nLab covering lifting property

Idea

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.)

Definition

Definition

(covering lifting property)

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A functor

$$F \;\colon\; \mathcal{C} \to \mathcal{D}$$

is said to have the covering lifting property if for every object $U \in \mathcal{C}$ and every cover $\{q_i \colon V_i \to F(U)\}$ of $F(U) \in \mathcal{D}$, there is a cover $\{p_j \colon U_j \to U\}$ of $U \in \mathcal{C}$ such that $\{f(U_j) \overset{f(p_j)}{\to} f(U)\}$ refines $\{ V_i \overset{q_i}{\to} f(U)\}$ (i.e., if every cover of the image of any $U$ under $f$ is refined by the image of a cover of $U$).

(MacLane-Moerdijk, p. 410)

Properties

Proposition

A functor between sites with covering lifting property (Def. ) induces a geometric morphisms between the corresponding sheaf toposes covariantly,

$$Sh(\mathcal{C}) \underoverset {\underset{F_{\ast}}{\longrightarrow}} {\overset{L \circ F^\ast}{\longleftarrow}} {\bot} Sh(\mathcal{D})$$

with inverse image given by pre-composition $F^\ast$ with $F$ followed by sheafification $L \,\colon\, PSh(\mathcal{C}) \longrightarrow Sh(\mathcal{C})$.

(MacLane-Moerdijk, Theorem VII.10.5, The Elephant, Proposition C2.3.18).

References

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

• Peter Johnstone, Sketches of an elephant: a topos theory compendium