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Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories
Category Theory · Submitted 13 Sep 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.86 | 0.15 | 0.16 | asserted 1.00 |
| s1 | Unital (monoidal) bimodule monoids yield (monoidal) categories and the corepresentation is for indexed enrichments of categories. | result 0.55 | 0.33 | 0.08 | asserted 1.00 |
| s2 | The original Baez--Dolan construction constructed algebras over operads. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s3 | 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. | contribution 0.98 | 0.96 | 0.12 | asserted 1.00 |
| s4 | 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. | result 0.92 | 0.77 | 0.38 | asserted 1.00 |
| s5 | In one direction, we prove that such a plus construction is based on the new notion of {Unique Factorization Category} (UFC). | contribution 0.55 | 0.89 | 0.19 | asserted 1.00 |
| s6 | We also prove that the resulting FC has special properties, like being cubical. | result 0.98 | 0.90 | 0.07 | asserted 1.00 |
| s7 | This explains why there is no monoid formulation for cyclic or modular operads or props, but there is for operads and properads. | interpretation 0.95 | 0.61 | 0.17 | asserted 0.99 |
| s8 | 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. | result 0.90 | 0.94 | 0.14 | asserted 1.00 |
| s9 | We give a local presentation, as well as a global description, and a graphical version using decorated groupoid colored graphs. | contribution 0.66 | 0.93 | 0.06 | asserted 1.00 |
| s10 | The global presentation utilizes pasting diagrams from 2-categories or equivalently double categories. | method 0.71 | 0.74 | 0.05 | asserted 1.00 |
| s11 | In the special case of a UFC, we also present a graphical formalism with groupoid colored graphs. | contribution 0.55 | 0.90 | 0.05 | asserted 0.98 |
| s12 | This allows us to identify our plus constructions as the step-by-step generalization of the Baez-Dolan plus constructions. | interpretation 0.75 | 0.84 | 0.19 | asserted 0.99 |
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The paper
Original 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.
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How the literature uses it
2 arXiv papers cite this in 15 sentences
Papers citing it · 11 sentences without a confident label
Builds on it 1 paper
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The current paper generalizes classical construction [ref] in the following sense: Factorizable groupoid bimodules correspond to special monoidal categories called hereditary unique factorization categories (UFCs) [ref] , see Proposition [ref].
Plethysm Products, Element and Plus Constructions · Application to classical theory · 0.95
Uses its method or tool 1 paper
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If we let $_nd$ be the Feynman category of directed corollas with at least one in-tail and one out-tail, then we can use [ref] to express it as a plus construction: [equation] We will show how this leads to a monoid definition of a reduced properad.
Plethysm Products, Element and Plus Constructions · Reduced Properads · 0.90
Compares with it 1 paper
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Later in Section [ref], we will compare this to the unique factorization categories of [ref] .
Plethysm Products, Element and Plus Constructions · Multiplication · 0.89
Cites it as background 1 paper
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In literature, plus constructions have been proposed for polynomial monads (cf. [ref] , [ref] ), for Feynman categories (cf. [ref] ), for operadic categories (cf. [ref] ), and for symmetric monoidal categories (cf. [ref] ).
Moment categories and operads · Plus construction · 0.95
What it cites · 50 sentences without a confident label
Builds on 2 papers
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This extends the definition of [ref] to arbitrary (symmetric (monoidal)) categories.
Feynman Categories · Bi–module monoids, plus constructions co–representing them and algebras · 1.00
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This extends the definition of [ref] to arbitrary (symmetric (monoidal)) categories.
Feynman categories and Representation Theory · Bi–module monoids, plus constructions co–representing them and algebras · 1.00
Compares with 2 papers
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This allows for a characterization which is analogous to that of [ref] .
Feynman Categories · Reduced version of the strong plus construction: $^+$ · 0.92
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This allows for a characterization which is analogous to that of [ref] .
Feynman categories and Representation Theory · Reduced version of the strong plus construction: $^+$ · 0.92
Cites as background 7 papers
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Although the notion of graphs we will use is available in several places [ref] , we will recall the basic structures here, as we will generalize them.
Feynman categories and Representation Theory · Graphs and Feynman categories · 0.91
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Although the notion of graphs we will use is available in several places [ref] , we will recall the basic structures here, as we will generalize them.
Feynman Categories · Graphs and Feynman categories · 0.91
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The non–decorated groupoid colored graphs are also at the heart of Koszul duality [ref] .
Koszul Feynman Categories · Introduction · 0.90
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