Speaker:

Nelson Martins-Ferreira

Tuesday, July 23, 2013 - from 17:30
to 17:55

Abstract:

In modern terms, the main result in [1] establishes a categorical
equivalence between preorders and A-spaces. A preorder is simply a
reflexive and transitive relation while an A-space is a topological
space in which any intersection of open sets is open. In [2], p.61,
Ern\'e writes Hence the question arises: How can we enlarge
the category of A-spaces on the one hand and the category of
quasiordered sets on the other hand, so that we still keep an
equivalence between the topological and the order-theoretical
structures, but many interesting 'classical' topologies are included
in the extended definition? and proposes the notions of B-spaces
and C-spaces.
With a different motivation in mind we will show that the category
of fibrous preorders (to be introduced) is equivalent to the
category of topological spaces. A fibrous preorder is a
generalization of a preorder and was obtained as an attempt in
finding a description for a topological space in terms of internal
categorical structures.
[1] Alexandrov, P., Diskrete Raume, Mat. Sbornik (N.S.)
\textbf{2} (1937), 501--518.
[2] Ern\'e, M., The ABC of order and topology, in: Herrlich,
H., and Porst, H.-E. (eds), Category Theory at Work, Heldermann
Verlag, Berlin, 1991, 57--83.