On Wed, 27 Aug 2003, Tom Leinster wrote:
Date: Wed, 27 Aug 2003 16:51:55 +0200 (CEST) From: Tom Leinster <leinster@ihes.fr> To: categories@mta.ca Subject: categories: Uniform spaces
Hello,
Does anyone know of any account of the basic properties of the category of uniform spaces? I'm after things like (co)limits, cartesian closure, and (co)limit-preservation by the forgetful functor to Top. Bourbaki gets me some of the way, but his decision not to use categorical language and the resulting circumlocutions make it a struggle.
Thanks, Tom
Hi Tom, The category UNIF of uniform spaces, without a separation axiom, is topological over sets, and hence complete and cocomplete, with concrete limits and colimits. UNIF is not cartesian closed. Cook and Fischer, Math. Ann. 173 (1967), 290-306, defined uniform convergence structures of a set X as sets \scrF of filters on XxX satisfying five axioms. With the obvious definition of uniform continuity, sets with a uniform convergence structure in this sense form a topological category over sets, but Gazik, Kent and Richardson in Bull.Austral.Math.soc 11 (1974), 413-424, showed that this category is not cartesian closed. In LNM 378, 591-637, I replaced the Cook-Fischer axiom that the principal filter generated by the diagonal of XxX is in \scrF by the less demanding axion that the principal filter generated by (x,x), for every x \in X, is in \scrF. This is now part of the accepted definition of uniform convergence spaces. In Bull.Austral.Math.Soc. 15 (1976), 461-465 my student R.S. Lee showed that the category of uniform convergence spaces with this definition is cartesian closed; this is not the cartesian closed hull of UNIF. For quasitoposes, we must go to semiuniform spaces which have partial morphisms -- relations (m,g) with m an embedding -- represented by one-point extensions. Semiuniform convergence spaces and their uniformly continuous maps form a quasitopos, but not the quasitopos hull of UNIF. This has been determined by Adámek and Reiterman, The quasitopos hull of the category of uniform spaes -- a correction, in the journal Topology and its Applications. For more information and literature, see my book Lecture Notes on Topoi and Quasitopoi. Oswald Wyler