nLab
local site

Definition

Definition

A site is called a local site if

• it has a terminal object $*$;

• the only covering family of $*$ is the trivial cover.

This appears as (Johnstone example C.3.6.3 (d)).

Properties

Proposition

The category of sheaves $\mathrm{Sh}(C)$ on a local site $C$ is a local topos.

Proof

Since $C$ has a terminal object, the global section functor $\mathrm{Sh}(C)\to \mathrm{Set}$ is given by evaluation on that object, hence is precomposition of sheaves with the inclusion $*\to C$. At the level of presheaves this has a right Kan extension functor, given by sending a set $S$ to the presheaf

$$\nabla S:U\mapsto S^{C(*,U)}.$$\nabla S : U \mapsto S^{C(*,U)} \,.

This is indeed a sheaf if $*$ is covered only by the trivial cover.

See (Johnstone example C.3.6.3 (d)).

Examples

• CartSp

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, (∞,1)-cohesive site

References

The definition appears as example C.3.6.3 (d) in

• Peter Johnstone, Sketches of an Elephant

!redirects local sites