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2019

Supplying bells and whistles in symmetric monoidal categories

Fong, Brendan, Spivak, David I

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It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in $\mathcal{C}$.

We define this formally and say that $\mathcal{C}$ supplies the algebraic structure.
For example, the category $\mathsf{Rel}$ of relations between sets has monoidal structures given by both cartesian product and disjoint union, and with respect to either one it supplies comonoids.
We prove several facts about the notion of supply, e.g.

Supplying bells and whistles in symmetric monoidal categories · Around

Built on

“A synthetic approach to stochastic maps, conditional independence, and theorems on sufficient statistics”, 2019
Original
Tobias Fritz · 1908
Earlier work this paper cites.
“Regular and relational categories: Revisiting ’Cartesian bicategories I”’
Original
Brendan Fong and David Spivak · 1909
Earlier work this paper cites.
“Coalgebras and Cartesian categories”
Thomas Fox · 1976
Earlier work this paper cites.
“Matrices, relations, and group representations”
Aurelio Carboni · 1991
Earlier work this paper cites.

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