CaTT contexts are finite computads

TL;DR

This paper compares two recent type-theoretic and recursive descriptions of weak ω-categories, establishing their equivalence for finite computads and connecting models of CaTT with computad algebras.

Contribution

It demonstrates a fully faithful functor from CaTT's syntactic category to computads and shows their equivalence on finite cases, linking models to monad algebras.

Findings

Fully faithful morphism from CaTT to computads

Equivalence of categories for finite computads

Connection between CaTT models and monad algebras

Abstract

Two novel descriptions of weak {\omega}-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are {\omega}-categories. The second is a recursive description of a category of computads together with an adjunction to globular sets, such that the algebras for the induced monad are again {\omega}-categories. We compare the two descriptions by showing that there exits a fully faithful morphism of categories with families from the syntactic category of CaTT to the opposite of the category of computads, which gives an equivalence on the subcategory of finite computads. We derive a more direct connection between the category of models of CaTT and the category of algebras for the monad on globular sets, induced by the adjunction with computads.