Poincar\'e duality isomorphisms in tensor categories

TL;DR

This paper generalizes Poincaré duality isomorphisms from vector spaces to objects in rigid pseudo-abelian tensor categories, broadening the understanding of duality in abstract categorical settings.

Contribution

It introduces a framework for Poincaré duality isomorphisms within rigid pseudo-abelian tensor categories, extending classical results beyond vector spaces.

Findings

Establishes natural Poincaré isomorphisms in tensor categories

Generalizes classical duality results to categorical objects

Provides a foundation for further categorical duality studies

Abstract

If for a vector space V of dimension g over a characteristic zero field we denote by $\wedge^iV$ its alternating powers, and by $V^\vee$ its linear dual, then there are natural Poincar\'e isomorphisms: $\wedge^i V^\vee \cong \wedge^{g-i} V$. We describe an analogous result for objects in rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor categories.