Outline
The goal of this seminar is to understand the work of Barwick, Glasman, and Haine on stratified étale homotopy theory, following their paper/book Exodromy [Exo]. This is a generalisation of monodromy representations of ℓ-adic local systems to constructible sheaves.
We will start with the version for stratified topological spaces, where MacPherson defined the exit path category, an analogue of the fundamental groupoid for stratified spaces. Next, we will discuss higher categorical versions due to Treumann and Lurie. We then spend some time reviewing classical étale homotopy theory, and proceed to the stratified étale homotopy type.
The ultimate goal is to get to the exodromy theorem for ℓ-adic constructible sheaves. As in the monodromy case, there is a topology that interacts with the ℓ-adic topology. To make sense of this in a higher categorical context, we need to talk about pyknotic objects, which some of you may know under the different guise of condensed mathematics.
The format is loosely inspired by Akshay's Thursday seminar. We aim for informal talks, and we can take as much time as we need. The seminar should be accessible to graduate students and postdocs, so we plan to spend substantial time on the basics of simplicial homotopy theory and higher category theory (in fact this is one of the reasons for picking this topic).
Talks will be on Tuesdays 1:30–3:00 (ET) through Zoom. A handful of IAS participants will be able to attend in person in S-101, although the speaker (and organiser) will not always be there. There are two ways to attend in person: register in advance by emailing me (Remy), or first come first served.
Pacing
We will start with an overview of the seminar and a discussion of MacPherson's result for stratified topological spaces. Very roughly, the goal of the first semester is to learn enough about simplicial sets, higher category theory, and higher topos theory in order to be able to read [Exo].
In the second semester, we will dive into the work of Barwick, Glasman, and Haine, leading up to a general exodromy result for ∞-topoi stratified over a finite poset or spectral topological space. We then turn to artihmetic aspects: étale homotopy theory, pyknotic and condensed mathematics, and ultimately the étale version of the exodromy correspondence.
The tentative plan outlined here is subject to change, and will be updated regularly. It is similar but not identical to that of the Heidelberg Oberseminar on the same topic (summer 2019).
References
The main references are:
[Exo]
C. Barwick, S. Glasman, P. Haine,
Exodromy.
Additional resources:
[Fri]
E. M. Friedlander, Étale homotopy of simplicial schemes.
Note that Treumann also has a thesis with the same title, which follows different numbering and contains more material.
Schedule
Because of the new member talks, we will not start until the first full week of October.
The schedule below will be updated regularly, and I'm open to suggestions!
Oct
13
Introduction to stratified spaces [Exo, Ch. 1], definition of exit path category [Treu, §7], 1-categorical exodromy equivalence [Exo, §0.3], sketch of proof [Treu, §6–7], [CP, §6].
Notes.
Oct
20
Simplicial sets, geometric realisation, singular simplices, Kan complexes and Kan fibrations, anodyne extensions, weak equivalences, examples [GJ, Ch. I], [Kerodon, Tag 00SZ]. Model categories [Hovey, Ch. 1], [GJ, §I.9, §II.1], [HTT, A.2.1], examples: simplicial sets, Quillen equivalence [Hovey, Def. 1.3.12] with the topological version [Hovey, Thm. 3.6.7], [GJ, Thm. I.11.4]. Injective and projective model structures on chain complexes of R-modules [Hovey, §2.3].
Part II: Bousfield localisation.
Oct
27
Define the nerve of a category and give some examples, describe the essential image in terms of the unique inner horn condition [HTT, Prop. 1.1.2.2], [Rezk, Prop. 5.7]. Define quasi-categories and compare with categories as well as Kan complexes.
Explain variants of the nerve construction: the simplicial nerve [HTT, 1.1.5.5], [Kerodon, Tag 00KM] and the differential graded nerve [Kerodon, Tag 00PK]. (These will be our main ways to produce quasi-categories.) Define the ∞-category of spaces [HTT, §1.2.16] and describe its low-dimensional simplices.
Pick some topics from [HTT, §1.2] to discuss, e.g. Hom spaces, homotopy category, joins and left and right cones.
Nov
10
Allen and Chuck will finish their talks.
Nov
17
Left, right, and inner fibrations, the covariant model structure, straightening/unstraightening in the unmarked setting [HTT, §2.1–2.2.3]. The main result is Theorem 2.2.1.2.
Nov
24
Define the Joyal model structure, prove that it is a model structure, and compare with simplicial categories [HTT, §2.2.4–2.2.5, §A.3.2]. The main results are Proposition 2.2.4.1, Theorem 2.2.5.1, and Proposition 2.2.5.8.
Dec
1
These play an important role in the theory, analogous to fibred categories (e.g. stacks) in 1-category theory.
Define Cartesian fibrations and discuss basic properties [HTT, 2.4.1.1–2.4.1.8], discuss the relation with (homotopy) fibres [HTT, §2.4.4], application to slice categories [HTT, §2.4.5], and determination of fibrant objects in the Joyal model structure [HTT, §2.4.6].
Dec
8
Discuss the Cartesian model structure [HTT, §3.1] and the marked version of straightening/unstraightening [HTT, §3.2].
Some of the main results: [HTT, Prop. 3.1.3.5 and 3.1.3.7], [HTT, Thm. 3.2.0.1], [HTT, Prop. 3.1.1.6]. Give proofs for some of these.
Dec
15
Define limits and colimits in quasicategories [HTT, §1.2.13] and their homotopy analogues in model categories [HTT, Rmk. A.2.8.8], and discuss the equivalence between these two notions [HTT, Thm. 4.2.4.1]. Compute some examples in the ∞-category of spaces.
Some other results that will be useful in the future: [HTT, Prop. 4.2.3.14 and Prop. 4.1.1.8], [HTT, Prop. 4.2.1.4–Cor. 4.2.1.8], and [HTT, Prop. 4.2.2.4 and Prop. 4.2.2.7]. Try to discuss some of these.
Spring 2021
Jan
12
Define the ∞-category of preseheaves of sets on an ∞-category [HTT, Def. 5.1.0.1], describe other models for this category [HTT, §5.1.1], show that limits and colimits exist and can be computed componentwise [HTT, §5.1.2], prove the Yoneda lemma and show that it preserves limits [HTT, §5.1.3], and show the universal property of the ∞-category of presheaves [HTT, Thm. 5.1.5.6].
Jan
19
Define adjoint functors [HTT, Def. 5.2.2.1], show that adjoints are unique up to homotopy [HTT, Rmk. 5.2.2.2], describe the equivalent point of view using a unit [HTT, Prop. 5.2.2.8], show that left adjoints preserve colimits [HTT, Prop. 5.2.3.5], and discuss some examples coming from the various model structures [HTT, Prop. 5.2.4.6].
Time permitting, show that adjoints uniquely determine each other up to contractible choice [HTT, Prop. 5.2.6.2].
Jan
26
Definitions of accessible and presentable ∞-categories, leading up to the adjoint functor theorem [HTT, Cor. 5.5.2.9].
Along the way, discuss some of the following key results: accessibility of functor ∞-categories [HTT, Prop. 5.4.4.3], accessibility of undercategories [HTT, Cor. 5.4.5.16], Simpson's characterisation of presentable ∞-catogries [HTT, Thm. 5.5.1.1], completeness of presentable ∞-categories [HTT, Cor. 5.5.2.4].
Feb
2
Definition of ∞-topos [HTT, Def. 6.1.0.4], statement of Giraud's axioms [HTT, Thm. 6.1.0.6] including all definitions [HTT, §6.1.1]. Define the ∞-topos of sheaves on a Grothendieck topology [HTT, Prop. 6.2.2.7], and maybe say a word on Grothendieck pretopologies (which are more familiar to algebraic geometers).
We will be interested in a special type of ∞-topoi called coherent ∞-topoi. Explain the definition [SAG, Def. A.2.1.6], show that a finitary Grothendieck topology gives a coherent ∞-topos [SAG, Prop. A.3.1.3], and show that morphisms of finitary Grothendieck topologies give coherent morphisms of ∞-topoi [Haine, Cor. 2.9]. Discuss key examples from algebraic geometry [Haine, Prop. 2.20 and Ex. 2.22]. See also [Exo, §3.3–3.7].
Feb
9
An overview of what's coming in the second half of this seminar: ∞-categorical exodromy equivalence, Galois categories, pyknotic/condensed mathematics, and the ℓ-adic exodromy theorem.
We will recall Alexandroff, Stone, and Hochster duality in the unstratified case [Exo, §1], and make a start to understanding the homotopy theory of stratified topological spaces [Exo, §2].
Feb
16
Complete Segal spaces form an altenative model for ∞-categories. Introduce them and say some words on how they relate to quasi-categories (after Joyal–Tierney).
This then gives an alternative viewpoint on the homotopy theory of statified spaces through spatial décollages [Exo, §2.6–2.8].
Feb
23
Try to explain key examples of ∞-topoi: topological spaces [Exo, Ex. 3.2.5], sites from algebraic geometry [Exo, Ex. 3.2.7, §3.7], spaces over a given ∞-groupoid [Exo, Ex. 3.2.8], posets [Exo, Ex. 3.12.15, 8.1.1]. In each case, discuss some of the adjectives (localic, bounded, coherent, Postnikov complete, hypercomplete): why is the small étale topos 1-localic? What is an example of an ∞-topos that isn't? State some of the implications between different properties [Exo, Ex. 3.3.6, Ex. 3.11.10, Ex. 3.11.12, Thm. 3.11.14, Cor. 3.11.17].
Explain basic constructions of ∞-topoi: points, global sections, étale geometric morphisms, products, fibre products, and filtered limits [Exo, §3.1, Cor. 3.9.4].
State some other results without proof: [Exo, Ex. 3.3.5, Thm. 3.8.9]. Time permitting, state the conceptual completeness and Deligne completness results [Exo, §3.11].
(It may be useful to recall truncated and connective objects [HTT, Def. 5.5.6.1, Def. 6.5.1.10].)
Mar
2
Define the shape functor [Exo, Def. 4.2.1] and its profinite analogue [Exo, Def. 4.4.2]. Time permetting, say some words about the proof of ∞-categorical Stone duality [Exo, Thm. 4.4.3], [SAG, Thm. E.2.4.1].
This leads to the notion of a Stone ∞-topos [Exo, Def. 4.4.4]. Discuss some of the alternative characterisations [Exo, Thm. 4.4.10 and Thm. 4.4.14] and deduce the monodromy equivalence for Stone ∞-topoi [Exo, Prop. 4.4.18].
In the exodromy story for stratified spaces, Stone spaces will play the role of trivially stratified spaces (i.e. with only one stratum).
Mar
9
Oriented pushouts of ∞-topoi correspond to 'recollements' of a space (or ∞-topos) from a complementary open and closed set inside it. Oriented pullbacks generalise a construction that Deligne used to study the nearby cycles functor.
Describe recollements [Exo, §5.1] and their universal property as oriented pushouts [Exo, §5.2, Lemma 5.1.14]. Descibe the mapping ∞-topos and path ∞-topos [Exo, §5.3] and use them to construct oriented fibre products of ∞-topoi [Exo, §5.4]. Describe Deligne's generating site for the oriented fibre product [Exo, §5.5] and show that it is again bounded coherent [Exo, Lemma 5.5.19].
Time permitting, discuss the étale base change property for oriented fibre products [Exo, Prop. 5.6.5].
Mar
16
Local ∞-topoi are modelled after henselian local rings in étale cohomology, and play a similar foundational role in a lot of arguments.
Define local ∞-topoi [Exo, §6.2] and localisations [Exo, Def. 6.3.7 and Prop. 6.5.3], and show compatibility with fibre products [Exo, Prop. 6.4.2]. State the result that localisations preserve bounded coherence [Exo, Lemma 6.6.4] (and sketch a proof if you have time).
Define the Beck–Chevalley conditions for base change squares [Exo, Def. 7.1.1] and state the main base change result [Exo, Thm. 7.1.7]. Time permitting, either give some examples [Exo, §7.2–7.4] or say some words about the proof [Exo, §7.5–7.7].
Mar
23
Define ∞-topoi stratified over a spectral topological space [Exo, Def. 8.2.1] and explain what this means in the key case where the base is a Sierpinski space [Exo, 5.1.4, Ex. 8.2.6]. Say some words about constructibility and coherence, e.g. [Exo, Lemma 8.3.9].
Introduce glueing squares [Exo, Def. 8.5.1], explain the two key examples [Exo, 8.5.2 and 8.5.8], and show how this turns spatial décollages into toposic décollages [Exo, Ex. 8.6.5 and Prop. 8.6.6]. On the other hand, a stratified ∞-topos also gives a toposic décollage via the nerve functor [Exo, Cons. 8.7.1]. Sketch the proof that this is an equivalence [Exo, Thm. 8.7.3].
Finally, apply this to profinite stratified spaces and show that the obtained nerve functor is fully faithful [Exo, Prop. 8.8.6].
Mar
30
Introduce spectral ∞-topoi and characterise them [Exo, Prop. 9.2.5] using Lurie's characterisation of Stone ∞-topoi [Exo, 4.4.10].
Then proceed to the main results of the seminar: Hochster duality for ∞-topoi [Exo, Thm. 9.3.1], recognition of constructible objects [Exo, Prop. 9.5.4], and finally the exodromy equivalence [Exo, Thm. 10.1.8].
Time permitting, cover any other aspects of §9–10 that you find interesting.
Apr
6
Define the Galois category of a scheme [Exo, Def. 12.1.3] and describe it explicitly in terms of Galois groups [Exo, 12.1.5]. Deduce from the previous talk the exodromy correspondence for schemes [Exo, Thm. 12.1.6].
Review the étale homotopy type in the sense of Artin–Mazur [Exo, §11.1] and Friedlander [Exo, §11.4], and explain some examples [Exo, §11.2]. Show how the Galois category recovers the étale homotopy type [Exo, Thm. 12.5.1] (you may need to explain §10.2 if the previous speaker hasn't).
Finally, prove the stratified Riemann existence theorem [Exo, Cor. 12.6.6–12.6.7].
Apr
13
Introduce pyknotic objects and pyknotic categories following [Exo, §13.3] and the references therein, and compare with condensed mathematics of Clausen–Scholze. Show how objects like Qₗ and its algebraic closure can naturally be viewed as pyknotic rings.
Extend the exodromy equivalence for sheaves of spaces on a scheme to perfect complexes over a finite ring [Exo, §13.2, §13.6], and finally prove the ℓ-adic exodromy theorem [Exo, Thm. 13.8.8].
