By the general properties of any
Input 0, we have $X_{n}\in\mathcal{C}$ for all $n\geq 0$. Then
$\mathcal{D}(X)$ with its symmetric monoidal structure is constructed as the
limit of the cosimplicial $\infty$-category
$n\in\boldsymbol{\Delta}\mapsto\mathcal{D}(X_{n})$, where the transition maps
are given by $\ast$-pullback. (Of course, the proof of Theorem 4.9 also
accounts for the ambiguity arising from the choice of a particular atlas,
roughly by taking a further limit over all possible atlases, and also handles
the $\ast$-pullbacks.)

Returning to the situation at hand, we already know that
$\mathcal{D}_{\mathrm{\acute{e}t}}(-,\Lambda)$ is a v-sheaf of symmetric
monoidal $\infty$-categories on the category of _all_ small v-stacks.
Moreover, the enhanced operation map constructed in Theorem 3.14 satisfies
$\mathcal{D}(X)\cong\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$ for all
$X\in\mathcal{D}\mathrm{ia}^{\mathrm{sep.spat}}$ by design, compatibly with
the symmetric monoidal structure and with $\ast$-pullbacks. Since in each of
our three iterations of DESCENT the atlas maps were chosen to be v-covers, it
is now clear by the argument in the previous paragraph that the enhanced
operation map obtained at the output of each successive iteration still
satisfies
${}_{\tilde{\mathcal{C}}}\mathrm{EO}(X)\cong\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$
(for
$\tilde{\mathcal{C}}\in\\{\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}},\mathcal{D}\mathrm{ia}^{\mathrm{lsep.lspat}},\mathcal{V}\mathrm{stk}^{\mathrm{dct}}\\}$)
compatibly with the symmetric monoidal structures and with $\ast$-pullbacks.
This gives the result. ∎

######  Proposition 4.13.

Let $g:X\to Y$ be a map of decent v-stacks which is representable in locally
spatial diamonds and compactifiable of locally finite dim.trg, so in
particular $g$ is fine. Then the functor $g_{!}$ obtained from (2) coincides
with the functor $Rg_{!}$ constructed in [11, Section 22].

Taken together, Propositions 4.12 and 4.13 complete the proof of Theorem 2.6.

###### Proof.

Let $A\in\mathcal{D}_{\mathrm{\acute{e}t}}(X,\Lambda)$ be some object. Let
$Y_{\bullet}\to Y$ be a v-hypercover by small coproducts of separated spatial
diamonds, so we get a pullback square

$\textstyle{X_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{\bullet}}$$\scriptstyle{v_{\bullet}}$$\textstyle{Y_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{\bullet}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

of small v-stacks. There is a natural equivalence
$\mathcal{D}_{\mathrm{\acute{e}t}}(Y,\Lambda)\simeq\mathcal{D}_{\mathrm{\acute{e}t,cart}}(Y_{\bullet},\Lambda)$,
identifying $g_{!}A$ with the coCartesian section
$n\in\boldsymbol{\Delta}\mapsto
g_{n!}v_{n}^{\ast}A\in\mathcal{D}_{\mathrm{\acute{e}t}}(Y_{n},\Lambda)$,
functorially in $A$. This identification is an immediate consequence of the
proper base change isomorphism $u_{n}^{\ast}g_{!}\simeq g_{n!}v_{n}^{\ast}$.
Comparing this with the discussion on p. 133-134 of [11], we are reduced to
showing that $g_{n!}\simeq Rg_{n!}$ for all $n\geq 0$. In other words, after
resetting the notation, we’ve reduced the general case of the proposition to
the special case where $Y$ is a small coproduct of separated spatial diamonds.

We immediately reduce further to the case where $Y$ is a separated spatial
diamond, so $X$ is a separated and quasiseparated locally spatial diamond and
$g:X\to Y$ is compactifiable of locally finite dim.trg. In particular, $g$ is
a fine morphism in the category
$\mathcal{D}\mathrm{ia}^{\mathrm{qs.lsep.lspat}}$, so the relevant $g_{!}$
functor is already obtained from Output I of our first iteration of DESCENT.
If moreover $g$ is quasicompact, then $g$ is a fine morphism of separated
spatial diamonds, so we are actually in the setting of Input I of our first
iteration, in which case we already know that $g_{!}\simeq Rg_{!}$ by Theorem
3.14.

Otherwise, let $\mathcal{U}$ be the (filtered) collection of all quasicompact
open subdiamonds $U\subset X$; note that each $U$ is automatically separated
and spatial. For each $U\in\mathcal{U}$, let $j_{U}:U\to X$ be the evident
map, and set $g_{U}=g\circ j_{U}$, so in particular $g_{U}$ is a fine morphism
of separated spatial diamonds. Now, on one hand it is completely formal to see
that

$\displaystyle g_{!}A\simeq$
$\displaystyle\,g_{!}\mathrm{colim}_{U\in\mathcal{U}}j_{U!}j_{U}^{\ast}A$
$\displaystyle\simeq$
$\displaystyle\,\mathrm{colim}_{U\in\mathcal{U}}g_{!}j_{U!}j_{U}^{\ast}A$
$\displaystyle\simeq$
$\displaystyle\,\mathrm{colim}_{U\in\mathcal{U}}g_{U!}j_{U}^{\ast}A.$

Here the first and third lines are trivial, and the second line follows from
the fact that $g_{!}$ commutes with all colimits. On the other hand, in [11]
$Rg_{!}A$ is _defined_ as
$\mathrm{colim}_{U\in\mathcal{U}}Rg_{U!}j_{U}^{\ast}A$, cf. [11, Definition
22.13] and the discussion immediately afterwards. By the discussion in the
previous paragraph, we already know that $g_{U!}\simeq Rg_{U!}$ for all
$U\in\mathcal{U}$, so comparing these expressions, we conclude that
$g_{!}\simeq Rg_{!}$ as desired. ∎