Introduction

The stable motivic homotopy categories of schemes are nice ∞-categories, which admit a whole six-functor formalism. For a smooth morphism of schemes f: X→Y over a base S, the usual pullback functor f*: SH(Y)→SH(X) admits a left adjoint f_♯, the additive pushforward; they are packaged in a functor from an ∞-category of spans.

Recently, Bachmann and Hoyois discovered a multiplicative pushforward, or norm g_⊗: SH(X)→SH(Y), for g: X→Y finite étale (generalises the tensor product). These satisfy a distributivity property, and are encoded by a functor from an (∞, 2)-category of bispans. The norm functors categorify Rost’s multiplicative transfers on Grothendieck–Witt rings and are an enhancement of motivic E_∞-ring spectra; similar distributivity for additive and multiplicative transfers on genuine G-spectra played a key role in the solution of the Kervaire invariant one problem by Hill, Hopkins, and Ravenel. In this course, we introduce the notion of bispans, whose universal property categorifies distributivity in commutative (semi)rings, following Elmanto-Haugseng. We will then discuss some important examples in motivic homotopy theory. Most part will be of interest to quite general audiences.

Syllabus

1. Some higher categorical preliminaries (collecting some important facts about ∞-category, to be used as our built-in language)

2. The (∞, 2)-category of spans and bispans (discussing their universal properties)

3. Perspectives on motivic homotopy theory

4. Norm structures

5. Examples (from representation theory/equivariant homotopy theory/spectral DM stack/K-theory...)

Reference

[1] Tom Bachmann, Marc Hoyois, Norms in motivic homotopy theory, Astérisque 425. Paris: Société Mathématique de France (SMF). ix, 207~p. (2021).

[2] Elden Elmanto, Rune Haugseng, On distributivity in higher algebra I: The universal property of bispans, Compositio Mathematica, Volume 159 , Issue 11 (2023) , pp. 2326-2415.

[3] Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies (Book 170), Princeton University Press, 2009.

[4] Jacob Lurie, Kerodon, an online resource for homotopy-coherent mathematics. Available at https: //kerodon.net.