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Let $\mathbf{C}$ be a closed symmetric monoidal category (I probably need even less than this; the examples I have in mind are simply the category of modules over a commutative ring and the category of sets) and let $\omega$ be an (arbitrary) object in $\mathbf{C}$ which I will term the “dualizing object”. Define a contravariant functor $D\colon \mathbf{C} \to \mathbf{C}$ taking $X$ to $[X,\omega]$ (internal Hom) and $X\to Y$ to the composition map $[Y,\omega] \to [X,\omega]$: let us call $DX$ the “dual” of $X$.

Now let $T = D^2$ be the covariant functor taking an object to its “bidual”. Call $\eta\colon 1_{\mathbf{C}}\to T$ the natural transformation $\eta_X \colon X \to D^2X = [[X,\omega], \omega]$ obtained from the evaluation map $X \otimes [X,\omega] \to \omega$. And define a natural transformation $\mu\colon T^2 \to T$ by letting $\mu_X \colon D^4 X \to D^2 X$ be $D(\eta_{DX})$.

Fact: $(T,\eta,\mu)$ is a monad.

This is probably a well-known observation, and it is certainly not difficult (although it is tedious to check, or at least I found it tedious, having to go up to the sexiesdual(!) $D^6X$ of $X$).

  • Does this monad have a name? (The “bidual monad” perhaps?) Is there a standard reference for it?

  • What are some “natural” occurrences, if any, of algebras for this monad?

(This came up to me by asking myself whether the sequence of iterated biduals $T^n X$ of an object stabilizes: the fact that $T$ is a monad says that, somehow, even though $T^2$ and $T$ are not the same, there is still a form of idempotency to $T$ in the existence of $\mu$.)

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    $\begingroup$ Let me point out that there's a non-tedious way to prove that $T$ is a monad: in any such case, $[-,\omega]$ is left adjoint to $[-,\omega]$. It is a bit weird because of the op's, but it's true and a simple computation: $\hom_C(x,[y,\omega]) = \hom_C(x\otimes y, \omega) = \hom_C(y,[x,\omega]) = \hom_{C^{op}}([x,\omega],y)$. Therefore your monad is just the monad of this adjunction $\endgroup$
    – Maxime Ramzi
    Commented Jun 26, 2021 at 12:04
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    $\begingroup$ In particular there are comparisons $C_T \to C^{op}\to C^T$, where $C_T$ and $C^T$ are the Kleisli and Eilenberg-Moore categories of $T$ respectively; that gives you natural occurrences of algebras $\endgroup$
    – Maxime Ramzi
    Commented Jun 26, 2021 at 12:07
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    $\begingroup$ A special case that is well known: If $C$ is a topos with its cartesian monoidal structure and $\omega$ is the subobject classifier (for example $C=Sets$ and $\omega=\{0,1\}$) then the category of $T$-algebra is equivalent to $C^{op}$ through the adjunction in Maxime Ramzi comments. $\endgroup$
    – Simon Henry
    Commented Jun 26, 2021 at 13:55
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    $\begingroup$ Another research direction that exploits this is known as continuation passing. There are some (old) references in my first ASD paper but this by no means my subject. Some proper computer scientists might want to comment. $\endgroup$
    – Paul Taylor
    Commented Jun 26, 2021 at 18:13
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    $\begingroup$ @MaximeRamzi's comment hits the nail on the head, but if you want a reference for a particular yet typical case, then the case of k-Vect and the contravariant dualization functor, and the fact one has an adjunction, is in MacLane's CWM. I don't have my copy at hand to give a precise reference, though. BTW as alluded to by various people the Banach space version of this was known to people like Linton and Semadeni, and is one reason why some of us like to work in mathematical frameworks where the Hahn-Banach theorem is true :) $\endgroup$
    – Yemon Choi
    Commented Jun 27, 2021 at 17:29

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It has been explained by Maxime Ramzi in the comments that this monad simply arises from the adjunction $[-,\omega] \vdash [-,\omega]^{\mathrm{op}}$.

As for the name, it's called the double dualization monad. The classical reference is

A. Kock, On double dualization monads, Math. Scand. 27 (1970), 151-165, pdf

The double dualization monad classifies algebra structures on a given object, see Theorem 3.2 in Kock's paper. The special case for $\mathbf{Set}$ appeas as Proposition 3.14 in

E. Manes, Monads of sets, Handbook of algebra. Vol. 3. North-Holland, 2003, 67-153, link

The double dualization monad also appears in Linton's "contravariant representation theorem", see for example Theorem 3.53 in Manes' article. The classical reference for this is

F.E.J. Linton, Applied functorial semantics I, Annali di Matematica Pura ed Applicata 86 (1970), 1–14, pdf

The double dualization monad for $\mathbf{Vect}_k$ and the object $k$ has been discussed at MO/104777.

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    $\begingroup$ PS: I don't know why Ben MacAdam deleted his answer - it also contained the reference to Kock's paper. $\endgroup$
    – Martin Brandenburg
    Commented Jun 27, 2021 at 19:39
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    $\begingroup$ There is also my whole research programme, Abstract Stone Duality, that re-axiomatises recursively based locally compact spaces using this monad. $\endgroup$
    – Paul Taylor
    Commented Jun 30, 2021 at 12:39
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This is called the double dual monad, and has been studied since (at least) the 70’s. It’s often looked at in categorical treatments of functional analysis-type things.

Here is a paper by Kock: https://www.mscand.dk/article/download/10995/9016 And a more recent paper by Lucyshyn-Wright: http://www.tac.mta.ca/tac/volumes/29/31/29-31.pdf

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  • $\begingroup$ Please would you give a reference for the "categorical treatments of functional analysis-type things". $\endgroup$
    – Paul Taylor
    Commented Jun 30, 2021 at 11:37

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