Categorical Quantum Models and Logics (Pallas by Chris Heunen

By Chris Heunen

This dissertation reports the common sense at the back of quantum physics, utilizing classification concept because the primary software and conceptual consultant. to take action, ideas of quantum mechanics are modeled categorically. those express quantum versions are justified by means of an embedding into the class of Hilbert areas, the conventional formalism of quantum physics. particularly, advanced numbers emerge with no need been prescribed explicitly. analyzing good judgment in such different types leads to orthomodular estate lattices, and moreover presents a ordinary atmosphere to contemplate quantifiers. ultimately, topos idea, incorporating express good judgment in a elegant means, we could one learn a quantum method as though it have been classical, specifically resulting in a singular mathematical thought of quantum-

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The following theorem characterises algebraically when a monoidal product is a coproduct, without any reference to universal properties, in a way reminiscent of [87] (see also [84]). 2 Theorem A symmetric monoidal structure (⊕, 0) on a category C provides finite coproducts if and only if the forgetful functor cMon(C) → C is an isomorphism of categories. P ROOF Suppose that (⊕, 0) provides finite coproducts, with the coherence maps α, λ and ρ induced by the coproducts. Denote the forgetful functor cMon(C) → C by U , and define F : C → cMon(C) on objects as F (X) = (X, ∇, u), where ∇ = [idX , idX ] : X ⊕ X → X, and u is the unique morphism 0 → X.

Projections are given by πi (x1 , x2 ) = xi , the coprojections are κ1 (x) = (x, 0) and κ2 (x) = (0, x). The zero-object is the 0-dimensional vector space {0}. 12 Example The categories preHilbK and HilbK inherit finite biproducts from VectK . The inner product on X1 ⊕ X2 is given by (x1 , x2 ) | (y1 , y2 ) X1 ⊕X2 = x 1 | y1 26 X1 + x2 | y2 X2 . 3. Biproducts This prescription shows that infinite biproducts might be problematic. Indeed, they cannot exist in HilbK , as the following lemma shows.

This is a well-defined morphism in Hilb: 2 2 (f )(ϕ)(y) 2 = ϕ(f1 (m)) y∈Y y∈Y |ϕ(f1 (m))|2 ≤ y∈Y m∈f −1 (y) 2 m∈f2−1 (y) |ϕ(f1 (m))|2 ≤ = 2 m∈M |ϕ(x)|2 < ∞. x∈X To verify that this assignment is functorial, suppose that Y o o N G g2 G Z is p1 o P G p2 G Z , another morphisms. Their composition g ◦ f is given by X o where P = {(m, n) ∈ M × N | f2 (m) = g1 (n)}, with p1 (m, n) = f1 (m) and p2 (m, n) = g2 (n). Hence ( 2 (g) ◦ 2 (f ))(ϕ)(z) = g1 ϕ(f1 (m)) = n∈g2−1 (z) m∈f2−1 (g1 (n)) ϕ(f1 (m)) (m,n)∈P,g2 (n)=z ϕ(p1 (m, n)) = ( 2 (g ◦ f ))(ϕ)(z).

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