$ \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 $(\infty,\infty)$-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 Eckhart 206 and 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).

Draft Schedule

M Tu W
0930 Talk 1 0930 Talk 5 0930 Talk 8
1100 Talk 2 1100 Talk 6 1100 Talk 9
1200 Lunch 1200 Lunch 1200 Lunch
1400 Talk 3 1400 Talk 7 1400 Talk 10
1530 Coffee 1530 Coffee 1530 Coffee
1600 Talk 4 1600 Spivak Colloquium 1600 Talk 11

Titles

Aoki. (Commutative) higher motives.

Gepner/Heine 1. From $(\infty,\infty)$-categories to oriented categories.

Gepner/Heine 2. Towards the categorification of homotopy theory.

Gepner/Heine 3. Hypercompletion, Postnikov towers, and coinductive equivalences.

Masuda 1. Categorical spectra and stability of oriented categories.

Masuda 2. Categorical spectra with adjoints.

Spivak. Accounting for our inventiveness.

Stefanich. Higher algebraic geometry.

Abstracts

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.

Bibliography

Aoki, Higher presentable categories and limits, 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.

Masuda, The algebra of categorical spectra, pdf.

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 Eckhart 206 and Ryerson 251.

Restaurants

There are no official meals planned for this workshop.

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.