Fixed Points, Self-Reference, and 2D-TQFT as One Categorical Duality — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Three fixed-point theorems (Lawvere, Brouwer, Borsuk–Ulam) and Frobenius algebras/2D-TQFT form a single categorical-topological structure: self-reference, fixed points, and topological quantum field theory are manifestations of the same adjunction/duality machinery. | MATH: Lawvere: in a Cartesian closed category, if f: A → B^A is surjective (or has a weak point-surjectivity), then every endomorphism g: B → B has a fixed point; formally ∃a: g(f(a)) = f(a). Brouwer: every continuous f: D^n → D^n has a fixed point; proved via π₁(S¹) ≅ ℤ (fundamental group) or combinatorially via Sperner's lemma. Borsuk–Ulam: for continuous f: S^n → ℝ^n, ∃x with f(x) = f(−x). Frobenius algebra: (A, μ: A⊗A→A, η: k→A, Δ: A→A⊗A, ε: A→k) satisfying (μ⊗id)∘(id⊗Δ) = Δ∘μ = (id⊗μ)∘(Δ⊗id) — the "Frobenius relation." 2D-TQFT: a symmetric monoidal functor Z: Cob₂ → Vect_k, where Cob₂ has objects = closed 1-manifolds (circles) and morphisms = 2D cobordisms; classification theorem: 2D-TQFTs ↔ commutative Frob Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225166570