nLab
covering lifting property

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Category Theory

Contents

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 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)

Properties

Proposition

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 F^\ast}{\longleftarrow}} {\bot} Sh(\mathcal{D})

with inverse image given by pre-composition with ff followed by sheafification.

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

References

Last revised on July 16, 2018 at 07:58:31. See the history of this page for a list of all contributions to it.