nLab pseudomonoid




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


  • 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.


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


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

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

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

Last revised on January 31, 2024 at 09:04:57. See the history of this page for a list of all contributions to it.