Alphabeta MathFrobenius Algebras and Oriented Two-Dimensional Sewing
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1 · Prerequisites
Abstract Algebra
Categorification and Geometric Representation Theory
Category Theory
Differential Geometry
- 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
- Sard Theorem and Transversality
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Vector Fields Flows and Lie Derivatives
Differential Topology
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- 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
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
The oriented open-closed surface category with colored free boundary
Definition
Frobenius objects and typed colored knowledgeable sewing data
Lemma
A Frobenius functional determines a unique Frobenius coproduct
Lemma
Face-relative Morse decomposition into the ten sewing generators
Lemma
Elementary colored surface sewings give the Frobenius relations
Lemma
Typed algebraic derivations of the Frobenius sewing moves
Definition
The marked boundary-relative open-closed normal word
Lemma
The complete colored boundary-relative surface normal-form theorem
Theorem
Closed oriented two-dimensional TQFTs are commutative Frobenius algebras
Theorem
Typed colored knowledgeable Frobenius data classify open-closed sewing
Sources
- Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras, arXiv:math/0510664v3
- Joachim Kock, Frobenius algebras and 2D topological quantum field theories, complete minicourse notes
- Gregory W. Moore and Graeme Segal, D-branes and K-theory in 2D topological field theory, arXiv:hep-th/0609042v1
- Joachim Kock, Frobenius algebras and 2D topological field theories, complete minicourse notes