nLab
little site

Idea

Let $C$ be a category with a pretopology $J$ (i.e. a site) and $a$ an object of $C$. As an analogy with sheaves on a topological space $X$, which are defined on the site $\mathrm{Op}(X)$ of open sets of $X$, we can try to define sheaves on $a$, using the elements of covering families of $a$ from $J$. This is called the little site of $a$, in contrast to the big site of $a$ which is the slice category $C/a$ with its induced topology.

The topos of sheaves on the little site is the petit topos of $a$.

A little site may sometimes be called a small site, but it's probably best to save that name for a site which is a small category.

Definition

David Roberts: The following is experimental, use at own risk, although I’m sure it has been thought about before.

Consider the subcategory $J/a$ of $C/a$ with objects $u_0\to a$ such that $u_0$ is a member of some covering family $U=\{u_i\to a\}$. Given two such objects $u_0\to a$, $v_0\to a$, and covering families $U$, $V$ that contain them, there is a covering family $W=\mathrm{UV}$ which is the pullback (or at least a weak pullback) of $U$ and $V$ in $C$. There is then some element $w$ of $W$ such that there is a square

$$\begin{array}{ccc}w& \to & v_0\\ \downarrow & & \downarrow \\ u_0& \to & a\end{array}$$\array{ w & \to & v_0 \\ \downarrow & & \downarrow \\ u_0 &\to & a }

so $J/a$ is ‘a bit like’ the category of opens of a space (it’s probably cofiltered, but I haven’t checked that there are weak equalisers).

Now the morphisms of $J/a$ are those triangles

$$\begin{array}{ccc}{v}_0& \to & u_0\\ & \searrow & \downarrow \\ & & a\end{array}.$$\array{ v_0 & \to & u_0 \\ & \searrow& \downarrow \\ & & a }\, .

such that $v_0\to u_0$ is an element of a covering family of $u_0$, so the arrows $w\to u_0$ and $w\to v_0$ really are morphisms of $J/a$. Then we say a covering family of $u_0\to a$ is a collection of triangles that, when we forget the maps to $a$, form a covering family of $u_0$ in $C$. This is at the very least a coverage, and so we have a site.

To be continued…

Related concept

• little site

• big site