HDA
HigherDimensional Algebra
John Baez
April 6, 2010
Here's the outline for a project I've been wanting to
do for a long time: a series of books on
higherdimensional algebra and its applications to topology,
representation theory, combinatorics, and quantum physics.
I may or may not actually finish, or even really start, this project.
Some of the sections have links
to more detailed outlines of those sections. But, it's all
incredibly unfinished.

THE BASIC IDEAS

The Dimensional Ladder

Introduction

The Ladder of nCategories

The Periodic Table

Categories

Categories of Mathematical Objects

Categories as Mathematical Objects

Categories from Spaces

Functors

The Category of Categories

Natural Transformations

Equivalence of Categories

Functor Categories

Cat as the Primordial 2Category

2Categories

Strict 2Categories

Weak 2Categories (Bicategories)

Monads and Adjunctions

Limits and Colimits

BiCat as the Primordial 3Category

2Cat versus BiCat  Coherence Theorems

Enriched Categories

Internal Categories

3Categories

Strict 3categories

Semistrict 3categories

Tricategories

Weak Monads and Adjunctions

Weak Limits and Colimits

Enrichment and Internalization

The Lax World

TriCat as the Primordial 4Category

Coherence Theorems

4Categories

Case Studies

Braids and Tangles

Quantum Groups

Real Numbers, Complex Numbers, Quaternions and Octonions

Electrical Circuits

Logic Gates

2Braids and 2Tangles

Logic, Topology, and Symmetry

The Idea of Galois Theory

Properties, Structure and Stuff

Weak Limits and Colimits

Case Studies

Categorified Arithmetic

Introduction

Various kinds of categories with colimits

Categories with finite coproducts

Categories with finite colimits

Categories with all small colimits (cocomplete categories)

Various kinds of monoidal categories

Monoidal categories

Braided monoidal categories

Symmetric monoidal categories

Various kinds of categorified rigs

Distributive categories

2Rigs

Rig categories

Structure Types

Homotopy Cardinality

Stuff Types and nStuff Types

Case Studies

Feynman Diagrams

Representations of the Classical Groups

SchurWeyl Theory

qDeformation

Euler characteristic versus homotopy cardinality

Categories as Theories

Category Representations

Finite Product Theories (Algebraic Theories)

Finite Limit Theories (Essentially Algebraic Theories)

Topoi as Theories

Operads and PROPs

Compact PROPs

Topological Quantum Field Theories

Conformal Field Theories

The History of HigherDimensional Algebra

LESSONS FROM HOMOTOPY THEORY

Simplicial Objects

Cohomology and the Bar Construction

Kan Complexes

Model Categories

Homotopy Limits and Colimits

Postnikov Towers

Iterated Loop Spaces

The Associahedron and Little kCubes Operads

Spectra

Case Studies

Higher Gauge Theory

Brave New Algebra

nStacks

Elliptic Cohomology

TOWARD A THEORY OF WEAK ∞CATEGORIES
Stay tuned!
© 2010 John Baez  all rights reserved
baez@math.removethis.ucr.andthis.edu