## Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory: Categorical Foundations of Conformal Field TheoryIn 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. |

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### Contents

1 | |

Chapter 2 Some Comments on Conformal Field Theory | 5 |

Chapter 3 Weighted Pseudo Limits in a 2Category | 9 |

Chapter 4 Weighted Pseudo Colimits in the 2Category of Small Categories | 21 |

Chapter 5 Weighted Pseudo Limits in the 2Category of Small Categories | 31 |

Chapter 6 Theories and Algebras | 39 |

Chapter 7 Pseudo TAlgebras | 61 |

Chapter 8 Weighted Pseudo Limits in the 2Category of Pseudo TAlgebras | 73 |

Chapter 9 Biuniversal Arrows and Biadjoints | 81 |

Chapter 10 Forgetful 2Functors for Pseudo Algebras | 113 |

Chapter 11 Weighted Bicolimits of Pseudo TAlgebras | 129 |

Chapter 12 Stacks | 137 |

163 | |

167 | |

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### Common terms and phrases

2-category of pseudo 2-category of small 2-monad 2-theory a e Obj admits pseudo admits weighted pseudo algebras axiom for pseudo bicategories bijection bilimits biuniversal arrow Chapter coherence 2-cells coherence diagrams coherence isos colimit commutative monoids composition axiom composition of 2-cells conformal field theory cotensor products define definition directed graph End(I End(X enriched in groupoids equivalence of categories example forgetful 2-functor function functor F Grothendieck covers Grothendieck topology Hence hom sets iso 2-cell isomorphism of categories j e Obj left biadjoint LEMMA Let F monoids with cancellation Mor A(FX Mora Morc(W morphism f morphism of pseudo morphism of theories natural isomorphism notation objects Proof pseudo cone pseudo functor pseudo morphisms pseudo natural transformation pseudo T-algebras PseudoCone(F rigged surfaces stack strict 2-functor strictly 2-natural structure maps terminal object Theorem theories enriched trivial unit axiom universal vertical composition weighted pseudo limits