Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Limits and colimits

limits and colimits

Contents

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.)

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.

Similarly: