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Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory
 
Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory
eBook ISBN:  978-1-4704-0464-2
Product Code:  MEMO/182/860.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory
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Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory
eBook ISBN:  978-1-4704-0464-2
Product Code:  MEMO/182/860.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1822006; 171 pp
    MSC: 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 2-theories, their pseudo morphisms, bilimits, bicolimits, biadjoints, stacks, and related concepts.

    These 2-categorical 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 2-theory of commutative monoids with cancellation. A conformal field theory is a morphism of stacks of such structures.

    This paper begins with a review of 2-categorical concepts, Lawvere theories, and algebras over Lawvere theories. We prove that the 2-category of small pseudo algebras over a theory admits weighted pseudo limits and weighted bicolimits. This 2-category is biequivalent to the 2-category of algebras over a 2-monad 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 2-functor 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 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.

    Readership

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Some comments on conformal field theory
    • 3. Weighted pseudo limits in a 2-category
    • 4. Weighted pseudo colimits in the 2-category of small categories
    • 5. Weighted pseudo limits in the 2-category of small categories
    • 6. Theories and algebras
    • 7. Pseudo $T$-algebras
    • 8. Weighted pseudo limits in the 2-category of pseudo $T$-algebras
    • 9. Biuniversal arrows and biadjoints
    • 10. Forgetful 2-functors for pseudo algebras
    • 11. Weighted bicolimits of pseudo $T$-algebras
    • 12. Stacks
    • 13. 2-theories, algebras, and weighted pseudo limits
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1822006; 171 pp
MSC: 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 2-theories, their pseudo morphisms, bilimits, bicolimits, biadjoints, stacks, and related concepts.

These 2-categorical 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 2-theory of commutative monoids with cancellation. A conformal field theory is a morphism of stacks of such structures.

This paper begins with a review of 2-categorical concepts, Lawvere theories, and algebras over Lawvere theories. We prove that the 2-category of small pseudo algebras over a theory admits weighted pseudo limits and weighted bicolimits. This 2-category is biequivalent to the 2-category of algebras over a 2-monad 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 2-functor 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 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.

Readership

  • Chapters
  • 1. Introduction
  • 2. Some comments on conformal field theory
  • 3. Weighted pseudo limits in a 2-category
  • 4. Weighted pseudo colimits in the 2-category of small categories
  • 5. Weighted pseudo limits in the 2-category of small categories
  • 6. Theories and algebras
  • 7. Pseudo $T$-algebras
  • 8. Weighted pseudo limits in the 2-category of pseudo $T$-algebras
  • 9. Biuniversal arrows and biadjoints
  • 10. Forgetful 2-functors for pseudo algebras
  • 11. Weighted bicolimits of pseudo $T$-algebras
  • 12. Stacks
  • 13. 2-theories, algebras, and weighted pseudo limits
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.