Theta-Categories and Tannakian duality

TL;DR

This paper introduces $ heta$-categories, a refined structure of symmetric monoidal $$-categories, and establishes a Tannakian duality linking these categories with fpqc-stacks, connecting Tannakian $ heta$-categories to schematic homotopy types over any base ring.

Contribution

It defines $ heta$-categories and proves a Tannakian duality theorem relating them to fpqc-stacks and schematic homotopy types in arbitrary characteristic.

Findings

$ heta$-categories refine symmetric monoidal $$-categories.

Established a Tannakian duality linking $ heta$-categories with fpqc-stacks.

Connected Tannakian $ heta$-categories to schematic homotopy types.

Abstract

We introduce a notion of $\Theta$-categories, which is a refinement of the notion of symmetric monoidal $\infty$-categories. We use this notion to prove a Tannakian duality statement, relating $\Theta$-categories with fpqc-stacks by means of a certain stack of fiber functors in the context of $\Theta$-categories. This provides, over a base ring of arbitrary characteristic, a strong link between Tannakian $\Theta$-categories and the schematic homotopy types.