Frobenius structures in star-autonomous categories

TL;DR

This paper generalizes the concept of Frobenius structures from quantales to objects in star-autonomous categories, establishing conditions under which nuclear objects have Frobenius endomorphism monoids and vice versa.

Contribution

It introduces Frobenius structures in arbitrary autonomous categories and characterizes nuclear objects via Frobenius endomorphism monoids in star-autonomous categories.

Findings

Frobenius structures characterize nuclear objects in autonomous categories.

Monoids of endomorphisms of nuclear objects have Frobenius structures.

Conditions for an object to be nuclear based on Frobenius structures are established.

Abstract

It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear.