Appendix D: Coulon-Floer vs Coulon and/or Coulon
This appendix explores the relationships and distinctions between the classical Coulon framework and the Coulon-Floer (CF) extension in both combinatorial and topological contexts.
1. Definitions
Coulon System: A combinatorial or algebraic structure C = <S, F, φ> where S is a set, F a functor mapping subsets to invariants, and φ a cochain functional.
Coulon-Floer System: CF = <S, F, φ, η, η_q>, extending C with deformation parameters controlling classical (η) and quantum (η_q) phase transitions.
Functorial Mapping: CF maps combinatorial chains to cohomological invariants, preserving composition and identity under deformation.
2. Correspondence and Extensions
Coulon → Coulon-Floer: Add geometric/topological deformation via η.
Coulon-Floer → Coulon: Restrict to classical combinatorial invariants (η = 0, η_q = 0).
Quantum Extension: Introduce η_q ≠ 0 producing multiplicative, possibly non-commutative scaling of permutation/combination counts.
Homotopy Extension: Construct cochain homotopy H: C_k → C_{k+1} mapping Coulon chains into CF complexes, preserving invariants under δH + Hδ = Id - ε.
https://stacker.news/items/1416883
This appendix explores the relationships and distinctions between the classical Coulon framework and the Coulon-Floer (CF) extension in both combinatorial and topological contexts.
1. Definitions
Coulon System: A combinatorial or algebraic structure C = <S, F, φ> where S is a set, F a functor mapping subsets to invariants, and φ a cochain functional.
Coulon-Floer System: CF = <S, F, φ, η, η_q>, extending C with deformation parameters controlling classical (η) and quantum (η_q) phase transitions.
Functorial Mapping: CF maps combinatorial chains to cohomological invariants, preserving composition and identity under deformation.
2. Correspondence and Extensions
Coulon → Coulon-Floer: Add geometric/topological deformation via η.
Coulon-Floer → Coulon: Restrict to classical combinatorial invariants (η = 0, η_q = 0).
Quantum Extension: Introduce η_q ≠ 0 producing multiplicative, possibly non-commutative scaling of permutation/combination counts.
Homotopy Extension: Construct cochain homotopy H: C_k → C_{k+1} mapping Coulon chains into CF complexes, preserving invariants under δH + Hδ = Id - ε.
https://stacker.news/items/1416883