nLab
petit topos

Idea

Recall that a Grothendieck topos $T$ is a category of sheaves $T=\mathrm{Sh}(S)$ on some site $S$.

If $S=\mathrm{Op}(X)$ is the category of open subsets of a topological space $X$ (or some manifold or the like) then one calls $T$ a petit topos.

If $S$ on the other hand is a category of all test spaces in some sense, such as

• the category Top of all topological spaces;

• the category Diff of all smooth manifolds

etc. one call $T$ a gros topos .

Objects in a gros topos may be thought of as spaces modeled on $S$ in the sense described at motivation for sheaves, cohomology and higher stacks.

Also the objects in a petit topos $\mathrm{Sh}(\mathrm{Op}(X))$ are a kind of generalized spaces, but generalized spaces over $X$ on which the rigid structure of morphisms in $\mathrm{Op}(X)$ (only inclusions of subsets, no more general maps) induces a correspondingly rigid structure so that they are not all that general. In fact, $\mathrm{Sh}(\mathrm{Op}(X))$ is equivalent to etale spaces over $X$.

Examples

For instance if $S=$ Diff we may think of objects in $\mathrm{Sh}(\mathrm{Diff})$ as smooth spaces and of objects in the (infinity,1)-category of (infinity,1)-sheaves on $S$ as smooth infinity-stacks.