TEXTOS DE MATEMÁTICA - SÉRIE B - Nº 21

SCHOOL ON CATEGORY THEORY AND APPLICATIONS

This volume consists of the lecture notes of the three courses to be given at the School on Category Theory and Applications, that will take place at the Department of Mathematics of the University of Coimbra, July 13-17, 1999.

We are grateful to John Baez, Maria Cristina Pedicchio and Vaughan Pratt for their willingness to give these courses and for making possible to have the lecture notes ready in advance.

We would like to thank the Department of Mathematics of the University of Coimbra for its support throughout the organization of this event and for accepting the publication of this volume in its series "Textos de Matemática - Série B".

Thanks are also due to Centro de Matemática da Universidade de Coimbra, Fundação para a Ciência e a Tecnologia, Fundação Luso-Americana para o Desenvolvimento and PRAXIS Project PCTEX/P/MAT/46/96 for financial support.

June 1999                  M. Sobral, M. M. Clementino, J. Picado, L. Sousa (ed.)

CONTENTS

n-Categories, John Baez   1

1. Introduction   3

2. n-Categories   7

3. Lessons from Homotopy Theory   12

4. Examples of Categorification   19

5. n-Categories and the Algebra of Opetopes   23

6. n-Coherent operad algebras   31

7. Conclusions   35

References   36

Chu Spaces, Vaughan Pratt   39

1. Introduction to Chu spaces   41

2. Special Realizations   47

3. General Realizations   53

4. Operations on Chu spaces   59

5. Axiomatics of Multiplicative Linear Logic   63

6. Transformational Logic   75

7. Full completeness of MLL for Chu spaces   87

Bibliography   99

Algebraic Theories, M. Cristina Pedicchio   101

Introduction   103

1. Algebraic categories   105

2. Exact categories   113

3. Lawvere Theorem for algebraic categories   121

4. Locally finitely presentable categories   129

5. Algebraic categories as locally finitely presentable ones   135

6. Algebraic functors and duality theorem   143

7. Mal'cev categories   151

Bibliography   159