# “Category” of Nonempty Metric Spaces and Contractive Maps?

The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as morphisms.

However, for my purposes I'd like to use the Banach fixed point theorem to get a category with a trace structure or Conway operators on it, which means I want to consider the contraction mappings on nonempty metric spaces -- that is, there should be $q < 1$ for each mapping $f$ such that $d_B(f(a), f(a')) \leq q \cdot d_A(a, a')$.

But nonempty metric spaces and contraction mappings don't form a category, since the identity function is not a contraction map! Is there some way of defining this kind of setup as a category? I'm happy to play games with the metrics (e.g., use ultrametrics, but bounds on them, that sort of thing), if it helps.

• It's fun to occasionally see a fairly "natural" definition which is not satisfied by the identity map. The notions of trace class or compact maps on Hilbert spaces is another example: the identity map on a Hilbert space is trace class iff the space is finite-dimensional. In this case I always felt that what was going on was that the functions you're interested in are somehow an "ideal" in the space of all functions. For example if f is trace class and g is continuous then f o g is trace class. Similarly if f is a contraction map and g is non-expansive then f o g is a contraction. – Kevin Buzzard Feb 18 '10 at 14:36
• Such notions of ideals have been worked out in arxiv.org/abs/math/9805102, for example. They cover Hilbert-Schmidt maps, such as in Andrew's answer, and trace class operators, as in Kevin's comment. One would think the ideal of contractions could be axiomatized similarly. – Chris Heunen May 21 '12 at 9:15