nLab
complete spread

Idea

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

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 XX is a locally connected locale LL together with a map of locales l:LXl\colon L\to X such that the connected components of opens l *Ul^*U for all opens UU in XX form a base for the locale LL.

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:LXl\colon L\to X has as objects pairs (U,c)(U,c), where UU is an open in XX and cc is a connected component of the open l *Ul^*U in LL.

A complete spread over a locale XX is a spread l:LXl\colon L\to X such that the unit of the adjunction between locally connected locales over XX and cosheaves of sets over XX 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 XX and the category of complete spreads over XX.

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.