Page 232 - Textos de Matemática Vol. 46

222 N. MARTINS-FERREIRA
that the category Top, of topological spaces and continuous maps, is sitting between the category of preorders and the category of fibrous preorders:
Preord // Top // FibPreord.
The main result of this work is the description of the subcategory of fibrous preorders which is equivalent to the category of topological spaces. Inspired by what are called spatial frames in point-free-topology (see e.g. [10]), the fibrous preorders arising in this way are called spatial fibrous preorders.
A fibrous preorder is a generalization of a preorder. It was obtained while looking for a simple description of topological spaces in terms of internal cat- egorical structures, that is a structure which can be defined in an arbitrary category with finite limits — as for instance the notion of internal category or internal groupoid, internal preorder or internal equivalence relation (see e.g. [2]).
This work is organized as follows: in section 2 we describe the category of fibrous preorders, by defining its objects and morphisms and an equivalence relation on each hom-set of fibrous morphisms; the equivalence of morphisms induces an equivalence between the objects (Proposition 2.5); at the end we recover the classical Alexandrov theorem stating that every A-space is equiv- alent to a preorder. In section 3 we introduce the notion of spatial fibrous preorder and prove that (up to equivalence) it defines a subcategory of the category of fibrous preorders which is isomorphic to the category of topological spaces. In section 4 we provide some examples illustrating how spatial fibrous preorders can be used to work with topological spaces described by systems of open neighbourhoods.
2. Fibrous preorders and fibrous morphisms
The following definition is a generalization of the notion of preorder, i.e. a reflexive and transitive relation. The word fibrous is a derivation of the word fibre and it is motivated by the presence of a morphism p: A ! B (see below), suggesting that A may be considered as a fibre over the base B. Moreover when p is an isomorphism the classical notion of preorder is recovered.
Definition 2.1. A fibrous preorder is a sequence R @ //A p //B
in which A and B are sets, p and @ are maps, R ✓ A⇥B is a binary relation (and as usual we simply write (a, b) 2 R as aRb) such that the following conditions hold:
(F1) p@(a, b) = b;