Pseudomonoids
A pseudomonoid is like a monoid except that the associativity and unitality laws need hold only up to coherent invertible 2-cells. Just as a monoid can be defined internally to any monoidal category, a pseudomonoid can be defined internally to any monoidal 2-category. The standard example is a pseudomonoid in \((\mathbf{Cat},\times)\), which is a monoidal category.
Literature
Books
- Aguiar & Mahajan, 2010: Monoidal functors, species and Hopf algebras (doi,
pdf), Appendix C.2: Pseudomonoids
- Unlike some other sources, gives explicit definitions of a pseudomonoid, (op)lax morphisms of pseudomonoids, and morphisms of those
Papers
- Day & Street, 1997: Monoidal bicategories and Hopf algebroids (doi)
- Sec. 3 defines a pseudomonoid in a Gray monoid , also known as a semistrict monoidal 2-category
- McCrudden, 2000: Balanced coalgebroids (pdf)
- Sec. 2 defines a pseudomonoid in a monoidal 2-category, “mildly generalizing” Day-Street
- Sec. 3-5 define braided, symmetric, and balanced pseudomonoids in braided, sylleptic, and balanced monoidal 2-categories, respectively
- A nice feature of this paper is its many examples of pseudomonoids
- Schäppi, 2014: Ind-abelian categories and quasi-coherent sheaves (doi, arxiv)
- Theorem 5.2: Tensor product of symmetric pseudomonoids in M is their bicategorical coproduct in the bicategory of symmetric pseudomonoids in M (nCat Cafe )
- Buckley et al, 2021: Oplax Hopf algebras (pdf, arxiv)
- Sec. 2.3 and Appendix A review monoidal bicategories and pseudo(co)monoids