Forgetful functor from complete metric space to metric space

Question

I am reading Mac Lane's Categories for the Working Mathematician. He mentioned that the usual completion of metric space is universal for the evident forgetful functor (from complete metric spaces to metric spaces). (p.57)

I am not sure what is this forgetful functor 'forgetting'. I think this functor is still keeping the underlying set and also the metric. So does this functor only forget the concept of Cauchy sequence? If so, how to formulate this sentence rigorously? Thank's in advance!

Answer 1

This forgetful functor doesn't really "forget" any structure the way most forgetful functors do: it just forgets the fact that your metric spaces happen to be complete. This doesn't change the morphisms, so this forgetful functor is really just the inclusion of complete metric spaces as a full subcategory of all metric spaces.

Answer 2

This falls within the large class of reflexive subcategories. This term describes the situation of a full subcategory $\mathcal A$ of a category $\mathcal C$ such that the fully faithful inclusion $i : \mathcal A \hookrightarrow \mathcal C$ has a left adjoint.

This is very common: abelian groups among groups, sheaves among presheaves, small categories among simplicial sets are the examples at the top of my head. You see that in each case the inclusion functor does not forget any structure, instead it forgets properties (commutativity, local glueing, Segal condition for each of the examples above).