Chapter 64: Internal Hom and Exponentials

the relevant chapter (Part XIV) earned the cartesian product × in E_τ (the relevant definition, I.D60), and Part XIII built the Yoneda embedding y : Cat_τ → PSh(Cat_τ) (the relevant definition, I.D54). This chapter constructs the internal hom (exponential) Q^P in the earned topos E_τ = PSh(Cat_τ) (the relevant definition, I.D64), satisfying the universal property:

for all presheaves A, P, Q. The presheaf formula is (Q^P)(X) = Nat(y(X) × P, Q), and in the thin setting of Cat_τ this simplifies: the exponential is determined by a support predicate on divisibility pairs. The Cartesian Closed Theorem (the relevant theorem, I.T28) proves that E_τ has all exponentials, completing its structure as a cartesian closed category. The Self-Enrichment Proposition (Proposition [prop:self-enrichment], I.P28) shows that E_τ is enriched over itself via the internal hom — the Hom-sets of E_τ live inside E_τ. The internal hom provides the function-space structure needed for the spectral coefficients of the relevant chapter and for the Global Hartogs Theorem (Part XV).