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

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. Operads are functors out of a particular Feynman Category (FC), viz. they are corepresented by it, and the plus construction was previously generalized to FCs. We now define several of these constructions in the general context of categories and (symmetric) monoidal categories, show that they are functorial — hence categorically good — 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 a {Unique Factorization Category} (UFC) — a new notion that we introduce. In the other direction, we 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. Additionally, 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 of these constructions, as well as a global description of the morphisms, and a graphical version using decorated groupoid colored graphs. Our considerations are in an enriched setting. The global presentation utilizes pasting diagrams from 2-categories or equivalently double categories, which is of independent interest. 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.