# Mathematical Sciences Research Institute

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# Seminar

HC & QS - Graduate Student Seminar: There are 3 kinds of symmetric monoidal $\infty$-category with duals and finite colimits & Manifold tensor categories March 12, 2020 (12:30 PM PDT - 01:30 PM PDT)
Parent Program: MSRI: Baker Board Room
Speaker(s) Tim Campion (University of Notre Dame), Christoph Weis (University of Oxford)
Description No Description
Video

#### 2-Weis

Abstract/Media

This will be livestreamed.  Join here: https://msri.zoom.us/j/627176824

(Tim Campion)

We will sketch a proof via string diagrams of the following fact:

Theorem: Let $C$ be a symmetric monoidal $\infty$-category with duals and finite colimits. Then $C$ splits as the product of 3 subcategories

$C = C_{ad} \times C_{st} \times C_{\neg ad}$

where

- $C_{ad}$ is an additive 1-category (e.g. finite-dimensional vector spaces);

- $C_{st}$ is a stable $\infty$-category (e.g. finite spectra or perfect chain complexes);

- $C_{\neg ad}$ is a semiadditive $\infty$-categorywhich is "anti-additive": its hom-spaces have no nontrivial invertible elements (many Span categories are examples).

Although this is a statement about $\infty$-categories, it specializes to a statement about ordinary 1-categories which is already somewhat surprising. Moreover, the bulk of the proof takes place in the homotopy category and proceeds via string diagram calculations; no knowledge of $\infty$-categories is required.

Time permitting, we will discuss the form of 1-dimensional tangle hypothesis needed to extend this spitting to braded monoidal categories, and given an application to motivic and equivariant homotopy theory.

(Christoph Weis)

Manifold tensor categories are a generalisation of fusion categories. Instead of requiring finitely many simple objects, one requires the simple objects to form a compact manifold. An example of such a category is the categorified (twisted) group ring of a Lie group showing up in the work of Freed-Hopkins-Lurie-Teleman on toral Chern-Simons-Theory (2009).

I will give the definition of Manifold tensor categories, indicate some partial progress on understanding them, and give a few fun examples along the way.