DescentforsheavesoncompactHausdorﬀspaces PeterJ.Haine October42022_2

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Descent for sheaves on compact Hausdor spaces

Peter J. Haine

October 4, 2022

Abstract

These notes explain some descent results for

∞

-categories of sheaves on compact Haus-

dor spaces and derive some consequences. Specically, given a compactly assembled

∞

-cat-

egory

ℰ

, we show that the functor sending a locally compact Hausdor space

𝑋

to the

∞

-cat-

egory

Shpost(𝑋;ℰ)

of Postnikov complete

ℰ

-valued sheaves on

𝑋

satises descent for proper

surjections. This implies proper descent for left complete derived

∞

-categories and that the

functor

Shpost(−;ℰ)

is a sheaf on the category of compact Hausdor spaces equipped with the

topology of nite jointly surjective families. Using this, we explain how to embed Postnikov

complete sheaves on a locally compact Hausdor space into condensed objects. This implies

that the condensed and sheaf cohomologies of a locally compact Hausdor space agree.

Contents

0 Introduction 2

0.1 Postnikov completion ................................. 2

0.2 Descent for Postnikov complete sheaves ....................... 3

0.3 The comparison between sheaf and condensed cohomology ............ 3

0.4 Linear overview .................................... 4

1 Background 5

1.1 Hypercompleteness .................................. 5

1.2 Postnikov completeness ................................ 7

1.3 Condensed/pyknotic mathematics .......................... 9

2 Proper descent 10

2.1 Proper descent for Postnikov sheaves ......................... 11

2.2 Consequences in shape theory ............................ 12

2.3 The necessity of Postnikov completion ........................ 13

3 The comparison functor 14

3.1 Descent for open covers ................................ 14

3.2 The comparison functor ................................ 17

3.3 Naturality of the comparison functor ......................... 18

4 Full faithfulness of the comparison functor 19

4.1 Full faithfulness for extremally disconected pronite sets ............. 19

4.2 Full faithfulness and descent ............................. 20

4.3 The necessity of Postnikov completion ........................ 22

References 23

arXiv:2210.00186v1 [math.AT] 1 Oct 2022

0 Introduction

The rst goal of these notes is to explain some descent results for

∞

-categories of sheaves on

locally compact Hausdor spaces. Our motivation comes from condensed/pyknotic mathemat-

ics developed by Clausen–Scholze [20;21;22;23], in our joint work with Barwick [2, Chapter

13; 3], and in Lurie’s work on ultracategories [17;18, Chapter 4]. Write

Comp

for the cate-

gory of compact Hausdor spaces. The category

Comp

has a Grothendieck topology where the

covering families are nite families of jointly surjective maps. Because of the simplicity of the

Grothendieck topology, the sheaf condition is very explicit: a presheaf on

Comp

is a sheaf if

and only if it carries nite disjoint unions of compact Hausdor spaces to nite products and

satises descent for surjections.

Our rst goal is to answer the following question:

0.1 Question.

Is the functor

Sh∶Compop →Cat∞

that assigns a compact Hausdor space

𝐾

the ∞-category Sh(𝐾)of sheaves of spaces on 𝐾a sheaf with respect to this topology?

Perhaps surprisingly, the answer to Question 0.1 is negative (see Corollary 2.18). Moreover, if one

replaces sheaves by hypersheaves, the answer to Question 0.1 is still negative. The reason for this

failure of descent is that every compact Hausdor space admits a surjection from a pronite set,

and the

∞

-category of sheaves on a pronite set satises a strong completeness property which

the

∞

-category of (hyper)sheaves on a general compact Hausdor space does not satisfy. So it is

not reasonable to ask for the

∞

-category of sheaves on a general compact Hausdor spaces to

be expressible as a limit of ∞-categories satisfying this completeness property.

0.1 Postnikov completion

Since this completeness property is central to these notes, before stating the main results, let us

briey introduce it. See §1.2 for more details.

0.2 Denition.

Let

𝑋

be a topological space. The Postnikov completion of the

∞

-category of

sheaves of spaces on on 𝑋is the inverse limit

Shpost(𝑋)≔lim⋯Sh(𝑋)≤𝑛+1 Sh(𝑋)≤𝑛⋯

τ≤𝑛+1 τ≤𝑛τ≤𝑛−1 

of the ∞-categories of sheaves of 𝑛-truncated spaces along the truncation functors.

Objects of Shpost(𝑋)are towers

⋯→𝐹𝑛+1 →𝐹𝑛→⋯→𝐹0

where

𝐹𝑛

is an sheaf of

𝑛

-truncated spaces on

𝑋

such that

τ≤𝑛𝐹𝑛+1 ⥲𝐹𝑛

. There is a natural left

adjoint

Sh(𝑋)→Shpost(𝑋)

sending a sheaf

𝐹

to its Postnikov tower

{τ≤𝑛𝐹}≥0

. We say that

Sh(𝑋)

is Postnikov complete if this functor Sh(𝑋)→Shpost(𝑋)is an equivalence.

0.3 Example.

The

∞

-topos of sheaves on a pronite set is Postnikov complete. On the other

hand, the ∞-topos of sheaves on the Hilbert cube 𝑖≥1[0,1]is not Postnikov complete.

For a presentable

∞

-category

ℰ

, we write

Shpost(𝑋;ℰ)

for the tensor product

Shpost(𝑋)⊗ℰ

With stable coecients this recovers the left-complete derived ∞-category of sheaves:

0.4 Example.

Let

𝑋

be a topological space and let

𝑅

be a ring. Write

D(𝑋;𝑅)

for the derived

∞

-category of the abelian category of sheaves of

𝑅

-modules on

𝑋

. Then

Shpost(𝑋;D(𝑅))

is the

left completion of

D(𝑋;𝑅)

with respect to the standard

-structure.

That is,

Shpost(𝑋;D(𝑅))

is the

limit of the diagram of ∞-categories

⋯D(𝑋;𝑅)≤𝑛+1 D(𝑋;𝑅)≤𝑛⋯

τ≤𝑛+1 τ≤𝑛τ≤𝑛−1

along the truncation functors with respect to the standard t-structure.

0.2 Descent for Postnikov complete sheaves

The following is the main descent result of these notes. Note that all compactly generated

∞

-cat-

egories are compactly assembled (see Recollection 1.10).

0.5 Theorem

(Corollary 2.8)

Let

ℰ

be a compactly assembled

∞

-category. Then for every proper

surjection of locally compact Hausdor spaces 𝑝∶𝑋↠𝑌, natural functor

Shpost(𝑌;ℰ) limShpost(𝑋;ℰ) Shpost(𝑋×𝑌𝑋;ℰ)⋯

pr∗

is an equivalence in

Cat∞

. Consequently, the functor

Shpost(−;ℰ)∶Compop →Cat∞

is a hyper-

sheaf of ∞-categories on the site of compact Hausdor spaces.

0.6 Example.

Let

𝑅

be a ring. Then the functor

D(−;𝑅)∶Compop →Cat∞

carrying a compact

Hausdor space to its left complete derived ∞-category is a hypersheaf. Hence the functor

D(−;𝑅)<∞∶Compop →Cat∞

that sends a compact Hausdor space

𝐾

to its bounded-above derived

∞

-category

is also a

hypersheaf of ∞-categories.

Passing to global sections shows that sheaf cohomology also satises proper descent.

0.7 Corollary.

Let

𝑅

be a connective

-ring spectrum and

𝑀

a bounded-above left

𝑅

-module

spectrum. The functor RΓsheaf(−;𝑀)∶Compop →LMod(𝑅)

is a hypersheaf.

Note that if

𝑅

is an ordinary ring, then the

∞

-category

LMod(𝑅)

is the derived

∞

-category

D(𝑅)

0.3 The comparison between sheaf and condensed cohomology

Part of our motivation for proving Theorem 0.5 is that it has a number of consequences. One

application is a generalization of work of Dyckho and Clausen–Scholze that compares sheaf

cohomology with condensed cohomology. Let

𝑋

be a locally compact Hausdor space. We can

also regard

𝑋

as an object of the

∞

-category

Sh(Comp)

via the restricted Yoneda embedding.

1We use homological indexing for our t-structures.

2What we write as D(𝐾;𝑅)<∞is often written as D+(𝐾;𝑅).

Dyckho [7, Theorem 3.11; 8] and Clausen–Scholze [23, Theorem 3.2] showed that if

𝐴

is an

abelian group, and 𝑋is compact then there is an isomorphism

H∗

sheaf(𝑋;𝐴)⥲H∗

cond(𝑋;𝐴)

from the sheaf cohomology of 𝑋to the cohomology of 𝑋regarded as an object Sh(Comp).

We extend this result in two directions: to locally compact Hausdor spaces and to very

general coecients. The comparison map between sheaf and condensed cohomology is induced

by a natural geometric morphism

𝑐𝑋,∗∶Sh(Comp)∕𝑋→Sh(𝑋)

given by sending a sheaf 𝐺∶Compop →Spc to the sheaf on 𝑋dened by

𝑐𝑋,∗(𝐺)(𝑈)≔MapSh(Comp)∕𝑋(𝑈,𝐺).

(See §3.2 for details.)

Since cohomology is computed by derived global sections, to show that the sheaf and con-

densed cohomologies of

𝑋

agree, it suces to show that

𝑐∗

𝑋

is fully faithful. Again, this is generally

only true after Postnikov completion (see Warnings 4.16 and 4.19).

0.8 Proposition

(Corollary 4.11)

Let

𝑋

be a locally compact Hausdor space and let

ℰ

be a

compactly assembled ∞-category. Then the pullback functor

𝑐∗,post∶Shpost(𝑋;ℰ)→Shpost(Comp∕𝑋;ℰ)

is fully faithful.

Proposition 0.8 implies, for example:

0.9 Corollary

(Corollary 4.12)

Let

𝑋

be locally compact Hausdor space. Let

𝑅

be a connective

E1-ring spectrum and let 𝑀be a bounded-above left 𝑅-module spectrum. Then the natural map

RΓsheaf(𝑋;𝑀)→RΓcond(𝑋;𝑀)

is an equivalence in the ∞-category LMod(𝑅)of left 𝑅-module spectra.

As a consequence, the condensed, singular, and sheaf cohomologies of a topological space ad-

mitting a locally nite CW structure all agree (see Remark 4.13 and (4.14)).

0.4 Linear overview

We imagine that the reader might be interested in condensed/pyknotic mathematics but not

necessarily familiar with all of the intricacies about the theory of

∞

-topoi. With this in mind, in

§1, we review the basics of hypercomplete and Postnikov complete

∞

-topoi; the familiar reader

can safely skip this section. Section 2 proves Theorem 0.5 and derives some consequences in

shape theory. In §3, we construct the comparison geometric morphism

𝑐∗∶Sh(Comp∕𝐿)→Sh(𝐿)

and record its basic properties. Section 4 is dedicated to proving Proposition 0.8.

Acknowledgments.

We thank Ko Aoki, Clark Barwick, Marc Hoyois, Jacob Lurie, Mark Mac-

erato, Zhouhang Mao, Denis Nardin, Piotr Pstrągowski, Marco Volpe, Sebastian Wolf, and Tong

Zhou for helpful comments and conversations around the contents of these notes. Special thanks

are due to Marc Hoyois and Jacob Lurie for explaining Example 2.15 to us. These notes are clearly

highly inuenced by Dustin Clausen and Peter Scholze’s ideas; we would like to thank them too.

We gratefully acknowledge support from the UC President’s Postdoctoral Fellowship and

NSF Mathematical Sciences Postdoctoral Research Fellowship under Grant #DMS-2102957.

1 Background

Recall the following fundamental results about the ∞-category of spaces.

(1)

Whitehead’s Theorem: A map

𝑓∶𝑋→𝑌

of spaces is an equivalence if and only if

𝑓

induces

a bijection on connected components and isomorphisms on homotopy groups at each base-

point. Said dierently,

𝑓

is an equivalence if and only if for each

𝑛≥0

, the induced map on

𝑛-truncations τ≤𝑛(𝑓)∶τ≤𝑛(𝑋)→τ≤𝑛(𝑌)is an equivalence.

(2)

Convergence of Postnikov towers: Every space

𝑋

is the limit of its Postnikov tower. That is, the

the natural map 𝑋→lim𝑛≥0τ≤𝑛(𝑋)is an equivalence.

The statement of Whitehead’s Theorem and the convergence of Postnikov towers make can be

formulated in an arbitrary

∞

-topos. However, even for the

∞

-topos of sheaves on a compact

Hausdor space, neither result need hold (see Example 2.15). The purpose of this section is

to review two completion procedures (hypercompletion and Postnikov completion) that force

Whitehead’s Theorem to hold and Postnikov towers to converge, respectively. In the higher-

categorical world, these give rise to three natural ‘sheaf theories’ (sheaves, hypersheaves, and

Postnikov complete sheaves) extending the classical theory of sheaves on a topological space.

They all have the same truncated objects, so the subtle dierences between these theories only

appears when considering ‘unbounded’ objects.

Subsection 1.1 reviews the basics of hypercompleteness; in the process, we set some no-

tation. In § 1.2 we review Postnikov completeness. In § 1.3, we recall the basic setup of con-

densed/pyknotic mathematics.

1.1 Hypercompleteness

In this subsection, we set up some notation and review the basics of hypercompletions of

∞

topoi. We refer the reader unfamiliar with hypercomplete objects and hypercompletion to [HTT,

§§6.5.2–6.5.4], [2, §3.11], or [11, §1.2] for further reading on the subject.

1.1 Notation.

Write

Spc

for the

∞

-category of spaces and

Cat∞

for the

∞

-category of

∞

-cate-

gories.

1.2 Notation. Let 𝒞be an ∞-site and ℰa presentable ∞-category. We write

PSh(𝒞;ℰ)≔Fun(𝒞op,ℰ)

for the

∞

-category of

ℰ

-valued presheaves on

𝒞

. We write

Sh(𝒞;ℰ)⊂PSh(𝒞;ℰ)

for the full

subcategory spanned by ℰ-valued sheaves. When ℰ=Spc, we simply write

PSh(𝒞)≔PSh(𝒞;Spc)and Sh(𝒞)≔Sh(𝒞;Spc).

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DescentforsheavesoncompactHausdorspacesPeterJ.HaineOctober4,2022AbstractThesenotesexplainsomedescentresultsforØ-categoriesofsheavesoncompactHaus-dorspacesandderivesomeconsequences.Specically,givenacompactlyassembledØ-cat-egoryE,weshowthatthefunctorsendingalocallycompactHausdorspaceXtotheØ-cat-eg...

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