arXiv subjects
Publications and source records attributed to Brian Day.
*-Autonomous categories were initially defined by M. Barr to describe a type of duality carried by many monoidal closed categories. Later they were generalised by the current author to include *-autonomous promonoidal categories. Together, these structures under "convolution" product give a clear indication of the usefulness of *-autonomy in quantum mathematics and related areas.
As a development of [2] and [3], we construct a "VN-bialgebra" in Vect_k for each k-linear split-semigroupal functor from a suitable monoidal category C to Vect_k. The main aim here is to avoid the customary compactness assumptions on generators of the domain category C (cf. [3]). Please note that the VN-bialgebras in Vect_k defined here are not necessarily von Neumann regular as k-algebras in the usual sense, the prefix "VN-" coming from the Set-based case of von Neumann regular semigroups.
Here we describe three straightforward examples of what was called a graphic Fourier transformation in [4]. At least two of these examples may be viewed simply as monoidal comonads on suitable monoidal closed functor categories, but the third example, which involves "centres" of monoidal closed functor categories, is generally not comonadic. For the first two examples (i.e., association schemes and RCFT's), a more elaborate "probicategory" set-up was envisaged in an earlier version of this note, but many readers missed the main point so it is simplified (hopefully) below.
We note an inversion property of the fusion map associated to many semibialgebras.
We describe what might be called the "Hall-fusion" bialgebra constructed from a promonoidal double, and mention the corresponding face version for probicategories.
We briefly relate the existence of a middle-four interchange map in a category with two monoidal structures, to the standard Cockett and Seely notion of a weakly distributive category.
We describe a $\C$-linear additive *-autonomous category of Banach spaces. Please note that a correction has been appended to the original version 1 which is maintained here for reference. Also, a proposed example of a *-autonomous category of topological $\C$-linear spaces has been added to version 2.
We describe a sufficient condition for the process of left Kan extension to be a conservative functor. This is useful in the study of graphic Fourier transforms and quantum categories and groupoids.
It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call "Frobenius monoidal", is sufficient.
We show that the (co)endomorphism algebra of a sufficiently separable "fibre" functor into Vect_k, for k a field of characteristic 0, has the structure of what we call a "unital" von Neumann core in Vect_k. For Vect_k, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group.
A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. We define the term "quantum category". The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's star-autonomous categories is more suitable than autonomy (= compactness = rigidity). This leads to our definition of quantum groupoid intended as a "Hopf algebra with several objects".