Topos-Theoretic Formulation of Axiom 0¶
Gemini, 2026-03-07
QS as Terminal Coalgebra¶
Under AFA, non-well-founded sets form the terminal coalgebra of the powerset functor. QS = terminal coalgebra. Any dynamical system maps uniquely into QS. QS is the universal container of uninstantiated potentiality.
CS as Geometric Morphism¶
CS projects the non-Boolean ambient topos (QS) into the Boolean subtopos (PS).
| Topos | RTSG |
|---|---|
| Ambient topos | QS |
| Boolean subtopos | PS |
| Geometric morphism | CS |
| Terminal coalgebra | QS universal potentiality |
| Evaluation map | Collapse = BRST H0(s) |
Advantages: CS becomes a natural transformation. Quantum logic (non-Boolean) in QS and classical logic (Boolean) in PS arise from subtopos structure. Terminal coalgebra guarantees universality.
Connection to Will Field (2026-03-07)¶
The GL action \(S[W]\) operates on the terminal coalgebra (QS). The geometric morphism (CS) projects to the Boolean subtopos (PS). The Will Field's U(1) symmetry is the categorical statement that the geometric morphism preserves the structural (but not phase) information.