Bifibrations of polycategories and classical multiplicative linear logic

TL;DR

This thesis develops the theory of bifibrations of polycategories, generalizing representability and applying it to lift models of logic, with concrete examples involving vector spaces and Banach spaces.

Contribution

It introduces bifibrations for polycategories, linking them to representability, and applies this framework to lift logical models from vector spaces to Banach spaces.

Findings

Established a polycategorical Bénabou-Grothendieck correspondence.

Demonstrated lifting of compact closed structures to $ ext{*}$-autonomous structures.

Provided an operational interpretation involving systems and norms.

Abstract

In this thesis, we develop the theory of bifibrations of polycategories. We start by studying how to express certain categorical structures as universal properties by generalising the shape of morphism. We call this phenomenon representability and look at different variations, namely the correspondence between representable multicategories and monoidal categories, birepresentable polycategories and $\ast$-autonomous categories, and representable virtual double categories and double categories. We then move to introduce (bi)fibrations for these structures. We show that it generalises representability in the sense that these structures are (bi)representable when they are (bi)fibred over the terminal one. We show how to use this theory to lift models of logic to more refined ones. In particular, we illustrate it by lifting the compact closed structure of the category of finite dimensional vector spaces and linear maps to the (non-compact) $\ast$-autonomous structure of the category of finite dimensional Banach spaces and contractive maps by passing to their respective polycategories. We also give an operational reading of this example, where polylinear maps correspond to operations between systems that can act on their inputs and whose outputs can be measured/probed and where norms correspond to properties of the systems that are preserved by the operations. Finally, we recall the B\'enabou-Grothendieck correspondence linking fibrations to indexed categories. We show how the B-G construction can be defined as a pullback of virtual double categories and we make use of fibrational properties of vdcs to get properties of this pullback. Then we provide a polycategorical version of the B-G correspondence.