On Postnikov completeness for replete topoi

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ON POSTNIKOV COMPLETENESS FOR REPLETE TOPOI

SHUBHODIP MONDAL AND EMANUEL REINECKE

Abstract.

We show that the hypercomplete

∞

-topos associated with any replete topos is Postnikov

complete, positively answering a question of Bhatt and Scholze; this will be deduced from the Milnor

sequences for sheaves of spaces on replete topoi that we construct. As a corollary, we generalize a

result of To¨en on aﬃne stacks.

1. Introduction

In their seminal work [

BS15

], Bhatt–Scholze introduced the pro-´etale site of a scheme, which simpliﬁes

many foundational constructions in the theory of

ℓ

-adic cohomology and has been fundamental to

the recent development of condensed mathematics [

BH19

Sch19

]. One of the key properties of the

associated 1-topos is captured by the notion of replete topoi, which they study in [

BS15

,§3]. We

recall this deﬁnition here.

Deﬁnition 1.1 ([

BS15

, Def. 3.1.1]).A 1-topos

is replete if for every diagram

≥0→ X

with the

property that

Fn+1 →Fn

is surjective for all

, the natural map

lim F→Fn

is surjective for every

Sheaves of spaces on a 1-topos

(see e.g. [

Lur09

,§6.5]) form an

∞

-topos, which we denote by

Shv∞

(

). Let us denote the associated hypercomplete

∞

-topos by

Shv∞

(

)

∧

. One may think of

objects of

Shv∞

(

)

∧

as sheaves of spaces that satisfy hyperdescent. In [

BS15

, Qn. 3.1.12], the authors

ask the following question:

Question 1.2. Let

be a replete topos. Do Postnikov towers converge in the hypercomplete

∞

-topos

Shv∞(X)∧?

One important feature of the above question is that it does not impose any ﬁniteness assumptions

(such as homotopy dimension

≤n

[

Lur09

, Def. 7.2.1.1] or cohomological dimension

≤n

[

Lur09

Cor. 7.2.2.30]) on the

∞

-topoi that appear in the previously known criteria for convergence of

Postnikov towers (see Remark 3.19 and Remark 3.20).

Certain stable analogs of Question 1.2 can already be found in the literature. In [

BS15

, Prop. 3.3.3],

the authors show that if

is a replete topos, the derived

∞

-category

(

Z) (or in other words,

hypercomplete sheaves of

Z-module spectra) is left complete. In [

Mat21

, Prop. A.10], the case of

sheaves of spectra on certain large sites (which satisfy an additional “quasi-compactness” condition and

admit countable ﬁltered limits) is addressed and used in the context of the arc-topology. Regarding

Question 1.2 itself, in [

BS15

, Prop. 3.2.3], the authors show that Postnikov towers converge in

Shv∞

(

)

∧

, when

is additionally assumed to be locally weakly contractible (see [

BS15

, Def. 3.2.1]).

The pro-´etale topos of a scheme is an example of a locally weakly contractible topos [

BS15

, Prop. 4.2.8].

In this paper, we answer Question 1.2 in general.

Theorem A. Let

be a replete topos. Then the hypercomplete

∞

-topos

Shv∞

(

)

∧

is Postnikov

complete.

Remark 1.3. In Theorem A, we prove that

Shv∞

(

)

∧

is Postnikov complete in the sense of Deﬁni-

tion 3.5; this is stronger than just requiring that the natural map

X→limnτ≤nX

is an equivalence

for every X∈ X .

2020 Mathematics Subject Classiﬁcation. 18N60, 18F10, 14F06.

arXiv:2210.14146v2 [math.AT] 27 May 2024

2 SHUBHODIP MONDAL AND EMANUEL REINECKE

Let us brieﬂy explain the main ingredients in the proof of Theorem A. Let

be a replete topos. We

need to prove that any tower in

Shv∞

(

)

∧

which “looks like” a Postnikov tower is in fact a Postnikov

tower (see Lemma 3.7). As explained in detail in the proof of Theorem A, this follows more or less

directly from Milnor sequences, which allow one to calculate homotopy groups of an N-indexed inverse

limit in the expected way.

However, it is quite subtle to have a theory of Milnor sequences in general since sheaﬁﬁcation does

not commute with inﬁnite limits (see [

To¨e06

, p. 21] and Remark 3.23). In fact, it is already unclear if

π0

commutes with N-indexed direct products in

Shv∞

(

). Let us consider the example arising from

condensed mathematics when

is the pro-´etale site of a point [

Sch19

, Def. 1.2]. In that case, one

can get around this issue by realizing that a sheaf is essentially the same as a presheaf on extremally

disconnected sets that sends ﬁnite disjoint unions to products [

Sch19

, Prop. 2.7]. Roughly speaking,

presheaves on the pro-´etale site of a point that send coproducts to products are essentially as good as

sheaves.

Motivated by this property, for a general replete topos we work with a class of abstract presheaves

called multiplicative presheaves (see Deﬁnition 2.5), which send arbitrary disjoint unions to products.

We observe in Proposition 2.14 that sheaﬁﬁcation commutes with N-indexed products of multiplicative

presheaves, overcoming the subtleties discussed above and leading to the existence of Milnor sequences

(see Proposition 3.22).

Remark 1.4. Let

be a replete topos. One might wonder whether

Shv∞

(

) is automatically

hypercomplete. In Example 3.29, we show that this is false even if one assumes

to be locally weakly

contractible. This also implies that

Shv∞

(

) is not Postnikov complete in general (see Remark 3.16).

Example 1.5. Many topoi naturally appearing in algebraic geometry are replete. Theorem A is

applicable in all such contexts, without having to make any assumptions on the ﬁniteness of homotopy

or cohomological dimension. For instance (after ﬁxing set-theoretic issues, see e.g. the footnote in

[

BS15

, Ex. 3.1.7]), topoi coming from the fpqc-topology,

-topology (see e.g. [

SP22

, Tag 0EVM]), and

the quasisyntomic topology [

BMS19

, Def. 4.10] are all replete, but in general not of ﬁnite cohomological

or homotopy dimension: as a concrete example, arc-descent for the ´etale cohomology of torsion sheaves

[BM21, Thm. 5.4] shows that

H∗(Spec(R)fpqc,Z/2) ≃H∗(Spec(R)v,Z/2) ≃H∗(Spec(R)´et,Z/2).

The latter is computed by the group cohomology ring

H∗

2), which is a symmetric algebra on

2) and thus nontrivial in all degrees (cf. Example 3.17). On the other hand, smaller topoi

whose covers are subject to certain ﬁniteness conditions (such as Zariski, ´etale, fppf) tend to be not

replete (cf. [BS15, Ex. 3.1.5]).

Example 1.6 ([

BS15

,§4.3]).Let

be a proﬁnite group. Let (

)

pro´et

denote the site of proﬁnite

sets with a continuous

-action, with covers given by continuous surjections. Then

Shv

(

)

pro´et

is replete. As a special case of Theorem A,

Shv∞

(

)

pro´et∧

is Postnikov complete. However,

this fails for non-replete variants of this topos that do not take the proﬁnite topology into account;

cf. Example 3.17.

To exhibit another consequence of Theorem A, let us mention the following application to aﬃne

stacks in the sense of [To¨e06, §2.2].

Corollary 1.7 (Corollary 3.28).Let

be an aﬃne stack over

Spec B

for any ring

. Then the

natural map F→limnτ≤nFis an equivalence.

Previously, the above corollary was only known under certain assumptions (Remark 3.27) and the

proof relied on the vanishing of cohomology groups of aﬃne schemes with coeﬃcients in unipotent

group schemes in degrees

1. As we see now, a more general result follows from Theorem A simply as

a consequence of repleteness. The Milnor sequences for replete topoi that appears in this paper will

also be used in [MR23].

ON POSTNIKOV COMPLETENESS FOR REPLETE TOPOI 3

Acknowledgments. We thank Bhargav Bhatt, Peter Haine, Akhil Mathew, and Peter Scholze for

helpful comments and conversations. Special thanks are due to the referee for many suggestions that

led to simpliﬁcations of the proofs and the exposition. We are grateful to the Max Planck Institute

for Mathematics (Bonn, Germany) and the Institute for Advanced Studies (Princeton, USA) for

their support during the preparation of this work. Additionally, Mondal acknowledges support from

the University of Michigan, the NSF Grant DMS #1801689, FRG #1952399 and the University of

British Columbia (Vancouver, Canada) and Reinecke acknowledges support from the NSF Grant DMS

#1926686.

2. Multiplicative presheaves on Grothendieck sites

In this section, we introduce a key class of objects for our paper, which we call “multiplicative

presheaves.” Roughly speaking, they capture the notion of a presheaf that takes arbitrary coproducts

to products. However, a site may fail to have arbitrary coproducts. To make the notion precise, we

will therefore need to phrase this condition in the appropriate category. Before formally introducing

multiplicative presheaves, let us ﬁx some notation.

Notation 2.1. Let

be a Grothendieck site. The Yoneda embedding gives a natural fully faithful

functor

h:T → PShv

(

). Composition with the sheaﬁﬁcation functor yields a functor

h♯:T → Shv

(

If there is no risk of confusion, we write

hX:=h

(

) and

h♯

X:=h♯

(

) for

X∈ T

. We may view a

presheaf

as a functor

P:Top →Set

. By right Kan extension along

h♯

, we obtain an extended

functor Ranh♯(P): Shv(T)op →Set.

Lemma 2.2. Let

be a Grothendieck site and

X∈ T

. Then for all

F∈Shv

(

), we have

Ranh♯(hX)(F) = HomShv(T)F, h♯

X; that is, Ranh♯hXis represented by h♯

Proof.

Follows from the universal property of right Kan extensions. For example, see [

Mac71

, Ex-

ercise X.3.2], which states that the right Kan extension of a (co)representable functor must be

(co)represented by the image of the former (co)representing object. □

The following two lemmas are well known and follow from the universal property of right Kan

extensions and the adjoint functor theorem.

Lemma 2.3. Let

be a Grothendieck site. If a functor

P: Shv

(

)

op →Set

preserves all small limits,

then P◦h♯:Top →Set is a sheaf. Further, there is a natural isomorphism Ranh♯(P◦h♯)≃P.

Lemma 2.4. Let

be a Grothendieck site. If

P:Top →Set

is a sheaf, then

Ranh♯

(

)

: Shv

(

)

op →

Set preserves all small limits. Further, there is a natural isomorphism Ranh♯(P)◦h♯≃P.

Now we are ready to introduce the notion of multiplicative presheaves.

Deﬁnition 2.5 (Multiplicative presheaves).Let

be a Grothendieck site. Let

be a presheaf on

We say that Pis a multiplicative presheaf if for every set of objects {Xj}j∈J∈ T for an indexing set

J, the natural map of presheaves `j∈JhXj→`j∈JhXj♯induces an isomorphism

HomPShv(T)a

j∈J

hXj♯

, P ∼

−−→ HomPShv(T)a

j∈J

hXj, P ≃Y

j∈J

P(Xj).

Remark 2.6. It follows directly from the deﬁnition that limits of multiplicative presheaves are again

multiplicative.

Example 2.7. By the universal property of sheaﬁﬁcation, every sheaf is an example of a multiplicative

presheaf.

The following proposition provides a useful source of multiplicative presheaves.

Proposition 2.8. Let

be a Grothendieck site and

P′: Shv

(

)

op →Set

is a product preserving

functor. Let P:Top →Set denote the functor P′◦h♯. Then Pis a multiplicative presheaf.

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ONPOSTNIKOVCOMPLETENESSFORREPLETETOPOISHUBHODIPMONDALANDEMANUELREINECKEAbstract.Weshowthatthehypercomplete∞-toposassociatedwithanyrepletetoposisPostnikovcomplete,positivelyansweringaquestionofBhattandScholze;thiswillbededucedfromtheMilnorsequencesforsheavesofspacesonrepletetopoithatweconstruct.Asaco...

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