There is a story I keep forgetting so I’d like to write it up here to fix it up in my brain.
Tambara modules are profunctors P: \cal C \nrightarrow D between \cal M-actegories \cal C, \cal D that are ‘lax equivariant’: they are equipped with a strength (here mc and md denote the action of m on c and d respectively)
which is dinatural in m and natural in c, and satisfies the laws you expect to satisfy relative to the monoidal structure on \cal M.
It has been proven that Tambara modules structures correspond to algebra structures on P for a monad \Psi on {\bf Prof}(\cal C, D). This monad is described by Pastro and Street in Doubles for monoidal categories:
We can visualize the right hand side in this definition using Mario Roman’s intuition that coends describe diagrams, thus understanding that \Psi freely wraps P in combs with residual in \cal M:
\Psi P(c,d) = \Bigg\{
Here the square boxes correspond to morphisms in \cal C and \cal D respectively, while the red bead denotes an element of P(c',d').
Now the fact is, strengths for P corresponds to a (unital and associative) algebra \Psi P \Rightarrow P. But where the heck does \Psi come from?
In Pastro and Street’s paper it comes as a the left adjoint of a far more pedestrian comonad, of which Tambara modules are coalgebras. But there’s also another cute way to see it, which motivates calling Tambara modules ‘modules’. IIRC, this is something Dylan Braithwaite noticed, using something I dreamt up almost as a joke when writing the big actegories paper. We took it further than what I will do here and sent a poster to this year’s CT conference.
The thing I dreamt up is a form of Day convolution for co/presheaves over actegories. Day convolution famously extends a monoidal structure on a category to one on its category of co/presheaves. If instead your base category \cal X is an \cal M-actegory, then you can make [\cal X, \bf Set] a [\cal M, \bf Set]-actegory, where [\cal M, \bf Set] is equipped with Day convolution. I called this extended action Day convolaction. You can call it ‘the free extension’ of the action, or still ‘Day convolution’.
Defining it is simple, because at the end of the day, Day convolution is just a left Kan extension, and these can be computed with a coend formula if you’re lucky enough to land in a (very) cocomplete category.
Thus we can perform the same trick (which Bartosz Milewski spelled out here) and define the action of a copresheaf M:\cal M \to \bf Set on a copresheaf X:\cal X \to \bf Set as follows:
This is freely putting together all things in M and X by using all possible ways to map into x from a given m':\cal M and x':\cal X. Again, picturing the coend gives a fairly intuitive meaning to this definition:
MX(x) = \Bigg\{
\Bigg\}/\text{\small dinaturality}
As with Day convolution, if we restrict to corepresentables we recover the action we started with (by doing Yoneda reduction twice):
Anyway, back to Tambara modules. One nice things about profunctors is that you can pretend they are just copresheaves if you really want, since a profunctor P: \cal C \nrightarrow D is indeed a copresheaf on \cal X:= C^{\rm op} \times D.
Moreover, if \cal C and \cal D are both left \cal M-actegories then \cal C^{\rm op} \times D is a left \cal M^{\rm op} \times M-actegory, in the way you expect (componentwise).
But then {\bf Prof}(\cal C, D) \cong [C^{\rm op} \times D, \bf Set] receives an action from [\cal M^{\rm op} \times M, \bf Set] \cong Prof(\cal M,M), by Day convolaction. If M:\cal M \nrightarrow M and P:\cal C \nrightarrow D, we can unravel the above definition to get the a definition of MP:
This starts looking a bit like the definition of \Psi, doesn’t it?
Except there we don’t have an M around, just a P. If we fix M = {\cal M}(-,=), the identity profunctor on \cal M, we get:
by Yoneda reduction \int^{n'} {\cal M}(m',n') \times {\cal D}(n'd',d) \cong {\cal D}(m'd',d). And this is exactly \Psi P!
What’s happening here is that whenever M is a monoid in the monoidal category which acts on a category, then M- becomes a monad on the actee–in fact, the ‘free M-module monad’. This is the microcosm principle of actions: the most general habitat to talk about a monoid acting on an object is a monoidal category (where monoids live) acting on a category (where objects live).
In our case, we’ve chosen a monoid, viz. \cal M(-,=), in {\bf Prof}(M,M) considered with its Day convolution monoidal structure (which, by a classical result of Day, means it’s a monoidal profunctor), thus we can conclude that \Psi = \cal M(-,=)-:{\bf Prof}(\cal C, D) \to {\bf Prof}(C,D) is a monad, and in fact the ‘free \cal M(-,=)-module’ monad, thus proving that Tambara modules are nothing but \cal M(-,=)-modules.
This is quite fun because it shows also that there’s an extra degree of freedom in the definition of Tambara modules, i.e. what they are a module of. Clearly the identity profunctor on \cal M is only one of many possible examples of monoidal profunctors. This is the subject of the aforementioned poster Dylan presented at CT23: you can consider arbitrary monoidal profunctors \cal M \nrightarrow M instead of the identity one, and actually you can even consider arbitrary monoidal profunctors \cal M \nrightarrow N between monoidal categories. These and bimodules thereof organize in a triple category which, speculatively, sits inside the triple category Christian Williams considered in his thesis.
This brings the generalization of Tambara modules from monoidal categories to actegories to a natural level of completion. It also has repercussions on the theory of optics, but that’s maybe for another time!
glad you like this, Chris! I’m also exhilarated by the full potential of double categories and their incredibly fertile kinds of maps.