Dirk Hofmann; Maria Manuel Clementino;
Lawvere Completeness in Topology
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 introduce the 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 it means weak sobriety while for the latter it 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; this structure permits us to define a Yoneda embedding in the realm of (T,V)-categories.
Applied Categorical Structures