Familial Monads as Higher Category Theories

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.