nLab covering lifting property



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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)

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

F:𝒞𝒟 F \;\colon\; \mathcal{C} \to \mathcal{D}

is said to have the covering lifting property if for every object U𝒞U \in \mathcal{C} and every cover {q i:V iF(U)}\{q_i \colon V_i \to F(U)\} of F(U)𝒟F(U) \in \mathcal{D}, there is a cover {p j:U jU}\{p_j \colon U_j \to U\} of U𝒞U \in \mathcal{C} such that {f(U j)f(p j)f(U)}\{f(U_j) \overset{f(p_j)}{\to} f(U)\} refines {V iq if(U)}\{ V_i \overset{q_i}{\to} f(U)\} (i.e., if every cover of the image of any UU under ff is refined by the image of a cover of UU).

(MacLane-Moerdijk, p. 410)



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

Sh(𝒞)F *LF *Sh(𝒟) 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 *F^\ast with FF followed by sheafification L:PSh(𝒞)Sh(𝒞)L \,\colon\, PSh(\mathcal{C}) \longrightarrow Sh(\mathcal{C}).

(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.