When is a linearly distributive category pivotal?

Micah Blake McCurdy – 9 April 2008

A linearly distributive category with negation can be thought of as a suitably generalised pivotal category, where the single dualising operation of a pivotal category is teased into two. Joyal and Street, in their treatment of pivotal categories, have shown that every pivotal category is equivalent to one whose dualising operation is strict; this result can be generalised to the case of linearly distributive categories with negation to produce retract equivalences between arbitrary such and "strict" ones, which have strict dualising operations, and whose de Morgan isomorphisms are identities. Furthermore, if one has lying to hand a cyclic structure on such an LDC, one can ask what conditions must be placed on it so that the obvious candidates for (one-sided) pivatal structures are, in fact, pivotal. As a pleasant complement to the previous result, which said that a cyclic structure represented a twist on a monoidal category if and only if it respected the tensor structure, here we find a pivotal structure if and only if the cyclicity respects the par structure. Regularly scheduled beer will resume, to celebrate the joyous easter season.