## Cyclicity is the appropriate commutativity for the passage from V to V-prof

### Micah Blake McCurdy – 19 May 2010

If V is a cyclic linearly distributive category (for instance, if it is a cyclic monoidal category) then V-categories, V-profunctors, and modulations of profunctors form another cyclic structure, namely, a cyclic linear bicategory. The same cannot be said for, say, braiding or symmetry, showing that cyclicity appears to be the correct notion of commutativity for this construction. Pursuing yet further generalizations, the cyclic linearly distributive V can be replaced with a cyclic linear bicategory W, and certain completeness assumptions on V/W can be discarded if we content ourselves with obtaining a cyclic polybicategory V-Prof instead of a cyclic linear bicategory.

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