Monoidal Functor Categories and Graphic Fourier Transforms

This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through ∗autonomous monoidal categories and related structures. There is a close resemblance to convolution products and the Wiener algebra (of transforms) in functional analysis. The analysis term “kernel” (of a distribution) has also been adapted below in connection with certain special types of “distributors” (in the terminology of J. Bénabou) or “modules” (in the terminology of R. Street) in category theory. In using the term “graphic”, in a very broad sense, we are clearly distinguishing the categorical methods employed in this article from standard Fourier and wavelet mathematics. The term “graphic” also applies to promultiplicative graphs, and related concepts, which can feature prominently in the theory. Introduction In the first section of this article symmetric ∗-autonomous monoidal categories V (in the sense of [1]) and enriched functor categories of the form P(A) = [A,V ] (cf. [13]), are used to describe aspects of the “graphic” upper and lower convolutions of functors from a promonoidal category A into V (in the sense of [6] for example), and their transforms. The particular ∗-autonomous monoidal structures that are of interest here have their tensor unit as the dualizing object: see §1. A formal notion of categorical “Fourier” transform of a functor from a monoidal functor category [A,V ] is introduced in §3. This notion is a particular type of Kan extension based on the idea of a multiplicative kernel between promonoidal categories. Roughly speaking, multiplicative kernels correspond, via the Kan extension process, to tensor-preserving functors between the resulting enriched monoidal functor categories (into V), the latter being in many ways analogous to function algebras into a ring. Then the transform of the convolution product of two functors becomes the (sometimes pointwise) tensor product of their transforms. Some basic examples of kernels are mentioned at the end of §2, including that of association schemes. In §3 we also look at transforms of functors in the context of the abstract Weiner (or Joyal-Wiener) category, constructed by analogy with the Wiener algebra in functional analysis. This category of transforms is, under very simple conditions, a monoidal category Received by the editors 2009-12-09 and, in revised form, 2011-02-23. Transmitted by Steve Lack. Published on 2011-03-03. 2000 Mathematics Subject Classification: 18D10, 18A25.