The covering lifting property on a functor between sites is a sufficient condition for it to induce a geometric morphism between the corresponding sheaf toposes covariantly, i.e. with direct image going in the same direction. Sometimes, one calls such a functor cocontinuous, cover-reflecting (e.g., the Elephant) or a comorphism of sites.
(As opposed to a morphism of sites, also known as a continuous functor, which induces a geometric morphism contravariantly, going the other way around.)
(covering lifting property)
is said to have the covering lifting property if for every object and every cover of , there is a cover of such that refines (i.e., if every cover of the image of any under is refined by the image of a cover of ).
A functor between sites with covering lifting property (Def. ) induces a geometric morphisms between the corresponding sheaf toposes covariantly,
with inverse image given by pre-composition with followed by sheafification .
(MacLane-Moerdijk, Theorem VII.10.5, The Elephant, Proposition C2.3.18).
Last revised on June 30, 2022 at 08:18:57. See the history of this page for a list of all contributions to it.