nLab pseudomonoid

Idea

The notion of 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.

See Street & Day (1997)

The archetypical example are monoidal categories, which are the pseudomonoids in the cartesian monoidal 2-category Cat. Similarly, monoidal enriched categories are pseudomonoids in VCat.

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. (However, it is typically better to work instead in a double category, where one may distinguish between tight pseudomonoids and loose pseudomonoids.)

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.

See also

• pseudoaction
• pseudomonad

References

The concept was introduced in:

• Ross Street, Brian Day, Monoidal bicategories and Hopf algebroids, Advances in Mathematics, 129 1 (1997) 99-157 [doi:10.1006/aima.1997.1649]

The terminology monoidale was introduced in:

• Dimitri Chikhladze?, A category of quantum categories, arXiv:0910.0512 (2009).

• Dimitri Chikhladze?, Stephen Lack, and Ross Street, Hopf monoidal comonads, arXiv:1002.1122 (2010).

• Daniel Schäppi, Ind-abelian categories and quasi-coherent sheaves, arXiv, 2014.

• Dominic Verdon, Coherence for braided and symmetric pseudomonoids, arXiv.