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234 N. MARTINS-FERREIRA
and (F4)-(F6). In particular, the additional structure of spatial fibrous pre- order is nothing but a comonoid structure in the monoidal category of fibrous preorders and fibrous morphisms, with an appropriate tensor product. This means that a fibrous preorder (R,A,B,@,p) is spatial if the canonical span
B ⇥ B oo B // 1 can be lifted to fibrous preorders. Further studies will then take place in FibPreord(C) and SpFibPreord(C) for an arbitrary ca- tegory C with finite limits. For instance if C is the category of finite sets then Preord(C) ' SpFibPreord(C) ' FibPreord(C), as it follows easily from Proposition 2.6.
Another possible application for the notion of (spatial) fibrous preorder is in the characterization of descent and e↵ective descent morphisms in the category of topological spaces and other kinds of categories of spaces. As the results of Manuela Sobral and her collaborators [3, 6, 7, 8, 11, 9, 12, 13, 14] show, there are some advantages in considering the notions of topological spaces as generalized preorders.
Acknowledgements
The author is indebted to George Janelidze and Manuela Sobral for their support during the period of maturing of the ideas presented in this work. Thanks are also due to the anonymous referee for many helpful contributions; the example of pre-topological spaces, in particular, is due to him or her.
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