symmetric monoidal (∞,1)-category of spectra
For a strict 2-monad (which classically is called a simply a “2-monad”), the 2-category is a strict 2-category, the functor is a strict 2-functor, and the transformations and are strict 2-natural transformation?s and satisfy their laws strictly. This is the same as a -enriched monad. Strict 2-monads live naturally in strict 3-categories.
For a fully weak 2-monad, is a weak 2-category (such as a bicategory), is a weak (aka pseudo) 2-functor, and and 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 to be a strict 2-category and a strict 2-functor, but and only pseudo natural. This sort of pseudomonad lives naturally in a Gray-category.
One can consider various 2-categories of algebras/modules 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.
At the strictest level, for a strict 2-monad we can consider the -enriched Eilenberg-Moore category, which consists of strict algebras (see algebra over a monad), strict morphisms, and strict transformations between these. In 2-categorical literature, it is usually denoted . 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). See lax morphism for further discussion. With strict algebras and these various types of morphism, we obtain 2-categories (the pseudo case), , and for lax and colax respectively. One can also define an F-category of strict and lax morphisms together (or strict and pseudo, or pseudo and lax), and a double category which includes both lax and colax morphisms.
For example, ordinary (non-strict) monoidal categories are the strict algebras for a strict 2-monad on , 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 .
We can, however, also consider pseudo algebras for a 2-monad; see pseudoalgebra for a 2-monad. If the 2-monad is not strict, then this is usually the only sensible course. Pseudoalgebras for a strict 2-monad usually give an “unbiased” weaker notion of the structure specified by . For example, the pseudoalgebras for are, not ordinary monoidal categories, but unbiased monoidal categories. (It is true, however, that the 2-category of unbiased monoidal categories and strong monoidal functors is strictly 2-equivalent, i.e. Cat-enriched equivalent, to the 2-category of ordinary biased monoidal categories and strong monoidal functors.)
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 .
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 ) 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.”
A strict 2-monad has an underlying monad , such that strict -algebras and strict -morphisms are the same as -algebras and -morphisms. (This is a special case of the the general theory of underlying ordinary categories for enriched categories.) Moreover, if a strict 2-category admits powers or copowers with the interval category, then any monad on its underlying ordinary category has at most one enrichment to a strict 2-monad. Thus, in this case “being a 2-monad” is a mere property of a monad; see the “unicity” paper of John Power below.
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
Max Kelly and Steve 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
Stephen 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.
John Power, Unicity of enrichment over Cat or Gpd, Appl. Categ. Struct. 2009, 1–7.
Relation to symmetric operads is discussed in
Joachim Kock, Data types with symmetries and polynomial functors over groupoids, 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science. (arXiv:1210.0828)