Compact Monoidal Categories from Linguistics to Physics

Abstract

This is largely an expository paper, revisiting some ideas about compact 2-categories, in which each 1-cell has both a left and a right adjoint. In the special case with only one 0-cell (where the 1-cells are usually called “objects”) we obtain a compact strictly monoidal category. Assuming furthermore that the 2-cells describe a partial order, we obtain a compact partially ordered monoid, which has been called a pregroup. Indeed, a pregroup in which the left and right adjoints coincide is just a partially ordered group (= pogroup).

A brief exposition of recent joint work with Preller and Lambek “Mathematical Structures in Computer Science”, 17, (2007) will be given here, investigating free compact strictly monoidal categories, which may be said to describe computations in pregroups. Free pregroups lend themselves to the study of grammar in natural languages such as English. While one would not expect to find a connection between linguistics and physics, applications of (free) compact symmetric monoidal categories to physics have been made by Coecke “The Logic of Entanglement” (2004), Abramsky and Coecke “Proceedings of 19th IEEE Conference on Logic in Computer Science”, pp. 415–425 (2004), Abramsky and Duncan “Mathematical Structures in Computer Science”, 16, 469–489 (2006), Selinger “Electronic Notes in Theoretical Computer Science”, 170, 139–163 (2007).

Compact symmetric monoidal categories had already been studied by Kelly and Laplaza “Journal of Pure and Applied Algebra”, 19, 193–213 (1980), who called them “compact closed” and by Barr “Lecture Notes in Mathematics”, 752 (1979), “Journal of Pure and Applied Algebra”, 111, 1–20 (1996), “Theoretical Computer Science”, 139, 115–130 (1995), who called them “compact star-autonomous”. I had intended to show that Feynman diagrams for quantum electro-dynamics (QED) could be described by certain compact Barr-autonomous categories, but was disappointed to find that these reduced to a rather degenerate case, that of partially ordered groups (= pogroups). Still, I will reluctantly present an extension of this idea from QED to the Standard Model. Finally, I will briefly review the transition from 2-categories to the bicategories of Bénabou “Lecture Notes in Mathematics” 47, 1–77 (1967), using methods of Bourbaki “Algebre multilineaire” (1948) and Gentzen (see Kleene “Introduction to metamathematics” (1952)), which may also be of interest in physics.