This post originally appeared on LocalCharts.

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

February 2, 2024
Matteo Capucci

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: \[Q/{\cal M}(C,D) = \int _{MM'} \int _{C'D'} \mathcal C(C', MC) \times {\cal M}(M,M') \times \mathcal D(M'D, D') \to Q(C',D')\\ \cong \int _{M} \int _{C'D'} \mathcal C(C', MC) \times \mathcal D(MD, D') \to Q(C',D')\\ \cong \int _{M} Q(MC,MD) = \Theta Q(C,D).\]