On Cyclic Star-Autonomous Categories

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.

💡 Research Summary

The paper introduces and thoroughly investigates the notion of a cyclic star‑autonomous category, a refinement of the usual (unbraided) star‑autonomous categories in which the left dual ⁎p and the right dual p⁎ of every object p are linked by a coherent natural isomorphism ϕp : ⁎p → p⁎. The authors first recall the standard definition of a star‑autonomous category: a symmetric monoidal closed category (C, ⊗, I) equipped with a dualizing object ⊥ such that each object p has a left dual ⁎p = p⊸⊥ and a right dual p⁎ = ⊥⊸p, together with the canonical isomorphism p⊸q ≅ ⁎q⊗p. In the classical setting the two duals are treated independently.

Cyclicity is imposed by requiring a family of natural isomorphisms ϕ = {ϕp} that satisfy two coherence equations:

1. Tensor compatibility – for any objects p and q, the diagram expressing ϕp⊗ϕq = ϕ(p⊗q) commutes. This guarantees that the cyclicity respects the monoidal product.

2. Internal hom compatibility – for any p, q, the diagram expressing ϕ(p⊸q) = (ϕp)⊸(ϕq) commutes. This ensures that the cyclicity interacts correctly with the closed structure.

These conditions are shown to be both necessary and sufficient for the family ϕ to be a coherent cyclicity. The authors develop a graphical calculus (Penrose‑style string diagrams) that makes the coherence conditions transparent and prove that any two parallel composites built from ⊗, ⊸, and ϕ are equal, establishing a strong coherence theorem for cyclic star‑autonomous categories.

The second major contribution is the observation that cyclic star‑autonomous categories provide the natural setting for enriched profunctors. If V is a cyclic star‑autonomous category, a V‑enriched profunctor Φ : A ⇸ B between V‑enriched categories A and B can be defined as the coend
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