nLab affine monad

Idea

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.

Definition

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:

1. The unit map $\eta_1:1\to T1$ is an isomorphism;
2. The following diagram commutes for all objects $A$ and $B$,

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.)

Strongly and weakly affine monads

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:

1. The internal monoid $T1$ (with its canonical monoid structure) is a group;
2. For all objects $A$, $B$ and $C$, the associativity diagram is a pullback.

Every (commutative) strongly affine monad is affine, and every affine monad is weakly affine.

Properties

• 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.

See also

• monoidal monad, strong monad, commutative monad
• Markov category
• joint and marginal distributions

References

• 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)