**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