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A↑↑Β iff A↑Β and A≠B. 6 Definition Let |YX | = {(a,z) / a∈ X, a is finite, z∈ |Y| }. Moreover, let (a,z) ↑ (a',z') iff i. a ↑↑ a' [mod X] ⇒ z ↑↑ z' [mod Y] and ii. a ↑ a' [mod X] ⇒ z ↑ z' [mod Y]. Then YX is the arrow domain (exponent object). Exercises 1. Prove that conditions (i) and (ii) may be stated equivalently as (a,z) = (a',z') or z↑↑z' or not a↑a'. 2. Prove that every element of YX is a trace of some stable function from X to Y , and conversely that if F: X→Y is stable then tr(F)∈YX.

The elements of |X| are called points, and the relation ↑ is called coherence. The coherent domain associated with (|X|,↑ ) is the collection X of subsets of P(|X|) whose points are pairwise coherent. The elements of X are ordered by set-inclusion. Coherence is extended to X in the obvious way, that is: A ↑ B iff A∪B∈X. Exercise Prove, when X is a coherent domain, that 1. ∅∈X 2. X is closed under directed union 3. 2 Definition Let X, Y be two coherent domains. A function F: X → Y is stable iff i).

Prove that conditions (i) and (ii) may be stated equivalently as (a,z) = (a',z') or z↑↑z' or not a↑a'. 2. Prove that every element of YX is a trace of some stable function from X to Y , and conversely that if F: X→Y is stable then tr(F)∈YX. 3. Let f,g : X → Y be two stable functions. Define f ≤B g (Berry's order) iff ∀x,y∈X x ⊆ y ⇒ f(x) = f(y)∩g(x) Prove that f ≤B g if and only if Tr(f) ⊆ Tr(g). Let moreover ≤p be the pointwise order. Prove that: 25 2. Constructions i. f ≤B g ⇒ f ≤p g ii. f↑g ⇒ (f ≤Bg ⇔ f ≤p g) 4.

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Categories, types and structures by Asperti A.


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