Alphabeta MathModular Categories, Verlinde Formula, and TQFT: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
Algebraic Topology
- Bocksteins Steenrod Squares and Cohomology Operations
- Chern and Pontryagin Classes by Splitting and Complexification
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Generalized Cohomology and the Atiyah Hirzebruch Spectral Sequence
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- Spectra and Stable Homotopy Groups
- Stiefel Whitney and Euler Classes by Universal Constructions
- The Serre Spectral Sequence and Applications
- Thom Spectra and Unoriented Bordism Detection
- Topological Vector Bundles and Grassmannian Classification
Braid Groups
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- Exactness and the Member Calculus
- Limits and Colimits
- Monoidal Categories and Monoidal Functors
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Strictification and Mac Lanes Coherence Theorem
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor and Fusion Categories
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
- Chains, Antichains, Sperner and Dilworth
- Finite Counting, Factorials and Binomial Coefficients
- Formal Power Series
- Graphs, Walks and Connectivity
- Lattice Paths and Catalan Numbers
- Linear Recurrences and Rational Generating Functions
- Plane Graphs, Euler's Formula and the Five Colour Theorem
- Trees, Forests and Spanning Trees
Commutative Algebra
Complex Analysis
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- Mittag-Leffler and Runge's Theorem
- Periods, Jacobians, and Abel--Jacobi Theory
- Riemann Surfaces, Branched Maps, and Differentials
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Coxeter Groups
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Sard Theorem and Transversality
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- The Gauss Bonnet Theorem for Riemannian Surfaces
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
- Characteristic Numbers and Cobordism Obstructions
- Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations
- Exotic Smooth Structures and Milnor Spheres
- Formal Immersions and the Smale Hirsch Theorem
- Gradient Like Vector Fields and Morse Trajectories
- Handle Cancellation Slides and Elementary Moves
- Handle Decompositions Duality and Rearrangement
- Intersection Pairings Self Intersection and Euler Classes
- Isotopy Extension and Embedding Theory Beyond Whitney
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Oriented and Mod Two Intersection Numbers
- Reeb Stability and Global Foliation Constructions
- Smooth Cobordism Relations Groups and Rings
- Smooth Surgery Traces and Handle Trading
- Stable Unstable Manifolds and Morse Smale Transversality
- Sublevel Deformation and the Handle Attachment Theorem
- The Hirzebruch Signature Theorem
- The Whitney Trick and Surgery Below the Middle Dimension
- Thom Spaces Normal Data and Collapse Maps
- Vanishing Cycles, Novikov and Taut Foliations
Functional Analysis
Group Theory
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
Number Theory
Partial Differential Equations
Quantum Algebra and Representation Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- 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
Topology
- Classification of Compact Connected Surfaces
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Ascoli–Arzelà Theorem
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Counterexample
Symmetric Rep(Z/2) has singular Hopf-link matrix
Example
The pointed Z/2 semion metric group
Example
The ell=3 framing anomaly and Kirby blow-up
Example
A genus-two RT dimension from factorization
Example
The sl2 modular data at ell=4
Example
The sl2 modular data at ell=3
Sources
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §8.13
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §3.1
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Example 8.13.5
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§3.3 and 5.7
- Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity, §3.3
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§3.3 and 5.8
- Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §3.3