nLab
display locale

Display locales

Display locales

Idea

The category of cosheaves of sets on a locale XX is equivalent to the category of complete spreads over XX. The equivalence functor sends a cosheaf DD of sets over XX to its display locale over XX.

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 etale locale? construction plays in the equivalence between sheaves of sets on XX and etale locales? over XX.

Definition

The display locale of a precosheaf DD on a locale XX is a locale over XX (i.e., an object in the slice category Loc/XLoc/X) given by the base change of the morphism

Alex(elem(D))Alex(elem(1 X))Alex(elem(D))\to Alex(elem(1_X))

(induced by the unique morphism D1 XD\to 1_X) along the morphism XAlex(elem(1 X))X\to Alex(elem(1_X)).

Here 1 X1_X denotes the terminal precosheaf on XX.

The functor elem:Precosheaf(X)Posetelem\colon Precosheaf(X) \to Poset computes the category of elements of a precosheaf on XX.

The functor Alex:PosetLocaleAlex\colon Poset \to Locale computes the Alexandroff locale of a poset PP by sending it the locale whose frame consists of downward closed subsets of PP.

The morphism XAlex(elem(1 X))X\to Alex(elem(1_X)) has as its underlying morphism of frames the map that sends a downward closed subset of XX to its supremum.

Adjunction

The display locale functor

Cosheaf(X)LocConLoc/XCosheaf(X) \to LocConLoc/X

is right adjoint to the cosheaf of connected components functor

LocConLoc/XCosheaf(X).LocConLoc/X \to Cosheaf(X).

(The above adjunction also makes holds for precosheaves if we generalize the cosheaf of connected components construction accordingly, see Section 2 in Funk.)

This adjunction exhibits the category of cosheaves of sets on XX as a (full) reflective subcategory of locally connected locales over XX. The essential image of the inclusion is known as the category of complete spreads over XX.

References

Last revised on April 27, 2020 at 08:57:20. See the history of this page for a list of all contributions to it.