TITLE: Category of Uniform spaces
QUESTION [8 upvotes]: I suspect that the category of uniform spaces and uniformly continuous maps and the full subcategory of complete uniform spaces are both bicomplete and  cartesian closed.  Can anyone comfirm or deny, with reference if possible?

REPLY [2 votes]: There's a professor at my university (Mike Rice) who I sometimes talk to about uniform spaces. When completeness came up he pointed me to the following two papers by himself and Gloria Tashjian:

"Cartesian-closed Coreflective Subcategories of Uniform Spaces Generated by Classes of Metric Spaces", Topology and Applications 15 (1983), 301-312.
"Cartesian-closed Coreflective Subcategories of Uniform Spaces", Trans. Amer. Math. Soc., 276, 1 (1983), 289-300.

The second paper gives an example which shows Unif is not cartesian-closed. We're not sure if also works to show CompleteUnif is not cartesian-closed. Another interesting example is the coreflective hull of $[0, 1]$.
If you want a full subcategory of Unif which is cartesian closed you can of course work in compactly-generated spaces (in Unif the compactly generated spaces are k-spaces with the fine uniformity), but Professor Rice pointed out to me that in Unif this subcategory can act very different from k-spaces inside Top.