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

TL;DR

The paper generalizes Baez–Dolan plus constructions to arbitrary (symmetric) monoidal categories, introducing Unique Factorization Categories (UFCs) and prehereditary UFCs to classify when plus-constructions yield Feynman categories. It shows that plus constructions corepresent (monoidal) Bimodule Monoid structures via plethysm, enabling algebras over module-theoretic objects to be described as functors or as monoids in plethysm. A dual track of formalisms is developed: generators-and-relations (with localization and reduction) and a graphical, groupoid-colored graph framework that encodes the same data, including both standard and hyper (twisted) versions needed for twists in bar/cobar/Koszul contexts. The work unifies operad-like theories (operads, props, properads) under the UFC/FC paradigm, explains why some operadic structures arise as plus-constructions while others do not, and provides explicit graphical and indexing formalisms that connect to cospans, spans, and decorated graph categories. The results have broad implications for categorical algebra, operad-like theories, and the graphical calculus used in higher category theory and Koszul duality.

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.

Figures (13)

• Figure 1: Equivariance relation of $\mu$ under $\mathcal{P}_+(f, F)$
• Figure 2: The diagram for the condition of prehereditary and the definition of connected components
• Figure 3: Pictorially, one can write a morphisms as a line diagram for a bijection with at most one tie around each strand, where we think of $T(Y)$ as bands tied at the bottom, as shown in (A). Here, the strand on the right is untied. Composition amounts to joining the strands and retying the ties if they overlap as we see in (B).
• Figure 4: A composition of two morphisms.
• Figure 5: Schematic of compositions. First tensor along rows then compose the rows in the column direction. Note the target of a row being the source of a column allows for different tensor decompositions, which in general can be of different length. The interstices are to mark the relative "cuts" of these tensors. In a prehereditary UFC the tensor lengths agree ---per unique decomposition.

Theorems & Definitions (239)

• Definition 1.1
• Remark 1.2
• Remark 1.3: Link to double categories
• Definition 1.4
• Definition 1.5
• Definition 1.7
• Remark 1.8
• Definition 1.9
• Remark 1.10
• Proposition 1.11