category of metric spaces with non-expansive maps

 analysis
Notation Met\Met Objects metric spaces Morphisms non-expansive maps ff (also called short maps or 11-Lipschitz maps), meaning d(f(x),f(y))≤d(x,y)d(f(x),f(y)) \leq d(x,y) for all x,yx,y Related Ban\Ban, Metc\Met_c, Met∞\Met_{\infty}, PMet\PMet External nLab link
In contrast to continuous maps, which only refer to the induced topology, non-expansive maps are more closely related to the metrics themselves. This category is badly behaved, though, especially when compared with Met∞\Met_{\infty}.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: singleton space
  • initial object: empty metric space
  • products: [finite case] direct products with the metric d(x,y)=sup⁡idi(xi,yi)d(x,y) = \sup_i d_i(x_i,y_i)

Special morphisms