Sheaf representation of monoidal categories

Axioms for the category of Hilbert spaces and linear contractions

“…Theorem 26. A dagger rig category 𝐃 satisfying axioms ( 1)-( 11) is equivalent to 𝐂𝐨𝐧 ℝ or 𝐂𝐨𝐧 ℂ , and the equivalence preserves daggers and is strong symmetric monoidal for both ⊗ and ⊕ and preserves the distributors of axiom (2).…”

“…In addition to being an important characterisation in its own right, this theorem is also a stepping stone towards trying to characterise related categories: Hilbert spaces and completely positive morphisms, going towards foundations of quantum information theory; Hilbert spaces and unitaries, going towards foundations of quantum computing; and Hilbert modules and adjointable morphisms, going towards unitary representations and foundations of quantum field theory [2].…”