Electronic ISBN:  9781470404642 
Product Code:  MEMO/182/860.E 
171 pp 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 182; 2006MSC: Primary 18; Secondary 81;
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

Chapters

1. Introduction

2. Some comments on conformal field theory

3. Weighted pseudo limits in a 2category

4. Weighted pseudo colimits in the 2category of small categories

5. Weighted pseudo limits in the 2category of small categories

6. Theories and algebras

7. Pseudo $T$algebras

8. Weighted pseudo limits in the 2category of pseudo $T$algebras

9. Biuniversal arrows and biadjoints

10. Forgetful 2functors for pseudo algebras

11. Weighted bicolimits of pseudo $T$algebras

12. Stacks

13. 2theories, algebras, and weighted pseudo limits


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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.

Chapters

1. Introduction

2. Some comments on conformal field theory

3. Weighted pseudo limits in a 2category

4. Weighted pseudo colimits in the 2category of small categories

5. Weighted pseudo limits in the 2category of small categories

6. Theories and algebras

7. Pseudo $T$algebras

8. Weighted pseudo limits in the 2category of pseudo $T$algebras

9. Biuniversal arrows and biadjoints

10. Forgetful 2functors for pseudo algebras

11. Weighted bicolimits of pseudo $T$algebras

12. Stacks

13. 2theories, algebras, and weighted pseudo limits