Pseudo Limits, Bi-adjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory

The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2-category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this definition. The structure present on the class C of rigged surfaces is captured by these concepts of 2-category theory. Here a rigged surface is a real, compact, not necessarily connected, two dimensional manifold with complex structure and analytically parameterized boundary components. Isomorphisms of such rigged surfaces are holomorphic diffeomorphisms preserving the boundary parameterizations. These rigged surfaces and isomorphisms form a groupoid and are part of the structure present on C. Concepts of 2-categories enter when we describe the operations of disjoint union of two rigged surfaces and gluing of two rigged surfaces along boundary components of opposite orientation. One needs a mathematical structure to capture all of these features. This has been done in [24]. One step in this direction is the notion of algebra over a theory in the sense of Lawvere [34]. We need a weakened notion in which relations are replaced by coherence isos. This weakened notion is called a pseudo algebra in this paper. Coherence diagrams are required in a pseudo algebra, but it was noticed in [24] that Lawvere’s notion of a theory allows us to write down all such diagrams easily. See Section 6 below. A symmetric monoidal category as defined in [39] provides us with a classical example of a pseudo algebra over the theory of commutative monoids. Theories, duality, and related topics are discussed further in [1], [2], [3], [35], [36]. Unfortunately, pseudo algebras over a theory are not enough to capture the structure on C. The reason is that the operation of gluing is indexed by the variable set of pairs of boundary components of opposite orientation. The operation of disjoint union also has an indexing. We need pseudo algebras over a “theory indexed over another theory,” which we call a 2-theory. More precisely, the pseudo algebras we need are pseudo algebras over the 2-theory of commutative monoids with cancellation. See [24] and Section 12 below. The term 2-theory does not mean a theory in 2-categories.