Pseudo limits, bi -adjoints, and pseudo algebras: Categorical foundations of conformal field theory.

Abstract

In this paper we develop categorical foundations needed for a rigorous approach to the definition of conformal field theory outlined by Graeme Segal. We discuss pseudo algebras over theories and 2-theories, their pseudo morphisms, bilimits, bicolimits, bi-adjoints, stacks, and related concepts. These 2-categorical concepts are used to describe the algebraic structure on the class of rigged surfaces. A <italic>rigged surface</italic> is a real, compact, not necessarily connected, two dimensional manifold with complex structure and analytically parametrized boundary components. This class admits algebraic operations of <italic>disjoint union</italic> and <italic>gluing </italic> as well as a <italic>unit</italic> given by the empty rigged surface. These operations satisfy axioms of commutivity, associativity, unitality, transitivity, distributivity, and unit cancellation up to coherence isomorphism. Furthermore, these coherence isomorphisms satisfy coherence diagrams. These operations, coherences, and their diagrams are neatly encoded as a <italic> pseudo algebra over the 2-theory of commutative monoids with cancellation </italic>. A <italic>conformal field theory</italic> is a morphism of stacks of such structures. This thesis begins with a review of 2-categorical concepts, Lawvere theories, and algebras over Lawvere theories. We prove that the 2-categories of small categories and small pseudo algebras over a theory admit weighted pseudo limits and weighted bicolimits. The 2-category of pseudo algebras over a theory is bi-equivalent to the 2-category of algebras over a 2-monad with pseudo morphisms. We prove that a pseudo functor admits a left bi-adjoint if and only if it admits certain bi-universal arrows. An application of this theorem implies that the forgetful functor for pseudo algebras admits a left bi-adjoint. We introduce stacks for Grothendieck topologies and prove that the traditional definition of stacks in terms of descent data is equivalent to our definition via bilimits. The final chapter contains a proof that the 2-category of pseudo algebras over a 2-theory admits weighted pseudo limits. This result is relevant to the definition of conformal field theory because bilimits are necessary to speak of stacks.