With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Recall that for a monoidal monad (or equivalently commutative monad) on a monoidal category, both the Kleisli category and the Eilenberg-Moore category inherit a monoidal structure.
An affine monoidal monad, on a cartesian monoidal category, preserves the monoidal unit (the terminal object), and so it makes the Kleisli and Eilenberg-Moore categories semicartesian.
Let $(C,\times,1)$ be a cartesian monoidal category. A commutative monad $(T,\mu,\eta)$ with monoidal structure map $\nabla$ is called affine if any of the following equivalent conditions hold:
where $\pi_1:A\times B\to A$ and $\pi_2:A\times B\to B$ are the product projection maps.
(Note: This should be generalizable to monads on cartesian multicategories.)
A strong monad on a cartesian monoidal category is called strongly affine (Jacobs’16) if and only if for all objects $A$ and $B$, the following diagram is a pullback, where $\sigma$ denotes the strength of the monad, and $\pi_1$ the product projection.
A commutative monad on a cartesian monoidal category is called weakly affine (FGPT’23) if and only if any of the following equivalent conditions hold:
Every (commutative) strongly affine monad is affine, and every affine monad is weakly affine.
The Eilenberg-Moore category of an affine commutative monad is a semicartesian monoidal category. In particular, it is a model of affine logic.
The Kleisli category of an affine commutative monad is a Markov category (Fritz’20). This makes affine commutative monads suitable as probability monads, modelling joint and marginal distributions.
Similarly:
The Kleisli category of a strongly affine commutative monad is a positive Markov category (FGGPS’23).
The Kleisli category of a weakly affine commutative monad is a weakly Markov category, see FGPT’23.
Anders Kock, Bilinearity and cartesian closed monads, Mathematica Scandinavica 29(2), 1971.
Bart Jacobs, Semantics of weakening and contraction, Annals of Pure and Applied Logic 69(1), 1994. (full text)
Bart Jacobs, Affine Monads and Side-Effect-Freeness, Proceedings of CMCS, 2016. (pdf)
Tobias Fritz, Paolo Perrone, Bimonoidal Structure of Probability Monads, Proceedings of MFPS, 2018, (arXiv:1804.03527)
Kenta Cho, Bart Jacobs, Disintegration and Bayesian Inversion via String Diagrams, Mathematical Structures of Computer Science 29, 2019. (arXiv:1709.00322)
Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics, Advances of Mathematics 370, 2020. (arXiv:1908.07021)
Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, Paolo Perrone and Dario Stein, Dilations and information flow axioms in categorical probability. Mathematical Structures in Computer Science, 2023. (arXiv:2211.02507).
Tobias Fritz, Fabio Gadducci, Paolo Perrone and Davide Trotta, Weakly Markov categories and weakly affine monads, Proceedings of CALCO, LIPIcs 10, 2023. (arXiv)
Last revised on February 9, 2024 at 11:28:59. See the history of this page for a list of all contributions to it.