nLab complete spread

Idea

A complete spread over a locale $X$ is a map of locales $L\to X$ that is in the same relation to an etale map of locales $L\to X$ as a cosheaf of sets over $X$ is to a sheaf of sets over $X$.

Definition

The original definition of complete spreads is due to R. H. Fox in 1957 and is slightly different from the definition for locales presented below.

A spread over a locale $X$ is a locally connected locale $L$ together with a map of locales $l\colon L\to X$ such that the connected components of opens $l^*U$ for all opens $U$ in $X$ form a base for the locale $L$.

See Proposition 4.3 in Funk for other equivalent characterizations of spreads.

Recall that the category of elements of the cosheaf of connected components of $l\colon L\to X$ has as objects pairs $(U,c)$, where $U$ is an open in $X$ and $c$ is a connected component of the open $l^*U$ in $L$.

A complete spread over a locale $X$ is a spread $l\colon L\to X$ such that the unit of the adjunction between locally connected locales over $X$ and cosheaves of sets over $X$ given by the display locale functor and the cosheaf of connected components functor is an isomorphism.

Properties

Complete spreads form precisely the essential image of the display locale functor, which therefore becomes an equivalence between the category of cosheaves of sets on a locale $X$ and the category of complete spreads over $X$.

The inverse functor is given by the cosheaf of connected components construction.

References

• Jonathon Funk, The display locale of a cosheaf.

• Marta Bunge, Jonathon Funk, Singular Coverings of Toposes?.