Cofree Tambara modules

In my latest post on Tambara modules, I’ve shown you that if \mathcal C, \mathcal D are \mathcal M-actegories then the free Tambara module construction \Psi : \bf Prof(\mathcal C, \mathcal D) \to Tamb(\mathcal C, \mathcal D) is basically the free \mathcal M-action construction, where \mathcal M denotes the hom profunctor on \mathcal M and ‘action’ means ‘Day convolaction’.

Recall Day convolaction extends an \mathcal A-actegory structure on \mathcal X to an [\mathcal A^{op}, \bf Set]-actegory structure on [\mathcal X^{op}, \bf Set]. One can use this actegory structure as a way for monoids in [\mathcal A^{op}, \bf Set] to act on objects of [\mathcal X^{op}, \bf Set]. Above, I’m talking about this instanced for \mathcal A = \mathcal M \times \mathcal M^{op} and \cal X = C \times D^{op}—thus getting an action of \bf Prof(\cal M,M) on \bf Prof(\cal C,D)—and then looking at actions of the monoid \mathcal M(-,=): \cal M \nrightarrow M.

It was shown by Pastro and Street, but also by Mario Romàn and others, that \Psi \dashv U \dashv \Theta, where U is the forgetful functor from Tambara modules to profunctors, and \Theta is the functor:

\Theta P(C,D) = \int_M P(MC, MD).

In fact, this functor is the first one usually introduces when starting Pastro-Street theory of Tambara modules, since it’s very easy to see that coalgebras of \Theta U are strengths.
Indeed, a strength is a natural family \mathsf{st}_M^{C,D}:P(C,D) \to P(MC,MD) and these maps are classified by the end above by definition!

Once we established Tambara modules are actions of \mathcal M, and that \Psi \dashv U is monadic, then \Theta has to be the cofree action construction! I’ve been overlooking this fact since I didn’t know that Day convolaction is always left-closed, meaning acting by P-:\bf Prof(\cal C,D) \to \bf Prof(\cal C,D) has parametric right adjoint -/P:\bf Prof(\cal C,D) \to \bf Prof(\cal C,D) (this is different from right-closed, where it’s receiving an action which has a right adjoint, see Janelidze-Kelly).

I’ll give a definition for profunctors straight away, but of course this works for general presheaves:

Definition. For P:\cal M \nrightarrow M monoidal profunctor and Q:\cal C \nrightarrow D, define Q/P:\cal C \nrightarrow D as

Q/P(C,D) = \int_{MM'} \int_{C'D'} \mathcal C(C', MC) \times P(M,M') \times \mathcal D(M'D, D') \to Q(C',D').

Then it’s easy to see that -/\cal M \cong \Theta, by using a couple of Yoneda reductions:

Q/{\cal M}(C,D) = \int_{MM'} \int_{C'D'} \mathcal C(C', MC) \times {\cal M}(M,M') \times \mathcal D(M'D, D') \to Q(C',D')\\ \cong \int_{M} \int_{C'D'} \mathcal C(C', MC) \times \mathcal D(MD, D') \to Q(C',D')\\ \cong \int_{M} Q(MC,MD) = \Theta Q(C,D).
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