Alphabeta MathHigher Categorical Symmetries and Defect Algebras
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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
- Finite Fields and Cyclotomic Extensions
- 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
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Dimension Constructible Images and Dimensions of Fibres
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
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
- Adjunctions Units and Counits
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- Exactness and the Member Calculus
- 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
- 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
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
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
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
- Delta Functors and Universality
- 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
- Yoneda Extensions and Homological Dimension
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
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
Scheme Theory
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
Definition
Finite semisimple module category over a fusion category
Definition
G-graded finite semisimple 2-vector space
Definition
Weighted RT background interface for Mod(C)
Definition
Separable algebra objects and bimodules in a fusion category
Definition
Weak linear 2-category
Lemma
The canonical algebra of a fusion category is Frobenius
Lemma
Surjective orbit tangents give finitely many open orbits
Lemma
A separability element contracts the normalized categorical Hochschild complex in positive degrees
Definition
Finite semisimple 2-category
Definition
Relative Deligne product in the braided module 2-category
Definition
Weak monoidal 2-category
Definition
Opposite algebra induced by a braiding
Lemma
Canonical algebra modules and coherent tensor sliding
Lemma
Right module internal ends and Eilenberg–Watts equivalence
Lemma
A uniform rank and action-multiplicity bound for module categories
Lemma
Module functor adjoints and separable-monad splittings in Mod(C)
Lemma
A separability idempotent constructs the balanced relative tensor product
Lemma
Finite semisimplicity for separable algebra modules and bimodules
Theorem
A braiding gives the tensor product algebra structure
Definition
Four-cocycle twisted monoidal structure on 2Vect_G
Definition
Fusion 2-category
Lemma
The braided tensor product of separable algebras is separable
Lemma
Nonzero global dimension and the canonical Frobenius scalar
Theorem
Relative tensor product gives associative composition of separable bimodules
Lemma
The canonical algebra of a characteristic-zero fusion category is separable
Lemma
Object duals in the separable-algebra model of Mod(C)
Theorem
A normalized group 4-cocycle gives a fusion 2-category 2Vect^ω_G
Theorem
A normalized 3-cochain gives the gauge equivalence of 2Vect^ω_G
Lemma
Categorical separability contracts the module deformation complex
Lemma
Module-functor categories are finite semisimple
Lemma
Bounded-rank module-structure varieties have finitely many open orbits
Theorem
Finite semisimple module categories are separable algebra modules
Theorem
The relative Deligne product is represented by the braided tensor algebra
Theorem
The algebraic 2-category Mod(C) is a fusion 2-category
Sources
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §1.4
- Décoppet, Extension Theory and Fermionic Strongly Fusion 2-Categories, §2.2
- Décoppet and Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories, §2.2
- Décoppet and Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories, §2.1
- Décoppet, Extension Theory and Fermionic Strongly Fusion 2-Categories, Example 2.2
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, Construction 2.1.19 (algebraic background)
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §§1.3 and 2.1.3
- 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, §§1.1-1.2
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Sections 7.20–7.21
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Theorem 9.1.4
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Proposition 9.1.1 and formula (9.2)
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §§1.4.1-1.4.3
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, Construction 2.1.19
- Décoppet, The Relative Deligne Tensor Product over Pointed Braided Fusion Categories, §1
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §2.1.1
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Proposition 7.18.5 and equation (7.57)
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§7.9–7.11
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Corollary 9.1.6
- Etingof, Nikshych, and Ostrik, On fusion categories, Section 8.1
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, Theorem 1.4.8 and Appendix A.3
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§7.11–7.12
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §§1.3 and 1.4
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Proposition 7.8.30 and §7.11
- 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, Definitions 2.1.5-2.1.6
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Sections 7.18–7.21
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §2.1.1 and Construction 2.1.19
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Sections 7.22 and 9.1
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§7.11–7.12 and §7.21
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Theorems 9.1.4–9.1.6
- Etingof, Nikshych, and Ostrik, On fusion categories, Sections 2.7 and 7.3
- Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§7.8–7.12 and §7.21
- Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, Construction 2.1.19 and Theorem 1.4.9