Taking profunctors preserves pseudo-cyclicity

Micah Blake McCurdy – 6 August 2008

We generalize the work of Rosenthal who proved the following result: Let V be a complete symmetric *-autonomous category, and let X be a category enriched in V. Then the category of V-enriched profunctors from X to itself bears a cyclic *-autonomous structure which satisfies what we call ``Rosenthal's axiom'', a certain coherence of the cyclic structure with the double-dualization isomorphisms. This result can be sharpened somewhat and thereby much better understood. Instead of working with *-autonomous structures, we work instead with linearly distributive structures with negation. If one removes the symmetry from V, replacing it with a pseudo-cyclic structure, then we can deduce that prof_V(X,X) bears a linearly distributive structure with negation which supports a pseudo-cyclic structure. Moreover, if the cyclicity of V satisfies Rosenthal's axiom, so too does prof_V(X,X). Note that a symmetric *-autonomous V can be thought of as bearing a cyclic structure (namely, the identity) which is pseudo-cyclic, pseudo-cocyclic, and moreover satisfies Rosenthal's axiom. This implies that even without weakening Rosenthal's hypotheses we can deduce that the cyclic structure on prof_V(X,X) is pseudo-cyclic and pseudo-cocyclic.