Frobenius Pairs and Atiyah Duality

TL;DR

This paper introduces Frobenius pairs as a generalization of Frobenius objects and demonstrates that Atiyah duality for smooth manifolds can be expressed through these structures in the stable homotopy category.

Contribution

It defines Frobenius pairs and shows their application to encapsulate Atiyah duality within the framework of monoidal categories.

Findings

Frobenius pairs generalize Frobenius objects in monoidal categories.

Atiyah duality can be represented as a commutative Frobenius pair.

The structure provides a new categorical perspective on duality in topology.

Abstract

We define a notion of "Frobenius pair", which is a mild generalization of the notion of Frobenius object in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a commutative Frobenius pair in the stable homotopy category of spectra.