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?
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asked Jan 23, 2011 at 20:14
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2 Answers
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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 $TX$.
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$\begingroup$ Thank you ~ this was already very helpful. The specific case i am dealing with is that of pseudoalgebras for a strict 2-monad. $\endgroup$ Commented Jan 24, 2011 at 7:30
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$\begingroup$ 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. $\endgroup$ Commented Jan 24, 2011 at 9:37
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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).
answered Jan 23, 2011 at 22:39