Higher-Dimensional 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 higher-dimensional 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.

1. THE BASIC IDEAS
   1. The Dimensional Ladder
      1. Introduction
         1. The Ladder of n-Categories
         2. The Periodic Table
      2. Categories
         1. Categories of Mathematical Objects
         2. Categories as Mathematical Objects
         3. Categories from Spaces
         4. Functors
         5. The Category of Categories
         6. Natural Transformations
         7. Equivalence of Categories
         8. Functor Categories
         9. Cat as the Primordial 2-Category
      3. 2-Categories
         1. Strict 2-Categories
         2. Weak 2-Categories (Bicategories)
         3. Monads and Adjunctions
         4. Limits and Colimits
         5. BiCat as the Primordial 3-Category
         6. 2Cat versus BiCat - Coherence Theorems
         7. Enriched Categories
         8. Internal Categories
      4. 3-Categories
         1. Strict 3-categories
         2. Semistrict 3-categories
         3. Tricategories
         4. Weak Monads and Adjunctions
         5. Weak Limits and Colimits
         6. Enrichment and Internalization
         7. The Lax World
         8. TriCat as the Primordial 4-Category
         9. Coherence Theorems
      5. 4-Categories
      6. Case Studies
         1. Braids and Tangles
         2. Quantum Groups
         3. Real Numbers, Complex Numbers, Quaternions and Octonions
         4. Electrical Circuits
         5. Logic Gates
         6. 2-Braids and 2-Tangles

   2. Logic, Topology, and Symmetry
      1. The Idea of Galois Theory
      2. Properties, Structure and Stuff
      3. Weak Limits and Colimits
      4. Case Studies

   3. Categorified Arithmetic
      1. Introduction
      2. Various kinds of categories with colimits
         1. Categories with finite coproducts
         2. Categories with finite colimits
         3. Categories with all small colimits (cocomplete categories)
      3. Various kinds of monoidal categories
         1. Monoidal categories
         2. Braided monoidal categories
         3. Symmetric monoidal categories
      4. Various kinds of categorified rigs
         1. Distributive categories
         2. 2-Rigs
         3. Rig categories
      5. Structure Types
      6. Homotopy Cardinality
      7. Stuff Types and n-Stuff Types
      8. Case Studies
         1. Feynman Diagrams
         2. Representations of the Classical Groups
         3. Schur-Weyl Theory
         4. q-Deformation
         5. Euler characteristic versus homotopy cardinality

   4. Categories as Theories
      1. Category Representations
      2. Finite Product Theories (Algebraic Theories)
      3. Finite Limit Theories (Essentially Algebraic Theories)
      4. Topoi as Theories
      5. Operads and PROPs
      6. Compact PROPs
      7. Topological Quantum Field Theories
      8. Conformal Field Theories

   5. The History of Higher-Dimensional Algebra

2. LESSONS FROM HOMOTOPY THEORY
   1. Simplicial Objects
   2. Cohomology and the Bar Construction
   3. Kan Complexes
   4. Model Categories
   5. Homotopy Limits and Colimits
   6. Postnikov Towers
   7. Iterated Loop Spaces
   8. The Associahedron and Little k-Cubes Operads
   9. Spectra
   10. Case Studies
       1. Higher Gauge Theory
       2. Brave New Algebra
       3. n-Stacks
       4. Elliptic Cohomology

3. TOWARD A THEORY OF WEAK ∞-CATEGORIES

Stay tuned!

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