Chapter 63: Wedge Product and the Co-Cartesian Structure

the relevant chapter earned the categorical product × in the earned topos E_τ (the relevant definition, I.D59) from the Cantor pairing of NF addresses (Part V). This chapter earns the second monoidal structure: the categorical coproduct ∧ (the relevant definition, I.D62), inherited from the join/lcm operation of Part VIII (the relevant definition, I.D32). The coproduct is pointwise: for presheaves P and Q, (P ∧ Q)(X) = P(X) ∨ Q(X) via Boolean disjunction on membership values. The distributivity theorem (the relevant theorem, I.T27) establishes that × distributes over ∧, exactly as multiplication distributes over addition in a ring. The bi-monoidal structure (the relevant definition, I.D63) equips E_τ with two compatible monoidal operations (E_τ, ×, ∧): × plays the role of multiplication, ∧ plays the role of addition. The internal hom will complete this to a cartesian closed structure.