## 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

Last revised on February 24, 2021 at 16:50:32. See the history of this page for a list of all contributions to it.