Tannakian categories: origins and summary

TL;DR

The paper traces the origins of tannakian categories from Grothendieck's motives and Weil cohomology to a robust duality theory: neutral tannakian categories over $k$ correspond to representations of an affine group scheme $G$, while non-neutral ones correspond to affine groupoid schemes and fibre-functor gerbes. It develops the gerbe-theoretic and 2-categorical framework linking tannakian categories to affine gerbes, introduces the motivic Galois group $\pi(\mathsf{T})$, and discusses polarizations that yield compact real forms. The motives section highlights the role of numerical motives and standard conjectures in achieving semisimplicity and polarization, while noting Deligne's suggested alternatives in light of unsettled conjectures. Collectively, the work provides a foundational, category-theoretic lens for understanding and unifying cohomology theories, motives, and their symmetry groups, with implications for the study of fundamental groups in algebraic geometry.

Abstract

We describe the origins of the theory of tannakian categories, and summarize its main results.

Theorems & Definitions (8)

• Theorem 1: I, 10.1
• Theorem 2: II, 3.1
• Theorem 3: II, 8.1
• Theorem 4: III, 1.1
• Theorem 5: IV, 3.3
• Theorem 6: V, 8.2
• Theorem 7: VI, 2.5
• Theorem 8: VI, 6.12