Contents

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 and at space.

Also the objects in a petit topos $\mathrm{Sh}(\mathrm{Op}(X))$ – a category of sheaves on the category of open subsets of a topological space $X$ – 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 $X$.

There is another notion of ‘large’ topos associated to a space $X$ (or more generally an object in a site), namely the topos of sheaves on the slice category $\mathrm{Top}/X$, with the obvious notion of covering family; a family is covering if it is so under the functor $\mathrm{Top}/X\to \mathrm{Top}$. This site is often referred to as the large site of $X$, as compared to the small site, which is $\mathrm{Op}(X)$ as above. The topos $\mathrm{Sh}(\mathrm{Top}/X)$ can be viewed as spaces modelled Top (or more generally some site), but parameterised by the representable sheaf $X$.

Definition

The petit topos of an object $a$ in a site $(C,J)$ is the category of sheaves on the small site of $a$.

Examples

For $X$ a topological space, the petit topos that it defines is the category of sheaves $\mathrm{Sh}(X):=\mathrm{Sh}(\mathrm{Op}(X))$ on the category of open subsets of $X$. A general object in this topos is an etale space over $X$. The space $X$ itself is incarnated as the terminal object $X=*\in \mathrm{Sh}(X)$.

A gros topos in which $X$ is incarnated is a category of sheaves on a site of test spaces with which $X$ may be probed. For instance for $C=$ Top, or Diff or CartSp with their standard coverages, $\mathrm{Sh}(C)$ is such a gros topos.

In good cases the intrinsic properties of $X$ do not depend on whether one regards it as an object of a petit or a gros topos. For instance at cohomology in the section Nonabelian sheaf cohomology with constant coefficients it is discussed how the nonabelian cohomology of a paracompact manifold $X$ with constant coefficients gives the same answer in each case.

References

Some aspects of an axiomatic characterization of petit vs. gros toposes is in

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

There is also something relevant in this article:

• Mathhieu Anel?, Grothendieck topologies from unique factorization systems (arXiv:0902.1130)

• Mamuka Jibladze, Homotopy types for “gros” toposes, thesis, pdf