nLab pseudomonoid

Idea

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.

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 $\mathbf{B}C$.

Variations

• A map pseudomonoid is a pseudomonoid whose multiplication and unit are maps, i.e. left adjoints. This is a more appropriate notion for monoidal bicategories whose morphisms are profunctors, since maps therein can be identified (modulo Cauchy completion) with functors.

Other more special kinds of pseudomonoid are generalizations of special kinds of monoidal categories, including:

• braided pseudomonoids
• symmetric pseudomonoids
• balanced pseudomonoids
• closed pseudomonoids
• $\ast$-autonomous, a.k.a. Frobenius pseudomonoids
• compact closed (or autonomous) pseudomonoids

Eventually these should probably have their own pages.

Properties

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.

References

• 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.