Page 233 - Textos de Matemática Vol. 46

FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 223
(F2) aRp(a);
(F3) @(a, b)Ry ) aRy;
for every a 2 A and b, y 2 B with aRb.
Clearly this is a generalization of a preorder. The axiom (F2) generalizes the reflexivity axiom and (F3) the transitivity axiom. Indeed, when p: A ! B is the identity map 1B : B ! B, then @ has to be the second projection, (F2) reads as aRa and (F3) as aRb and bRy implies aRy.
Concerning the definition of morphisms, one possibility would be to say that they are maps with a certain property, which in the case of topological spaces would correspond to the property of a map being continuous. However, by requiring that a morphism is a map with an extra structure, as considered here, we have the following three advantages: (a) we can still look at it as a property, by asking whether or not the extra structure exists; (b) in the case of topological spaces, and more specifically in metric spaces, one is able to distinguish between di↵erent notions of continuity, such as uniform continuity (see for instance the example of natural spaces in Section 4); (c) it is a more natural notion to define in an abstract category. Furthermore, in the case of preorders the extra structure is uniquely determined, up to a unique isomorphism of pullbacks, provided it exists.
For convenience of writing we will sometimes refer to the structure of a fi- brous preorder A = ( R @ // A p // B ) simply as a five-tuple (R,A,B,@,p).
Definition 2.2. Let A = (R,A,B,@,p) and A0 = (R0,A0,B0,@0,p0)) be two fibrous preorders. A fibrous morphism between A and A0 is a pair (f,') with f:B!B0 amapfromBtoB0 and':A0f !Aamapfrom
to A such that
and
A0f ={(a0,b)2A0⇥B|p0(a0)=f(b)} p'(a0, b) = b
'(a0,b)Ry ) a0R0f(y)
(2.1) (2.2)
foralla0 2A0 andb,y2Bwithp0(a0)=f(b).
In other words a fibrous morphism from A to A0 is a map f from B to B0 together with a span
A0oo⇡1 A0 ' //A f