nLab local site

1. 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)).

2. Properties

Proposition. The category of sheaves $Sh(C)$ on a local site $C$ is a local topos.

Proof. Since $C$ has a terminal object, the global section functor $Sh(C) \to Set$ 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

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

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

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

3. Examples

• CartSp

4. 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

5. References

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

• Peter Johnstone, Sketches of an Elephant

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