Bifibrations of polycategories and classical linear logic

In this talk we will explore four different ways of interpreting classical multiplicative linear logic categorically: as a *-autonomous category where the connectives are given by structures, as a *-representable polycategory where the connectives are expressed through universal properties, as a bifibred polycategory where the connectives are recovered by fibrational properties and as a Frobenius pseudomonoid internal to a polycategory where the connectives are defined by internal operations. We will relate these approaches through different correspondences. First we will see that *-representable polycategories are unbiased alternatives to the two-tensor polycategories with duals that have been introduced by Cockett and Seely and that has been proven to be equivalent to -autonomous categories. Then we will see that bifibred polycategories generalise those in the sense that a polycategory is *-representable iff it is bifibred over the terminal polycategory. Finally, we will present a polycategorical Grothendieck correspondence between bifibrations of polycategories and pseudofunctors into MAdj, the 2-polycategory of multivariable adjunctions. When restricted to bifibrations over the terminal polycategory we get back the correspondence between *-autonomous categories and Frobenius pseudomonoids in MAdj that was recently observed by Mike Shulman. If time permits we will also look at some refinements of *-autonomous categories by considering bifibred polycategories over a representable polycategory other than the terminal one.

03 Dec 2020 00:00 — 00:00