Thursday, 1 July 2010

ct.category theory - "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:AtoB such that dB(f(a),f(a))leqdA(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 dB(f(a),f(a))leqqcdotdA(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.

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