# nLab pseudomonoid

## Idea

A pseudomonoid in a monoidal 2-category is a categorification of the notion of a monoid object in a monoidal category.

## Definition

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 $C$ can be equivalently defined as a monad in the corresponding one-object 2-category $\mathbf{B}C$ (the delooping of $C$), so a pseudomonoid in a monoidal 2-category $C$ can equivalently be defined as a pseudomonad in the corresponding one-object 3-category $\mathbf{B}C$.

## References

Ross Street and Brian Day, Monoidal Bicategories and Hopf Algebroids.

Revised on November 7, 2016 01:51:34 by Jon Beardsley (71.35.171.182)