nLab
2-monad

A 2-monad is a monad on a 2-category, or more generally a monad in a 3-category. This concept manifests at varying levels of strictness:

• For a strict 2-monad (which classically is called a simply a “2-monad”), the 2-category $K$ is a strict 2-category, the functor $T:K\to K$ is a strict 2-functor, and the transformations $\mu$ and $\eta$ are strict 2-natural transformation?s and satisfy their laws strictly. Strict 2-monads live naturally in strict 3-categories.

• For a fully weak 2-monad, $K$ is a weak 2-category (such as a bicategory), $T$ is a weak (aka pseudo) 2-functor, and $\mu$ and $\eta$ are pseudo natural transformations that satisfy their laws up to specified isomorphisms satisfying coherence conditions. Weak 2-monads live naturally in fully weak 3-categories (or tricategories)

• In between we have various notions that are sometimes called pseudomonads. For instance, we could require $K$ to be a strict 2-category and $T$ a strict 2-functor, but $\mu$ and $\eta$ only pseudo natural. This sort of pseudomonad lives naturally in a Gray-category.

One can consider various 2-categories of algebras (pseudo algebras, $2$-algebras) for a 2-monad, depending on whether the algebras satisfy their laws strictly or weakly, and whether the morphisms commute with the algebra structure strictly or weakly.

Many common types of structure on categories are specified by strict algebras for a strict 2-monad, but usually the strict morphisms are too strict. There are three types of weak morphism: pseudo (which preserve the structure up to a specified coherent isomorphism), lax (which preserve it up to a noninvertible transformation) and colax or oplax (for which the transformation goes the other direction).

For example, ordinary (non-strict) monoidal categories are the strict algebras for a strict 2-monad on $\mathrm{Cat}$, but usually we care about pseudo, lax, and oplax monoidal functors rather than strict ones. Strict monoidal categories are the strict algebras for a different strict 2-monad on $\mathrm{Cat}$, and the pseudo algebras for this 2-monad are, not ordinary monoidal categories, but unbiased? monoidal categories.

There are also 2-monads that specify property-like structure. For instance, there is a 2-monad whose algebras are categories with finite products. Actually, its algebras are categories equipped with specified finite products, the strict morphisms of these algebras preserve these specified finite products on the nose, and the pseudo morphisms preserve them in the usual sense of “preserving finite products.” In this case, every functor between algebras is an oplax morphism, since there is always a canonical comparison map $F(A\times B)\to F(A)\times F(B)$.

Doctrines

2-monads (particularly on Cat) are also sometimes called doctrines, with the intuition in mind that they are an “algebraic theory” of structure on a category just as a monad (on $\mathrm{Set}$) is an algebraic theory of structure on a set. However, this use of terminology is arguably at variance with the original intuitive meaning of “doctrine.”

References

• R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, Jour. Pure Appl. Algebra 59 (1989), 1–41
• F. Marmolejo, Doctrines whose structure forms a fully faithful adjoint string, Theory and Applications of Categories 3 (1997), 23–44. http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html
• S. Lack, A coherent approach to pseudomonads, Adv. Math. 152 (2000), 179–202. http://www.maths.usyd.edu.au:8000/u/stevel/papers/psm.ps.gz
• G.M. Kelly and S. Lack, On property-like structures, Theory and Applications of Categories 3 (1997) 213–250. http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html
• S. Lack, Codescent objects and coherence, JPAA 175 (2002), 223–241.
• I. J. Le Creurer, F. Marmolejo, E. M. Vitale, Beck’s theorem for pseudo-monads, J. Pure Appl. Algebra 173 (2002), no. 3, 293–313.