∗-autonomous Categories in Quantum Theory

So-called ∗-autonomous, or “Frobenius”, category structures occur widely in mathematical quantum theory. This trend was observed in [3], mainly in relation to Hopf algebroids, and continued in [8] with a general account of Frobenius monoids. Below we list some of the ∗-autonomous partially ordered sets A = (A , p, j, S) that appear in the literature, an abstract definition of ∗-autonomous promonoidal structure being made in [3, §7]. Without going into much detail, we also note some features of the convolution [A ,V ] (defined in [1]) of a given such A with a complete ∗-autonomous monoidal category V . A monoidal functor category of this type is a completion of A , with an appropriate universal property; it is always again ∗-autonomous (as seen in [3] for example). The basic descriptions of promonoidal (equals premonoidal) structure and the resulting convolution product are given in [1] and [3].