Yoneda 完备度量空间范畴的完备性和余完备性

该文章探讨了以 Yoneda 完备度量空间为对象的范畴的完备性和余完备性. 证明了若态射是 Yoneda 连续映射或者 Yoneda 连续的非扩张映射, 则该范畴是完备且余完备的; 若态射是 Yoneda 连续的 Lipschitz 映射, 则该范畴是有限完备和有限余完备的, 但是它既不完备也不余完备. 最后证明了以实数值连续格为对象, Yoneda 连续的右伴为态射的范畴是完备的.

Completeness and cocompleteness of categories of Yoneda complete metric space

This note investigates completeness and cocompleteness of some categories having as objects the Yoneda complete metric spaces. It is shown that if the morphisms are chosen to be Yoneda continuous maps or Yoneda continuous nonexpansive maps, then the category is both complete and cocomplete; if the morphisms are chosen to be Yoneda continuous Lipschitz maps, then the category is finitely complete and finitely cocomplete, but neither complete nor cocomplete. Finally, it is shown the category of real-valued continuous lattice and Yoneda continuous right adjoints is complete.