nLab local model structure on simplicial sheaves

Idea

See model structure on simplicial presheaves for background, idea and introduction.

The local model structure on simplicial presheaves has a slight variant with presheaves replaced by sheaves.

This is originally due to Joyal. Later Jardine introduced the local model structure on simplicial presheaves and proved that both are Quillen equivalent.

Definition

For $C$ a site and $SSh(C)$ the category of sheaves on $X$ with values in simplicial sets, Joyal’s local model structure on $SSh(X)$ is (from a revisionist point of view) the restriction of the injective local model structure on simplicial presheaves from all presheaves to sheaves.

Related concepts

• model structure on simplicial presheaves

References

• J. F. Jardine, page 9 of: Fields lectures: simplicial presheaves (pdf)