nLab
concrete site

Idea

A concrete site is a site whose objects can be thought of as sets with extra structure: it is a category that is a concrete category and a site in a compatible way.

In a category of presheaves on a concrete site one can consider concrete presheaves.

Definition

Definition

A concrete site is a site $C$ with a terminal object $*$ such that

1. the functor ${\mathrm{Hom}}_C(*,-):C\to \mathrm{Set}$ is a faithful functor;

2. for every covering family $\{f_i:U_i\to U\}$ in $C$ the morphism

   $$\coprod _i{\mathrm{Hom}}_C(*,f):\coprod _i{\mathrm{Hom}}_C(*,U_i)\to {\mathrm{Hom}}_C(*,U)$$\coprod_i Hom_C(*,f) : \coprod_i Hom_C(*, U_i) \to Hom_C(*, U)

   is surjective.

Examples

• every (small) concrete category becomes a concrete site when equipped with the trivial coverage (every covering family consists of just an identity morphism);

  • for instance the simplex category $\Delta$, which is the site for simplicial sets.

• Any small subcategory of concrete categories such as Top, Diff, etc, with their standard coverages (by open cover)s;

  • for instance CartSp (covering families are open covers);

    which is the site for smooth spaces, including diffeological spaces.