The category Met has metric spaces as objects and short maps as morphisms. This is a category because the composition of two short maps is again short.
The monomorphisms in Met are the injective short maps, the epimorphisms are the dense image short maps (for instance, the inclusion: 
, which is clearly mono, so Met is not a balanced category !!), and the isomorphisms are the isometries.
The empty set (considered as a metric space) is the initial object of Met; any singleton metric space is a terminal object. There are thus no zero objects in Met.
The product in Met is given by the supreme metric mixing on the cartesian product. There is no coproduct.
We have a "forgetful" functor Met → Set which assigns to each metric space the underlying set, and to each short map the underlying function. This functor is faithful, and therefore Met is a concrete category.
Follows the Top article.
See the discussion page.