For S a set, S-sorted algebraic (or ‘Lawvere’) theories are, equivalently, finite-product categories whose objects are freely generated by S, finitary monads on Set/S, or monoids in a category of ‘S-coloured cartesian collections’.
When S is a suitable direct category, I will describe equivalences of categories between finitary monads on [Sop, Set], monoids in a category of ‘S-coloured cartesian collections’, and a certain category of contextual categories (in the sense of Cartmell) under Sop.
Examples of such S are the categories of semi-simplices, globes and opetopes. Opetopes will be a running example, and we will see that there are three idempotent finitary monads on the category of opetopic sets, whose algebras are, respectively, small categories, coloured planar Set-operads, and planar coloured combinads (in the sense of Loday).
This is partly joint work with Peter LeFanu Lumsdaine, and partly joint work with Cdric Ho Thanh.
This video is part of Masaryk University‘s Algebra seminar.
