Pseudomonad (category theory)

In mathematical category theory, a pseudomonad is a mathematical generalization of a monad. It is essentially the same notion as a pseudomonoid^[1] as introduced in Gray-monoid, and was introduced by Marmolejo (1997) for every Gray-category.^[2] A pseudomonad ${\displaystyle T=T(\mu ,\eta ,\tau ,\lambda ,\rho )}$ on a Gray-category or 2-category^[3] (or a more generalized notion weak 2-category^[4]) ${\displaystyle {\mathcal {C}}}$ consists of a 2-functor (in particular, if it is a functor on a weak 2-category, it is a pseudo-functor) ${\displaystyle T:{\mathcal {C}}\rightarrow {\mathcal {C}}}$ equipped with pseudonatural transformations ${\displaystyle \mu :T^{2}\rightarrow T}$ and ${\displaystyle \eta :\mathrm {Id} \rightarrow T}$ which satisfy the monad laws up to coherent invertible modifications.^[5][6] In monads, the identity and the associativity of composition hold strictly as equalities, whereas in the axiom of pseudomonads they hold only up to isomorphism, which satisfy the coherent axioms.^[7]

The 2-categorical analogue of Beck's monadicity theorem holds for the pseudomonads. Whereas the original theorem gives a necessary and sufficient condition for an adjunction to be monadic, the 2‑categorical analogue replaces the adjunctions and monads on ordinary categories that are the subject of the original theorem by pseudo-adjunctions and pseudomonads on 2-categories.^[8] The formal theory of monads can be developed in arbitrary 2-category, but to develop a formal theory of pseudomonads, move to the Gray-category.^[2] The analogue of distributive laws between monads also applies to pseudomonads. This was explicitly introduced by Marmolejo (1999) and is called the pseudodistributive law. Initially, it was thought that nine coherence axioms sufficed for the definition of a pseudodistributive law between pseudomonads, but this was later reduced to eight.^[9]

Let ${\displaystyle C}$ denotes a Gray-category or 2-category. Then a pseudomonad on ${\displaystyle C}$ consists of:

• A 1-cell is a 2-functor ${\displaystyle T:{\mathcal {C}}\rightarrow {\mathcal {C}}}$

• Two 2-cells are pseudonatural transformations ${\displaystyle \mu :T^{2}\rightarrow T}$ and ${\displaystyle \eta :\mathrm {Id} \rightarrow T}$

• Three invertible 3-cells of the following form:

.

It is possible to define an analogue of adjunctions in Gray categories; these are known as pseudoadjunctions or pseudo-adjunctions. Every pseudoadjunction gives rise to a pseudomonad. ^[2]

A pseudoadjuction ${\displaystyle F\dashv G}$ between Gray categories ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$ consists of:

• Two 1-cells: the functors ${\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}$ and ${\displaystyle G:{\mathcal {D}}\to {\mathcal {C}}}$
• Two 2-cells: maps ${\displaystyle t:\mathrm {id} _{\mathcal {D}}\to GF}$ and ${\displaystyle w:FG\to \mathrm {id} _{\mathcal {C}}}$
• Two 3-cells: Invertible maps ${\displaystyle \tau }$ and ${\displaystyle \omega }$, shown by the following diagrams:

The following pasting diagrams must be equal to the identity: ^[2]

• Doctrine (mathematics)
• Formal criteria for adjoint functors

• Le Creurer, I.J.; Marmolejo, F.; Vitale, E.M. (2002). "Beck's theorem for pseudo-monads". Journal of Pure and Applied Algebra. 173 (3): 293–313. doi:10.1016/S0022-4049(02)00038-5.
• Cheng, Eugenia; Hyland, Martin; Power, John (2003). "Pseudo-distributive Laws". Electronic Notes in Theoretical Computer Science. 83: 227–245. doi:10.1016/S1571-0661(03)50012-3.
• Gambino, Nicola; Lobbia, Gabriele (2021). "On the formal theory of pseudomonads and pseudodistributive laws". Theory and Applications of Categories. 37: 14–56. doi:10.70930/tac/4c53gqla.
• Lack, Stephen (2000). "A Coherent Approach to Pseudomonads". Advances in Mathematics. 152 (2): 179–202. doi:10.1006/aima.1999.1881.
• Marmolejo, F. (1997). "Doctrines whose structure forms a fully faithful adjoint string" (PDF). Theory and Applications of Categories [electronic only]. 3: 24–44. doi:10.70930/tac/z9m8u710 (inactive 25 January 2026). ISSN 1201-561X.{{cite journal}}: CS1 maint: DOI inactive as of January 2026 (link)
• Marmolejo, Francisco (1999). "Distributive laws for pseudomonads" (PDF). Theory and Applications of Categories. 5 (5): 81–147. doi:10.70930/tac/k5ldrv38 (inactive 25 January 2026).{{cite journal}}: CS1 maint: DOI inactive as of January 2026 (link)
• Marmolejo, F.; Wood, R. J. (2008). "Coherence for pseudodistributive laws revisited" (PDF). Theory and Applications of Categories. 20 (6): 74–84. doi:10.70930/tac/axzkv912.
• Shulman, Michael A. (2012). "Not every pseudoalgebra is equivalent to a strict one". Advances in Mathematics. 229 (3): 2024–2041. doi:10.1016/j.aim.2011.01.010.
• Tanaka, Miki; Power, John (2006). "Pseudo-distributive laws and axiomatics for variable binding". Higher-Order and Symbolic Computation. 19 (2–3): 305–337. doi:10.1007/s10990-006-8750-x.
  • Tanaka, Miki (2005). Pseudo-Distributive Laws and a Unified Framework for Variable Binding (PDF) (Thesis).
• Walker, Charles (2019). "Distributive Laws via Admissibility". Applied Categorical Structures. 27 (6): 567–617. arXiv:1706.09575. doi:10.1007/s10485-019-09567-9.

• Day, Brian; Street, Ross (1997). "Monoidal Bicategories and Hopf Algebroids". Advances in Mathematics. 129: 99–157. doi:10.1006/aima.1997.1649.
• Fiore, M.; Gambino, N.; Hyland, M.; Winskel, G. (2018). "Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures". Selecta Mathematica. 24 (3): 2791–2830. arXiv:1612.03678. doi:10.1007/s00029-017-0361-3.

• "Algebraic Geometry for Category Theorists". The n-Category Café.
• "modification". nLab.
• "pseudonatural transformation". nLab.
• "2Cat". nLab.