nLab (0,1)-presheaf

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 $P$ is an antitone predicate

$$F:P \rightarrow \Omega$$

from $P$ to the poset $\Omega$ of truth values, or equivalently, a monotone predicate

$$F:P^\op \rightarrow \Omega$$

from the opposite poset of $P$ to $\Omega$.

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

$$f:P \rightarrow S$$

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 (0,1)-presheaves on a posite forms a locale.

Related concepts

• lower set

• (0,1)-presheaf, (0,1)-sheaf

• presheaf

• (2,1)-presheaf

  presheaf of groupoids

  (2,1)-sheaf, 2-sheaf, indexed category, stack

• (∞,1)-presheaf

• (∞,n)-presheaf