$ \newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TP{\mathrm{TP}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} $

What

A highly focused workshop on recent developments in $\omega$-categories. On 16-17 March, it will take place at the NITMB in downtown Chicago. On 18 March it will take place at the University of Chicago in Ryerson 251.

Mini-courses

David Gepner (JHU) and Hadrian Heine (MPIM).

Naruki Masuda (NU).

German Stefánich (MPIM).

Research talks

Ko Aoki (MPIM).

Tim Campion.

Tomer Schlank (Chicago).

NITMB Colloquium

David Spivak (Topos Institute).

Schedule

M Tu W
0930 Gepner/Heine 1 0930 Stefanich 2 0930 Gepner/Heine 3
1100 Stefanich 1 1100 Gepner/Heine 2 1100 Stefanich 3
1200 Lunch 1200 Lunch 1200 Lunch
1400 Campion 1400 Masuda 2 1400 Aoki
1530 Coffee 1530 Coffee 1530 Coffee
1600 Masuda 1 1600 Spivak Colloquium 1600 Schlank

Titles

Aoki. (Commutative) higher motives. [pdf notes]

Campion. The universal property of the Gray tensor product. [pdf notes] [more notes]

Gepner/Heine 1. Towards the categorification of homotopy theory. [pdf notes] [more notes]

Gepner/Heine 2. Hypercompletion, Postnikov towers, and coinductive equivalences. [pdf notes]

Gepner/Heine 3. Homotopy posets and the oriented exact sequence. [pdf notes]

Masuda 1. Categorical spectra and stability of oriented categories. [pdf notes] [both talks]

Masuda 2. Categorical spectra 2. [pdf notes] [more notes]

Schlank. Height, additivity, Galois, and categorification. [pdf notes]

Spivak. Accounting for our inventiveness. [slides]

Stefanich. Higher algebraic geometry. [talk 1] [talk 2] [talk 3] [all talks]

Abstracts

Aoki. (Commutative) higher motives. Arithmetic geometry studies systems of polynomial equations by associating geometric objects called varieties. To study varieties, one uses various cohomology theories. Grothendieck introduced the idea of motives to capture the common cohomological content of these theories.

My thesis connects this theory with higher presentable category theory. More precisely, one of the main theorems shows that the stable presentably symmetric monoidal (∞,2)-category freely generated by a smooth structured ring (1-affine) stack with trivial homology coincides with the presentable (∞,2)-category of kernels arising in Morel–Voevodsky’s motivic homotopy theory. This confirms Scholze’s expectation.

I will begin by explaining classical 1-categorical motives and then describe how higher methods naturally arise in this context. The thesis also proves an analogous result for Scholze’s Berkovich motives, which provide a theory of motives for analytic varieties (i.e., allowing inequalities). If time permits, I will explain why this analytic version of the (∞,2)-category of 2-motives behaves better than its algebraic counterpart, in particular in light of my higher rigidity theorem.

Campion. The universal property of the Gray tensor product. A strong monoidal left adjoint ωCat^Gray $\rightarrow (A , \otimes, I)$ is a 1-cocategory object Γ in A whose coobject part is identified with I, such that for all $a \in A$, the double category $A( Γ \otimes Γ , a)$ has all companions.

Spivak. Accounting for our inventiveness. This talk is written to be heard in two registers: one that category theorists can trust and another that biologists can find motivating. These two audiences may seem to have very little in common. However, every mathematician is a biological entity, and our ability to “do mathematics,” e.g. to invent and employ abstractions, is one that biology had to produce in a stack of technologies that grounds out in bare physics. I’ll try to make clear that humans are not the first biological systems to be inventive: biology is a story rife with inventiveness at every stage. The key inventions all tend to increase portability of form so that it can take hold in new substrates, and this is the heart of abstraction. I’ll propose that the mechanism is accounting: a system develops sensitivities, works to bring them into coherence, and when the accounts settle, what emerges is a new portable capacity. This is sensemaking, and its product is abstraction.

I’ll then pivot to discuss polynomial functors in one variable, which I think is itself a highly portable abstraction. As a category, Poly is easy to generate (the free completely distributive category on a point), yet extraordinarily rich and computationally tractable, with a wide range of monoidal closed structures, (co)free (co)monads, and applications from dependent type theory to interacting dynamical systems. In the remainder of the talk I’ll discuss my attempt to model living systems, and suggest that the accounts are not yet settled: we do not have a formal account of our own physical ability to create new abstractions. I believe this is a problem both audiences can work on.

Stefanich. Higher algebraic geometry. The goal of this mini-course will be to give an introduction to a world of geometry in which one is allowed to consider, not just spectra of commutative rings, but also spectra of symmetric monoidal higher categories. In contrast to usual algebraic geometry, this is a setting where every object is affine in a suitable sense, and thus its geometry can be faithfully described in terms of higher categorical algebra. The talks will feature a mixture of general theory and examples; we will see along the way how to use this theory to geometrize arbitrary six-functor formalisms, and make precise the connection between invertible topological field theories and stable homotopy groups of spheres. Based on joint work with Peter Scholze.

Bibliography

Aoki, Higher presentable categories and limits, arXiv.

Aoki, Berkovich 2-motives and normed ring stacks, arXiv.

Aoki, Barthel, Chedalavada, Schlank, and Stevenson, Higher Zariski geometry, arXiv.

Campion, An $(\infty,n)$-categorical pasting theorem, arXiv.

Campion, The Gray tensor product of $(\infty,n)$-categories, arXiv.

Heine, On the categorification of homology, arXiv.

Gepner and Heine, Oriented category theory, arXiv.

Gepner and Heine, Homotopy posets, Postnikov towers, and hypercompletions of ∞-categories, arXiv.

Masuda, The algebra of categorical spectra, pdf.

Ozornova and Rovelli and Walde, Cores and localizations of (∞,∞)-categories, arXiv.

Stefanich, Categorification of sheaf theory, arXiv.

Stefanich, Presentable $(\infty,n)$-categories, arXiv.

Stefanich, Higher sheaf theory I: correspondences, arXiv.

Organizers

Benjamin Antieau, David Gepner, Naruki Masuda

Registration

The deadline to register has passed. Note that registration is required for access to the building and to the NITMB.

Funding

The deadline to apply for funding has passed.

Lodging

Participants, even those we fund, are asked to book their own lodging and be reimbursed. We will book lodging for speakers. Suitable nearby hotels include the Warwick–Allerton or the Omni Hotel.

Travel

The NITMB is accessible via Amtrak through Chicago Union Station and via plane through Chicago O’Hare or Midway airports. For more details, see the NITMB’s getting here page.

Location

The workshop will take place in the lecture hall at the NITMB on Monday and Tuesday, 16 and 17 March. On Wednesday 18 March it will take place at the University of Chicago’s Hyde Park campus in Ryerson 251.

Restaurants

There are no official meals planned for this workshop. However, the following are nearby restaurants that might be suitable:

  • Beatnik On The River
  • Beatrix
  • Bistronomic
  • Cafecito
  • Can’t Believe It’s Not Meat
  • Chicago Diner
  • Ema
  • Feed Your Head
  • Francesca’s On Chestnut
  • Jake Melnick’s Corner Tap
  • Labriola
  • Lucky Cross
  • Luxbar
  • Pierrot Gourmet
  • Pizzeria Portofino
  • Planta
  • Quartino Restorante
  • RL Restaurant
  • SIFR
  • Somerset
  • Sunny Side Up Restaurant
  • Tempo Cafe
  • The Original Pancake House
  • Wildberry Pancakes and Cafe

Acknowledgments

This conference is supported by the NSF grant DMS-2152235, FRG: higher categorical structures in algebraic geometry, by the Simons Collaboration on Perfection, by the NITMB, by Northwestern University, and by the University of Chicago.