Category of Uniform spaces

Question

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?

Answer 1

There are two possible meanings of "uniform space" in the literature. I'll follow Isbell's terminology where the definition of a uniform space includes the separation axiom, and one speaks of a pre-uniform space when it is not included.

The category of pre-uniform spaces and uniformly continuous maps has concrete limits and colimits: you can take the (co)limit of the diagram of underlying sets, in the category of sets, and then endow it with the initial (final) uniform structure.

The category of uniform spaces still has concrete limits; and colimits that are generally not concrete, using a standard uniform quotient of a pre-uniform space.

Neither category is cartesian closed. The exponential law $Z^{Y\times X}\ne (Z^Y)^X$ fails in general, unless $X$ is compact. For instance, a uniformly continuous map $I\times\Bbb R\to\Bbb R$ (that is, a uniform homotopy) is not the same as a uniformly continuous map $\Bbb R\to \Bbb R^I$ (that is, a homotopy through uniformly continuous maps $\Bbb R\to\Bbb R$). Indeed, $id:\Bbb R\to\Bbb R$ is not uniformly null-homotopic, but is null-homotopic through uniformly continuous maps.

All of the above is discussed in some form in Isbell's book "Uniform spaces". For a quick review see also section 2.B here. Beware of the tricky nature of sequential colimits, as noticed by Taras Banakh, The topological structure of direct limits in the category of uniform spaces.

As for complete uniform spaces, I believe they are closed under limits, but not closed under pushouts.

Added later: If you want something that feels like uniform spaces and is cartesian closed, I suggest the Cartesian closed hull of the category of uniform spaces by Jiří Adámek and Jan Reiterman. The objects of this hull are bornological uniform spaces, i.e. uniform spaces endowed with a collection of "bounded" sets; the morphisms are the uniformly continuous maps which preserve the bounded sets. Bornological uniform spaces are really cute: when all sets are designated as bounded, these can be identified with the usual uniform spaces, and when only subsets of compact sets are designated as bounded, these can be identified with compactly generated Tychonoff topological spaces.

Unfortunately Adámek and Reiterman have an error in the proof of Lemma 2.3 (in the last line of the proof of assertion (i)). I haven't seen this error discussed in the literature, but I believe it can be remedied by replacing $Hom(A^\ast,I)$ in the statement of the lemma with $Hom(A^\ast,I^\Lambda)$, where $\Lambda$ is a family of pseudo-metrics defining the uniformity of $A$. .

Other work in this direction includes The category of uniform convergence spaces is cartesian closed by R. S. Lee and some related work by her adviser Oswald Wyler; and Metrizable spaces in Cartesian-closed subcategories of uniform spaces and Productivity of α-Bounded Uniform Spaces by Gloria Tashjian, which I think clarifies her previous results with her advisor M. D. Rice.

Answer 2

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:

1. "Cartesian-closed Coreflective Subcategories of Uniform Spaces Generated by Classes of Metric Spaces", Topology and Applications 15 (1983), 301-312.

2. "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.