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

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*Thomas M. Fiore*

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.

#### Table of Contents

# Table of Contents

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

- Contents vii8 free
- Acknowledgements ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Some Comments on Conformal Field Theory 516 free
- Chapter 3. Weighted Pseudo Limits in a 2-Category 920
- Chapter 4. Weighted Pseudo Colimits in the 2-Category of Small Categories 2132
- Chapter 5. Weighted Pseudo Limits in the 2-Category of Small Categories 3142
- Chapter 6. Theories and Algebras 3950
- Chapter 7. Pseudo T-Algebras 6172
- Chapter 8. Weighted Pseudo Limits in the 2-Category of Pseudo T-Algebras 7384
- Chapter 9. Biuniversal Arrows and Biadjoints 8192
- Chapter 10. Forgetful 2-Functors for Pseudo Algebras 113124
- Chapter 11. Weighted Bicolimits of Pseudo T-Algebras 129140
- Chapter 12. Stacks 137148
- Bibliography 163174
- Index 167178 free