A pseudomonoid (sometimes also called a monoidale) in a monoidal 2-category is a categorification of the notion of a monoid object in a monoidal category.
A pseudomonoid in the cartesian monoidal 2-category Cat is precisely a monoidal category. The general definition can be extracted from this special case in a straightforward way. The precise definition can be found in Section 3 of the paper of Day and Street referenced below.
Just as a monoid in a monoidal category can be equivalently defined as a monad in the corresponding one-object 2-category (the delooping of ), so a pseudomonoid in a monoidal 2-category can equivalently be defined as a pseudomonad in the corresponding one-object 3-category .
Other more special kinds of pseudomonoid are generalizations of special kinds of monoidal categories, including:
Eventually these should probably have their own pages.
The 2-category of symmetric pseudomonoids in a symmetric monoidal 2-category has (weak) 2-coproducts given by the tensor product of underlying objects (analogously to how the category of commutative monoids in a monoidal category has coproducts given by the tensor product of the underlying objects). This is proven in Schaeppi, Appendix A.
Ross Street and Brian Day, Monoidal bicategories and Hopf algebroids, Advances in Mathematics, Volume 129, Issue 1, 15 July 1997, Pages 99-157.
Daniel Schäppi, Ind-abelian categories and quasi-coherent sheaves, arXiv, 2014.
Dominic Verdon, Coherence for braided and symmetric pseudomonoids, arXiv.
Last revised on September 27, 2022 at 23:03:44. See the history of this page for a list of all contributions to it.