The presheaf category on a monoidal category inherits the monoidal structure via the Day convolution. Moreover you can inherit (bi)closed monoidal structure.
In the study of Fourier analysis we can use the Fourier transform $F$ and its inverse as tools of simplification. That is, we have two sorts of tensor products: convolution of functions $\otimes_{hard}$ , and multiplication of functions $\otimes_{easy}$. From here we can simplify hard calculations as easy ones: $$A \otimes_{hard} B = F^{-1} ( FA \otimes_{easy} FB)$$
Abstracting over this principle I find myself asking a few questions:
- Can the Fourier transform be nicely categorified in a useful manner such that is groks with the Day convolution?
- For presheaf categories $P$ that carry a convolutional monoidal structure, what other sorts of categorical structure do we need to model a Fourier-like approach? I'm loosely referring to a Fourier-like transform as one which "turns convolution into multiplication" in a way that is advantageous/"easier".
Apologies that these are loose questions. A cursory search online doesn't find much for 1, and finds nothing for 2.
I have some vague ideas on 2. To model the Fourier transform, you could have a (strong? lax?) monoidal functor $F$ out of $P$ into some other category with a computationally easier tensor. Perhaps $F$ should form either an isomorphism or equivalence of categories so that this transform is invertible.
However, each of these hunches don't feel too principled and I find myself naively categorifying the existing approach without much insight. So I am here asking for some more resources that may have touched on this idea, suggestions on why such a thing might not be useful/interesting, or some ideas on a principled way to structure such a construction
