nLab
(0,1)-presheaf

Contents

Context

(0,1)-Category theory

Topos theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

A (0,1)-presheaf is a presheaf with values in the (0,1)-category of truth values. A 0-truncated (∞,1)-presheaf.

Definition

A (0,1)-presheaf on a poset or proset PP is an antitone predicate

F:PΩF:P \rightarrow \Omega

from PP to the poset Ω\Omega of truth values, or equivalently, a monotone predicate

F:P opΩF:P^\op \rightarrow \Omega

from the opposite poset of PP to Ω\Omega.

More generally, for a poset SS, a S-valued (0,1)-presheaf on PP is just an antitone

f:PSf:P \rightarrow S

so (0,1)-presheaves are just antitones.

(0,1)-category of (0,1)-presheaves

The (0,1)-category of a (0,1)-presheaf on a (0,1)-site forms a (0,1)-topos. In traditional order theoretic language, the poset (or proset) of a (0,1)-presheaf on a posite forms a locale.

Last revised on May 5, 2021 at 01:15:45. See the history of this page for a list of all contributions to it.