Alphabeta MathHigher Categorical Symmetries and Defect Algebras: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- Limits and Colimits
- Monads Comonads and Their Algebras
- Monoidal Categories and Monoidal Functors
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor and Fusion Categories
Combinatorics
Linear Algebra
Real Analysis
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Example
Matrix 2-vector spaces exhibit horizontal and vertical composition
Example
Typed object, 1-morphism, and 2-morphism defect composition
Example
Relative tensor products of bimodules over C times C
Example
A C4 coboundary changes the pentagonator and is gauged away
Example
The trivial group gives ordinary 2Vect
Sources
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §1.4.5
- Décoppet and Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories, §2.1
- Décoppet and Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories, §1
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §2.1
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, Construction 2.1.19
- Décoppet and Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories, §2.3
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, Construction 2.1.16
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §2.1.3
- Décoppet, Extension Theory and Fermionic Strongly Fusion 2-Categories, Example 2.2