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76 Chapter 12. Entourages and (quasi-)uniformities
Thus ∆(x) ∈ o(E) and, by Proposition 2.2.1, ∆ : L→o(E) is localic. Therefore, by 11.1, an o(E) is an entourage if and only if
{a ∈ L | (a, a) ∈ E} = 1.
12.1.2. Note. By the following general property of localic maps it follows that
an entourage may be equivalently described as an open o(E) for which E → U ≤ ∆∆∗(U) for all U ∈ L ⊕ L.
Proposition. For any localic map f : L → M, f∗(a) = 1 iff a → x ≤ ff∗(x) for every x ∈ M.
Proof. ⇒: Assume f∗(a) = 1. It suffices to show that, for every x, b ∧ a ≤ x implies b ≤ ff∗(x). So let b ∧ a ≤ x. Then f∗(b) ∧ f∗(a) ≤ f∗(x), that is, f∗(b) ≤ f∗(x). Thus b ≤ ff∗(x).
⇐: ff∗(a) ≥ a → a = 1 which implies, by Lemma 2.3, that f∗(a) = 1.
12.2.Compositionofentourages. Recallfrom[26]thatinanycartesiancategory C with images, a relation r : X ̸→ Y is a subobject r : R → X×Y; equivalently, a pair
(r1 :R→X,r2 :R→Y) ofmorphismswith<r1,r2 >:R→X×Y asubobject.Thecomposites◦rof
r:X̸→Y ands:Y ̸→Zistheimageof
< r 1 · s ′1 , s 2 · r 2′ > : R × Y S → X × Z ,
where(R×Y S,s′1,r2′)isthepullbackof(r2,s1). R×Y S
s ′1 x x x x F F F F F r 2′
R {{xxxxxxx FFFFFF## S
r1 ~~~ GGGGG r2 s1 xxxxx X ~~~~~~~~ GGGGGG## Y {{xxxxxxx
???? s2 ????? Z
This composition is associative iff the category C is regular ([26], Theorem 1.569). Since Loc is not regular this calculus of relations is not associative in