Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Given a category $\mathcal C$ together with a monad $T$ on $\mathcal C$, we get an adjunction $$\mathcal C^T(T-,-)\cong \mathcal C(-,\mathrm{For}-).$$

Is the same true for 2-monads on a 2-category?

share|cite|improve this question

2 Answers 2

up vote 4 down vote accepted

As Todd says, there are several flavours of 2-monad.

If you are interested in strict 2-monads, strict algebras for these, and strict morphisms, then yes you have an adjunction (even an enriched adjunction) as usual.

If you mean something weaker, then you will have something weaker than an adjunction. In particular, if you are considering pseudomorphisms of algebras - those which preserve the structure only up to (suitably coherent) isomorphism - then you'll have an equivalence between the category of algebra morphisms from a free algebra $TX$ to an algebra $B$ and the category of morphisms in the base 2-category from $X$ to (the underlying object of) $B$. So rather than an adjunction you'll get some sort of biadjunction. See Corollary 5.6 of the Blackwell-Kelly-Power paper for the case of strict monads, strict algebras, and pseudomorphisms, which is in fact the most important case for many purposes.

An important aspect of this is that free algebras $TX$ are flexible, which means among other things that any pseudomorphism from $TX$ to $B$ is isomorphic to a strict one. This is false for a general algebra $A$ in place of $TX$.

share|cite|improve this answer
Thank you ~ this was already very helpful. The specific case i am dealing with is that of pseudoalgebras for a strict 2-monad. – Garlef Wegart Jan 24 '11 at 7:30
That case (strict monads and pseudoalgebras) does come up sometimes in formal contexts. But if you are dealing with some concretes situtation involving a category or family of categories with some operations satisfying some equations either strictly or up to isomorphisms satisfying some equations, then you can describe such a structure as the strict algebras for some other 2-monad, and almost certainly this will be easier. – Steve Lack Jan 24 '11 at 9:37

Well, yes. There are several flavors of 2-monad, as explained in this nLab article, but I'm guessing you mean something like a pseudomonad $T$, and the 2-category $C^T$ of pseudoalgebras over $T$ (with the natural isomorphism replaced by a pseudonatural equivalence). The references in the nLab article may have what you need; I'd look specifically at the article by Blackwell, Kelly, and Power (I don't have the article at hand to confirm this).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.