Corpus ID: 225068412

# Sheet diagrams for bimonoidal categories

```@article{Comfort2020SheetDF,
title={Sheet diagrams for bimonoidal categories},
author={Cole Comfort and Antonin Delpeuch and Jules Hedges},
journal={arXiv: Category Theory},
year={2020}
}```
• Published 26 October 2020
• Mathematics
• arXiv: Category Theory
Bimonoidal categories (also known as rig categories) are categories with two monoidal structures, one of which distributes over the other. We formally define sheet diagrams, a graphical calculus for bimonoidal categories that was informally introduced by Staton. Sheet diagrams are string diagrams drawn on a branching surface, which is itself an extruded string diagram. Our main result is a soundness and completeness theorem of the usual form for graphical calculi: we show that sheet diagrams… Expand
2 Citations

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#### References

SHOWING 1-10 OF 14 REFERENCES
From fibered symmetric bimonoidal categories to symmetric spectra
In here we define the concept of fibered symmetric bimonoidal categories. These are roughly speaking fibered categories D->C whose fibers are symmetric monoidal categories parametrized by C and suchExpand
A Survey of Graphical Languages for Monoidal Categories
This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but alsoExpand
• Mathematics, Computer Science
• MFPS
• 2018
A conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability by endowing the usual probability monads with a monoidal and an opmonoidal structure, mutually compatible. Expand
STRICTIFICATION OF CATEGORIES WEAKLY ENRICHED IN SYMMETRIC MONOIDAL CATEGORIES
We show that categories weakly enriched over symmetric monoidal cate- gories can be strictied to categories enriched in permutative categories. This is a \many 0-cells" version of the stricticationExpand
Normalization for planar string diagrams and a quadratic equivalence algorithm
• Computer Science, Mathematics
• 2018
It is shown that left- and right-handed exchanges each give strongly normalizing rewrite strategies for connected string diagrams, which gives a linear-time solution to the equivalence problem in the connected case, and a quadratic solution in the general case. Expand
Generalised Proof-Nets for Compact Categories with Biproducts
• R. Duncan
• Mathematics, Computer Science
• ArXiv
• 2009
It is demonstrated how to represent quantum processes as proof-nets, and it is shown that the dynamic behaviour of a quantum process is captured by the cut-elimination procedure for the logic. Expand
The groupoid of finite sets is biinitial in the 2-category of rig categories
The groupoid of finite sets has a "canonical" structure of a symmetric 2-rig with the sum and product respectively given by the coproduct and product of sets. This 2-rigExpand
The geometry of tensor calculus, I
• Mathematics
• 1991
This paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals. Mathematics Subject Classifications (1991). 18D10, 52B11, 53A45 ,Expand
Algebraic Effects, Linearity, and Quantum Programming Languages
A new elementary algebraic theory of quantum computation, built from unitary gates and measurement is presented, and an equational theory for a quantum programming language is extracted from thegebraic theory. Expand
A complete language for faceted dataflow programs
We present a complete categorical axiomatization of a wide class of dataflow programs. This gives a three-dimensional diagrammatic language for workflows, more expressive than the directed acyclicExpand