Fully embedding (sober) topological spaces in a category of coalgebras with a sequence of three universal constructions
by
Aurelio Carboni

Starting with Sets, one can perform the category of relations, and in a recent work of Dawson, Paré and Pronk, the universal property of such a construction has been investigated: it is essentially the 2-category obtained by adding the adjoint to each map to a cartesian category, in such a way that Beck-Chevalley condition holds, and insisting that the resulting 2-category is in fact locally ordered (using axiom of choice in Sets for this last). Clearly, in such a construction some properties of Sets are lost, notably the cartesian product looses his cartesianess, and the (regular epi)-(mono) factorization of a map is not available anymore as a stable, proper factorization system. We show that freely adding the last property to the result of the first ``free'' construction, and cofreely adding the first property to the result, we get a category of coalgebras and the claimed full embedding, which is in fact a coreflection. Eventhough the category of coalgebras so constructed has a plain direct description, the description by universal properties, and in particular by the second one, is quite useful to investigate the full embedding and his properties.

The above is a report of developements from recent work of Bucalo and Rosolini.

Date received: June 27, 2002