Higher Dimensional Categories

The study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.

We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.

This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.

Contents:

• Introduction
• From Categories to Double Categories:
  • A Review of Basic Category Theory
  • Introducing Two-Dimensional Category Theory
  • Double Categories
  • Double Adjunctions
  • Double Limits
• Multiple Categories:
  • Weak and Lax Multiple Categories
  • Multiple Adjunctions
  • Monads and Algebras for Multiple Categories
• Appendices:
  • Applications in Homological Algebra and Algebraic Topology
  • Symmetric Cubical Sets and Cubical Categories
  • Solutions and Hints
• References
• Index

Readership: Graduate and postgraduate students, researchers.Category Theory;Double Categories;Multiple Categories0Key Features:

• Higher Dimensional Category Theory is usually studied in a globular form, more complicated than the present one. This is the only book based on 'multiple categories' and their extensions
• The main notions are presented in a concrete way, starting from examples taken from Algebra, Lattice Theory and Topology
• Then the theory is developed presenting other examples and some 250 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints