Memoirs of the American Mathematical Society 2006; 171 pp; softcover Volume: 182 ISBN10: 0821839144 ISBN13: 9780821839140 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/182/860
 In this paper we develop the categorical foundations needed for working out completely the rigorous approach to the definition of conformal field theory outlined by Graeme Segal. We discuss pseudo algebras over theories and 2theories, their pseudo morphisms, bilimits, bicolimits, biadjoints, stacks, and related concepts. These 2categorical concepts are used to describe the algebraic structure on the class of rigged surfaces. A rigged surface is a real, compact, not necessarily connected, two dimensional manifold with complex structure and analytically parametrized boundary components. This class admits algebraic operations of disjoint union and gluing as well as a unit. These operations satisfy axioms such as unitality and distributivity up to coherence isomorphisms which satisfy coherence diagrams. These operations, coherences, and their diagrams are neatly encoded as a pseudo algebra over the 2theory of commutative monoids with cancellation. A conformal field theory is a morphism of stacks of such structures. This paper begins with a review of 2categorical concepts, Lawvere theories, and algebras over Lawvere theories. We prove that the 2category of small pseudo algebras over a theory admits weighted pseudo limits and weighted bicolimits. This 2category is biequivalent to the 2category of algebras over a 2monad with pseudo morphisms. We prove that a pseudo functor admits a left biadjoint if and only if it admits certain biuniversal arrows. An application of this theorem implies that the forgetful 2functor for pseudo algebras admits a left biadjoint. 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 paper ends with a proof that the 2category of pseudo algebras over a 2theory admits weighted pseudo limits. This result is relevant to the definition of conformal field theory because bilimits are necessary to speak of stacks. Readership Table of Contents  Introduction
 Some comments on conformal field theory
 Weighted pseudo limits in a 2category
 Weighted pseudo colimits in the 2category of small categories
 Weighted pseudo limits in the 2category of small categories
 Theories and algebras
 Pseudo \(T\)algebras
 Weighted pseudo limits in the 2category of pseudo \(T\)algebras
 Biuniversal arrows and biadjoints
 Forgetful 2functors for pseudo algebras
 Weighted bicolomits of pseudo \(T\)algebras
 Stacks
 2Theories, algebras, and weighted pseudo limits
 Bibliography
 Index
