nLab idempotent monoidal functor

Idea

An idempotent monoidal functor is a functor $F : C \to D$ between monoidal categories with diagonals, that is, a monoidal functor which respects the diagonals on both sides.

In computer science, where lax monoidal functors are used to represent certain kinds of effects (as applicative functors), idempotent lax monoidal functors represent effects that can be executed one or more times with the same result: for example, reading from an immutable data source or raising an exception.

Definition

A (lax) monoidal functor $F : (C,\otimes) \to (D, \otimes)$, with monoidal structure $\nabla$, between monoidal categories with diagonals is idempotent if for all $A \in C$ the diagram

commutes.

Dually, if $F$ is an oplax monoidal functor, then it is idempotent if the diagram

commutes.

Note

This has little to do with either notion of idempotent monoid in a monoidal category: one yields strong monoidal functors while the other is not quite equivalent to this. Perhaps a better name would be diagonal monoidal functor, but this risks confusion with diagonal functor.

Properties

Theorem

For any lax monoidal endofunctor $F : (\mathcal{C}, \times) \to (\mathcal{C}, \times)$ on a cartesian monoidal category, $F$ is idempotent if and only if the following composite is the identity for all $A,B \in \mathcal{C}$:

Proof

See (Orchard 2014, proposition 3.3.9).

Examples

• Any lax monoidal functor induced by a strong idempotent monad is idempotent.

Related concepts

• monoidal functor

• strong monoidal functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

• braided monoidal functor

• symmetric monoidal functor

• idempotent monoidal functor

• idempotent monad

References

The (original?) definition is in

• Dominic Orchard, Programming contextual computations, PhD thesis, University of Cambridge, technical report 854 (2014) [pdf]