category of pseudo-metric spaces with non-expansive maps

  • notation: PMet\PMet
  • objects: pseudo-metric spaces
  • morphisms: non-expansive maps ff, meaning d(f(x),f(y))≤d(x,y)d(f(x),f(y)) \leq d(x,y) for all x,yx,y
  • Related categories: Met\Met

In contrast to metric spaces, we do not demand d(x,y)=0  ⟹  x=yd(x,y)=0 \implies x=y here.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

There are 7 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

Special objects

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

Special morphisms

  • isomorphisms: bijective isometries
  • monomorphisms: injective non-expansive maps
  • epimorphisms: surjective non-expansive maps
  • regular monomorphisms:
  • regular epimorphisms: surjective non-expansive maps f:X→Yf : X \to Y with the property that for all x,x′∈Xx,x' \in X, d(p(x),p(x′))d(p(x),p(x')) is the infimum of the sums ∑i=0n−1d(yi,xi+1)\sum_{i=0}^{n-1} d(y_i,x_{i+1}), where x=x0x=x_0, xi∼yix_i \sim y_i for 0≤i≤n0 \leq i \leq n, and yn=x′y_n = x'.