By Asperti A.
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Extra info for Categories, types and structures
3 Proposition Let F: C → D be a functor. If a
C = pD for some category D with pullbacks for every pair of arrows. Ct is the associated category of total maps. Since Ct is a subcategory of C, we compose total and partial morphisms by using the same operation of composition. For typographical reasons, we write a° instead of a⊥. The set-theoretic idea we try to formalize categorically is that, when an object a is “lifted” to a° by adding an extra least element, then any hom-set of total maps with target a° is isomorphic to the corresponding hom-set with target 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. Let X,Y be coherent domains. A stable function f: X→Y is linear iff : i. a ∪ b∈X ⇒ f(a ∪ b) = f(a) ∪ f(b) ii.