nLab
cosheaf of connected components

Idea

The display locale construction defines a fully faithful functor from cosheaves of sets on a locale XX to the slice category Loc/XLoc/X of locales over XX.

The construction in this article describes a functor in the opposite direction that yields the inverse of the above functor once restricted to the appropriate subcategory.

Thus, the functor described here plays the same role in the equivalence between cosheaves of sets on XX and complete spreads over XX as the sheaf of sections construction plays in the equivalence between sheaves of sets on XX and etale locales? over XX.

Definition

Suppose l:LXl\colon L\to X is a map of locales, where LL is a locally connected locale. Define a precosheaf λ l\lambda_l on XX as follows. Send an open UU in XX to the set of connected components of l *Ul^* U, which is a locally connected locale because so is LL. Send an inclusion of opens to the induced map on the sets of connected components.

Proposition

(Funk, Proposition 2.1.) This precosheaf is a cosheaf.

References

Created on April 18, 2020 at 02:00:43. See the history of this page for a list of all contributions to it.