Motivic homotopy theory of algebraic stacks

This semester the motives seminar will look at extensions of the stable homotopy category to a theory on suitable algebraic stacks, together with a six-functor formalism. Stacks as generalisations of schemes occur in many natural settings; one that is quite important is their role in the theory of moduli. Many invariants arising in modern enumerative geometry rely on constructing virtual fundamental classes in the Chow ring of suitable moduli stacks, if one wishes to refine these invariants to theories such as the Chow-Witt groups, hermitian K-theory, or the cohomology of the Witt sheaves, one needs to extend the foundations of these theories from schemes to stacks. Here we will look at a way of defining motivic stable homotopy categories for stacks using the Nisnevich topology, which should allow one to obtain these finer invariants.

The program consists of a quick overview of some of the necessary material about infinity categories before starting the main block on the category of stacks with good properties for the Nisnevich topology, and Chirantan Chowdhury's construction of the motivic stable homotopy category for such stacks. This will occupy most of the semester. We will also look at some related theories, such as Hoyois' construction of equivariant versions of the motivic stable homotopy category, as well as work of Khan-Ravi on other approaches to the theory. .

We will hold the seminar "in person'' with an online link via Zoom, so all those who we were happy to have take part from a distance will still be able to do so. We meet on Tuesdays, 16-18 Uhr in WSC-S-U-3.01. If you are interested in attending the seminar or giving one of the lectures, please contact me (Marc Levine) at

Previous Motives Seminars

Follow the Teaching link at the top of this page to find links to previous motives seminars.

Program: Motivic homotopy theory of algebraic stacks

For a detailed program, please follow the link Seminar Program. We will post lecture notes and videos of the lectures on this page.

Lecture 0 (12.10-Chirantan Chowdhury) Introduction and overview
Lecture 0 Video

Lecture 1 (19.10-Herman Rohrbach) Introduction to $\infty$-categories
Lecture 1 Video

We recall the basic notion of $\infty$-categories. We shall begin with recalling the definition and introduce basic terminologies like objects, morphisms, compositions, homotopy category, initial/final objects, limits and mapping spaces. The two main examples of $\infty$-categories that we are interested in arise from simplicial categories and $2$-categories. We shall end the talk by recalling the notion of fibrations of simplicial sets, categorical equivalence and Cartesian fibrations.

References: [Lur09, Chapter 1-2], [Lur18a, Chapter 1, 2.2,2.3] and [Rez, Part 4].

Lecture 2 (26.10-Marc Levine) Presentable stable $\infty$-categories

We define the notion of presentability and stability in the setting of $\infty$-categories. The talk shall start by recalling the notion of presheaves, Yoneda lemma and adjoint functors in the setting of $\infty$-categories. We then move on defining presentable $\infty$-categories by filtered categories and ind-objects. We briefly recall the notion of $\infty$-sheaves. The talk ends by defining the notion of stable $\infty$-categories and the $\infty$-category of presentable stable $\infty$-categories $\text{Pr}^L_{stb}$.

References: [Lur09], [Lur17] and [Lur18b].

Lecture 3 (02.11-Manuel Hoff) Symmetric monoidal $\infty$-categories and module objects.

We define the algebra and module objects in the setting of $\infty$-categories. In particular, we are interested in defining symmetric monoidal $\infty$-categories. In order to make sense of these notions, we start by defining $\infty$-operads which are generalized notions of colored operads. Symmetric monoidal $\infty$-categories are special kinds of $\infty$-operads. The talk ends by defining module objects over $\infty$-operads and explaning a higher categorical generalization of the fact that a morphism between algebra objects $A \to B$ gives $B$ an $A$-module structure.

References: [Lur17, Chapter 2,3] and [Rob14, Section].

Lecture 4 (09.11-Dhyan Aranha) Motivic homotopy theory of schemes

We define the motivic homotopy theory of schemes in the language of $\infty$-categories. We shall briefly recall the definition of unstable, pointed and stable homotopy theory. Then we recall the functoriality and six operations. We shall state the existence of the exceptional pushforward functors without proving them. This shall be later explained while discussing the enhanced operation map in Talk 10. The talk ends by stating properties like localization, homotopy purity, homotopy invariance and explaining the construction of $\alpha_f$ and purity transformation $\rho_f$.

References: [Rob14], [CD19] and [Hoy17].

Lecture 5 (16.11-Anneloes Viergever) Algebraic stacks

We discuss the notion of algebraic spaces and algebraic stacks. We also discuss some examples of algebraic stacks like local quotient stacks, moduli stack of vector bundles. We also define the deformation to the normal cone in the setting of algebraic stacks. The talk ends by discussing the notion of resolution property of algebraic stacks.

References:[Sta21], [Knu71], [Hei10], [LMB00] [Tot04] and [Gro17].

Lecture 6 (23.11-Pietro Gigli) Descent along sections

In this talk, we recall the $\infty$-categorical setup generalizing the classical statement that descent along morphisms admitting sections is automatic. The talks by recalling the classical statement in ordinary category theory using the notion of split forks. Then we move in explaining the skeletal descriptions of simplicial category $\Delta$, augmented simplicial category $\Delta_+$ and the split-simplicial category $\Delta_{-\infty}$. The talks end by stating the main result which is due to [Lur09, Lemma].

References:[Lur09, Section 6.1] and [Cho, Chapter 2].

Lecture 7 (30.11-Jan Hennig) Kan extensions, descent theory and localization of $\infty$-categories

In this lecture, we recall some important notions in higher category theory that we need for proving [Cho, Theorem 3.4.1], which allows us to extend sheaves from schemes to a large class of algebraic stacks. We start the talk by introducing the notion of Kan extensions in the setting of $\infty$-categories. Kan extensions allow us to associate limits of diagrams in a functorial manner. The second part of the lecture introduces the notion of descent theory and states specific conditions when $\infty$-sheaves can be realised by descent along Čech nerves of coverings. The last part of the lecture reviews localization of $\infty$-categories and explains the existence of localization along any class of morphisms.

References:[Lur09, Section 4.2,5.5], [Lur18b, Appendix A 3.1-3.3],[Cho, Appendix A] and [Lan21, Section 2.4].

Lecture 8 (07.12-Gabriela Guzman) Enhancement of sheaves along coverings with local sections

We prove [Cho, Theorem 3.4.1] and thus extend the stable homotopy functor $\mathcal{SH}^{\otimes}(-)$ from schemes to the $(2,1)$-category $\text{Nis-locSt}$. The talk starts by introducing $\mathcal{T}$-local sections in a site $(\mathcal{C},\mathcal{T})$ and stating some properties. We then move to defining the $(2,1)$-category of stacks admitting $\mathcal{T}$-local sections of which the category of qcqs algebraic spaces and the $(2,1)$-category $\text{Nis-locSt}$ are examples. We state[Cho, Theorem 3.4.1] and explain the proof using the theory of Kan extensions and localizations explained in the previous lecture. The lecture ends defining the functor $\mathcal{SH}^{\otimes}_{\text{ext}}(-)$ ([Cho, Corollary 3.5.3]) and constructing the four functors $f^*,f_*,-\otimes-$ and $\text{Hom}(-,-)$ on $\text{Nis-locSt}$.

References:[Cho, Chapter 3].

Lecture 9 (14.12-Alessandro D'Angelo) Compactification in the setting of $\infty$-categories

We briefly describe the idea of Deligne's compactification in $\infty$-categorical setting due to Liu and Zheng [LZ12]. The talks starts with briefly recalling Deligne's argument of constructing the exceptional pushforward $f_!$ in étale cohomology. The rest of talk introduces the terminology of multi-marked and multi-tiled simplicial sets which is needed to state the theorem of $\infty$-categorical compactification ([LZ12, Theorem 0.1]) and to construct the enhanced operation map (which shall be done in the next talk). The talk ends with a brief sketch of the idea of [LZ12, Theorem 0.1].

References:[Del73, Section 3][LZ12] and [Cho, Section 4.2, Appendix D].

Lecture 10 (21.12-Viktor Kleen) Enhanced operations for stable homotopy theory of algebraic stacks

We extend the exceptional pushforward and pullback functors $f_!$ and $f^!$ from schemes to $\text{Nis-locSt}$. This is extended by the so called enhanced operation map due to Liu and Zheng [LZ17, Section 2.2]. The talk starts with recalling the setting of six operations with smooth (and proper) base change and projection formula on the level of schemes and constructs the enhanced operation map. Then we explain how the enhanced operation map encodes the exceptional pushforward functors, base change and projection formula. The rest of talk deals with explaining extending the enhanced operation map from schemes to $\text{Nis-locSt}$ ([Cho, Proposition 4.4.2]) with a brief sketch of the proof.

References:[Cho, Chapter 4, Appendix D], [LZ17, Section 1, Section 3] and [Rob14, Section 9.4].

Lecture 11 (11.01.22) Six operations for $\mathcal{SH}_{\text{ext}}(\mathcal{X})$

In this talk, we prove relations between the six operations that we enocuntered in the previous talks in particular; localization, homotopy invariance and homotopy purity. The talk starts with proving smooth and proper base change and then move on to prove localization and homotopy invariance. We construct the natural transformations $\alpha_f$ and $\rho_f$. The talks ends with proving the homotopy purity theorem via the deformation to the normal cone.

References: [Cho, Chapter 5].

Lecture 12 (18.01) Equivariant motivic homotopy theory

We recall the equivariant motivic homotopy theory due to Hoyois [Hoy17]. The talk starts with defining the unstable homotopy category $H^G(S)$ and stating the importance of tame condition of group scheme while constructing the purity isomorphism. We state functoriality,smooth and proper projection formula and define the pointed motivic homotopy category $H^G_{\bullet}(S)$ and stating the unstable ambidexterity map. The talk ends with defining the equivariant stable homotopy theory $\text{SH}^G(S)$ and stating the six operations and descent properties of $\text{SH}^G(-)$.

References: [Hoy17].

Lecture 13 (25.01) Generalized cohomology theories of algebraic stacks

We recall the construction of generalized cohomology theories on scalloped algebraic stacks and also define limit-extended cohomology theories.

References: [KR21].

Lecture 14 (01.02) Just in case.



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