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
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.
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.
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.
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 $A,B \in \mathcal{C}$:
idempotent monoidal functor
The (original?) definition is in
Created on March 8, 2024 at 13:35:39. See the history of this page for a list of all contributions to it.