Chapter 61: The Subobject Classifier and the Earned Topos

the relevant chapter (Part XI) defined Ω_τ = Truth4 as the subobject classifier (the relevant definition, I.D41) — but that was a preview. With the presheaf topos PSh(Cat_τ) (the relevant definition, I.D57) and its Grothendieck topos structure (the relevant theorem, I.T24) now in hand, the preview becomes a theorem. The Ω_τ Subobject Classifier Theorem (the relevant theorem, I.T25) proves that Ω_τ = {T, F, B, N} is the subobject classifier of PSh(Cat_τ). We construct the characteristic morphism χ_S : X → Ω_τ (the relevant definition, I.D58), define the earned topos E_τ (the relevant definition, I.D59), and prove it is paraconsistent (Proposition [prop:non-boolean], I.P27): Ω_τ is a Boolean algebra, yet material implication does not satisfy explosion — a structural consequence of the explosion barrier (the relevant theorem, I.T13). The four truth values of Part XI are now forced by the topos structure. E_τ provides the foundation for Part XIV and the Global Hartogs Theorem (Part XV).