Familial Monads as Higher Category Theories

TL;DR

This paper provides a unified framework for describing higher category structures as algebras for familial monads, simplifying their characterization and relating various higher categorical concepts.

Contribution

It offers an explicit description of higher categories via familially representable monads, avoiding complex naturality proofs and connecting to categorical polynomials.

Findings

Explicit characterization of higher categories as algebras for familial monads

Descriptions of pullbacks, composites, and exponentiations in terms of classifying functors

Applications to higher category theory and cubical sets

Abstract

Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any higher category structure which arises as algebras for a familially representable monad on a presheaf category, then use this to describe several examples relating to higher category theory and cubical sets. The proof of this characterization avoids tedious naturality arguments by passing through the theory of categorical polynomials; along the way, we give descriptions of pullbacks, composites, and exponentiations of split opfibrations in terms of their classifying functors which may be of independent interest.