Monoidal weak omega-categories as models of a type theory

TL;DR

This paper develops a type-theoretic framework for monoidal weak omega-categories, extending previous models of weak omega-categories to include monoidal structures by translating between specialized type theories.

Contribution

It introduces the MCaTT type theory for monoidal weak omega-categories and establishes its correctness via translations with the CaTT type theory for omega-categories.

Findings

Defined the MCaTT type theory for monoidal weak omega-categories.

Proved the equivalence between models of MCaTT and omega-categories with one 0-cell.

Analyzed the interaction of models with structural rules in both type theories.

Abstract

Weak $\omega$-categories are notoriously difficult to define because of the very intricate nature of their axioms. Various approaches have been explored, based on different shapes given to the cells. Interestingly, homotopy type theory encompasses a definition of weak $\omega$-groupoid in a globular setting, since every type carries such a structure. Starting from this remark, Brunerie could extract this definition of globular weak $\omega$\nobreakdash-groupoids, formulated as a type theory. By refining its rules, Finster and Mimram have then defined a type theory called CaTT, whose models are weak $\omega$-categories. Here, we generalize this approach to monoidal weak $\omega$-categories. Based on the principle that they should be equivalent to weak $\omega$-categories with only one $0$-cell, we are able to derive a type theory MCaTT whose models are monoidal categories. This requires changing the rules of the theory in order to encode the information carried by the unique $0$-cell. The correctness of the resulting type theory is shown by defining a pair of translations between our type theory MCaTT and the type theory CaTT. Our main contribution is to show that these translations relate the models of our type theory to the models of the type theory CaTT consisting of $\omega$-categories with only one $0$-cell, by analyzing in details how the notion of models interact with the structural rules of both type theories.