Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories (2209.06121v2)

Abstract: Baez-Dolan type plus constructions serve three main purposes: They (1) corepresent categorical bimodules that are monoids with respect to a plethysm product, (2) allow to define functors as bimodule monoids, and thereby algebras over functors, (3) provide a theory of twists of monads. Unital (monoidal) bimodule monoids yield (monoidal) categories and the corepresentation is for indexed enrichments of categories. The original Baez--Dolan construction constructed algebras over operads. We define several of these constructions in the general context of categories and (symmetric) monoidal categories, show that they are functorial, and prove their corepresentation properties. One application is that the structures corepresented by an FC, like operads, props, etc. can be defined as plethysm monoids if and only if the corepresenting FC is a plus construction. In one direction, we prove that such a plus construction is based on the new notion of {Unique Factorization Category} (UFC). We also prove that the resulting FC has special properties, like being cubical. This explains why there is no monoid formulation for cyclic or modular operads or props, but there is for operads and properads. Using the bimodule monoids point of view, we prove that as monoidal bimodule monoids FCs are characterized by the fact that the functor constructing free algebras preserves the property of being strongly monoidal. We give a local presentation, as well as a global description, and a graphical version using decorated groupoid colored graphs. The global presentation utilizes pasting diagrams from 2-categories or equivalently double categories. In the special case of a UFC, we also present a graphical formalism with groupoid colored graphs. This allows us to identify our plus constructions as the step-by-step generalization of the Baez-Dolan plus constructions.