Lawvere completeness in topology
Abstract: It is known since 1973 that Lawvere's notion of
(Cauchy-)complete enriched category is meaningful for metric spaces: it
captures exactly Cauchy-complete metric spaces. In this paper we
introducethe corresponding notion of Lawvere completeness for
(T,V)-categories and show that it has an interesting meaning for
topological spaces and quasi-uniform spaces: for the former ones means
weak sobriety while for the latter means Cauchy completeness. Further,
we show that V has a canonical (T,V)-category structure which plays a
key role: it is Lawvere-complete under reasonable conditions on the
setting; permits us to define a Yoneda embedding in the realm of
(T,V)-categories.
