On Cyclic Star-Autonomous Categories

TL;DR

This paper explores cyclic star-autonomous categories, establishing coherence, their relation to enriched profunctors, and showing that all such categories are equivalent to those with trivial cyclicity isomorphisms.

Contribution

It provides coherence results for cyclic star-autonomous categories and demonstrates their natural role in enriched profunctor theory, also showing equivalence to categories with identity cyclicity.

Findings

Cohesion of cyclicity isomorphisms is established.

Enriched profunctors inherit a canonical cyclic structure.

Every cyclic star-autonomous category is equivalent to one with trivial cyclicity.

Abstract

We discuss cyclic star-autonomous categories; that is, unbraided star- autonomous categories in which the left and right duals of every object p are linked by coherent natural isomorphism. We settle coherence questions which have arisen concerning such cyclicity isomorphisms, and we show that such cyclic structures are the natural setting in which to consider enriched profunctors. Specifically, if V is a cyclic star-autonomous category, then the collection of V-enriched profunctors carries a canonical cyclic structure. In the case of braided star-autonomous categories, we discuss the correspondences between cyclic structures and balances or tortile structures. Finally, we show that every cyclic star-autonomous category is equivalent to one in which the cyclicity isomorphisms are identities.