By Ruark A.E.

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5. 6. HH HH HH HH f HH HH HH $ R G A g Ö In Sets, the initial object is the empty set ∅. In Pos, the initial object is the poset (∅, ∅). In Top, the initial object is the space (∅, {∅}). In VectK , the one-element space {0} is the initial object. In a poset, the initial object is the least element with respect to the ordering, if it exists. 15 A terminal object in a category C is a C-object 1 such that, given any other C-object A, there exists one and only one C-arrow from A to 1.

22) ˆ = 0. This implies that V (0) 3. Given a projection operator Pˆ we know that Pˆ 2 = Pˆ , therefore V (Pˆ )2 = V Pˆ 2 = V (Pˆ ). 23) V (Pˆ ) = 1 or 0. 24) It follows that Since quantum propositions can be expressed as projection operators (the reason will be explained later on in the book), what the last result implies is that, for any given state |ψ , the valuation function can only assign value true or false to propositions. The set of all eigenvectors of a self-adjoint operator Aˆ forms an orthonormal basis for H, thus we can define the resolution of unity in terms of the projection operators corresponding to the eigenvectors: 1ˆ = M Pˆm .

An iso arrow is always epic. Proof Consider an iso f such that g ◦ f = h ◦ f (f : a → b and g, h : b → c) g = g ◦ 1b = g ◦ f ◦ f −1 = (g ◦ f ) ◦ f −1 = (h ◦ f ) ◦ f −1 = h ◦ f ◦ f −1 = h therefore f is right cancellable. Note not all arrows which are monic and epic are iso, for example: 1. In poset, even though all functions are monic and epic, the only isos are the identity map. In fact, consider an arrow f : p → q, this implies that p ≤ q. If f is an iso, then f −1 : q → p exists, therefore, q ≤ p.