Monoidal categories

Monoidal monad

In category theory, a monoidal monad is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations and are monoidal natural transformations. In other words, is equipped with coherence maps and satisfying certain properties (again: they are lax monoidal), and the unit and multiplication are monoidal natural transformations. By monoidality of , the morphisms and are necessarily equal. All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal natural transformations. (Wikipedia).

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Related pages

Cartesian monoidal category | Strict 2-category | Hopf algebra | Power set | Monad (category theory) | Monoidal category | Monoidal adjunction | Bialgebra | Monoidal natural transformation | Category theory