nLab pseudoaction

Idea

Pseudoactions are categorified actions. In other words, they are coherent actions of pseudomonoids.

Examples

1. A pseudomonoid in $(\mathbf{Cat}, 1, \times)$ is exactly a monoidal category. The pseudoaction of a monoidal category on a category is an actegory.
2. A pseudomonoid in $(\mathbf{Fib}, 1, \times)$ is a monoidal fibration. Pseudoactions of a monoidal fibration on a fibration are considered in (Vasilakopoulou ‘18).
3. A pseudomonad is a pseudomonoid in the Gray monoid of endomorphisms of an object in a Gray-category (see (Marmolejo ‘99, Definition 3.1)). A pseudoalgebra of a pseudomonad is a pseudoaction of the pseudomonad on the functor picking out the carrier of the pseudoalgebra.

See also

• categorification
• pseudomonoid
• module

References

• Christina Vasilakopoulou, On enriched fibrations, Cahiers de Geometrie et Topologie Differentielle, vol. LIX-4, 2018, (cahiers), (arXiv:1801.01386)

• Francisco Marmolejo, Distributive laws for pseudomonads, Theory and Applications of Categories, 5 5 (1999) 81-147 [tac]