Could not include topos theory - contents
There are two different (related) relationships between Grothendieck topoi and a notion of generalized space. (Recall that a Grothendieck topos $T$ is a category of sheaves $T = Sh(S)$ on some site $S$.)
On the one hand, we can regard the topos itself as a generalized space. This tends to be a useful point of view when the site $S$ is the category of open subsets $Op(X)$ of a topological space $X$ (or some manifold or the like), or some other site which we regard as containing data from only “one space.” In this case, we refer to $T$ as a little topos, or (if we fail to translate the original French) a petit topos.
On the other hand, we can view a topos $T$ as a well-behaved category whose objects are generalized spaces. This tends to be a useful point of view when the site $S$ is a category of all test spaces in some sense, such as Top, Diff, or CartSp. In this case, we refer to $T$ as a big topos, or (in French) a gros topos.
These distinctions carry over in a straightforward way to higher topoi such as (∞,1)-topoi.
Objects in a big topos $Sh(S)$ may be thought of as spaces modeled on $S$, in the sense described at motivation for sheaves, cohomology and higher stacks and at space.
On the other hand, the objects of a petit topos, such as $Sh(X)$, can also be regarded as a kind of generalized spaces, but generalized spaces over $X$ on which the rigid structure of morphisms in $Op(X)$ (only inclusions of subsets, no more general maps) induces a correspondingly rigid structure so that they are not all that general. In fact, $Sh(Op(X))$ is equivalent to the category of etale spaces over $X$—i.e. spaces “modeled on $X$” in a certain sense. More generally, for any topos $E$, the objects of $E$ can be identified with local homeomorphisms of toposes into $E$.
From the “little topos” perspective, it can be helpful to think of a “big topos” as a “fat point,” which is not “spread out” very much spatially itself, but contains within that point lots of different types of “local data,” so that even spaces which are “rigidly” modeled on that point can have a lot of interesting cohesion and local structure. (One should not be misled by this into thinking that a big topos has only one point, although it is usually a local topos and hence has an initial point.)
If $X$ is a topological space, then the canonical little topos associated to $X$ is the sheaf topos $Sh(X)$. On the other hand, if $S$ is a site of probes enabling us to regard $X$ as an object of a big topos $H = Sh(S)$, then we can also consider the topos $H/X$ as a representative of $X$. These two toposes are often called the little topos of $X$ (or petit topos of $X$) and the big topos of $X$ (or gros topos of $X$) respectively.
There might be some debate about whether $H/X$ is, itself, “a little topos” or “a big topos.” While it certainly contains information about the space $X$ specifically, its objects are not “spaces locally modeled on $X$” but rather spaces locally modeled on the big site $S$ which happen to have a map to $X$. The standard phrase “the big topos of $X$” is the most descriptive.
Note that if $X$ is actually an object of the site $S$, then $H/X$ can be identified with the topos of sheaves on the slice site $S/X$ (and otherwise, it can be identified with the topos of sheaves on the category of elements of $X\in Sh(S)$). This site $S/X$ is often referred to as the big site of $X$, as compared to the little site, which is $Op(X)$ (or appropriate replacement). The topos $Sh(S/X)$ can thus be viewed as spaces modelled on $S$, but parameterised by the representable sheaf $X$.
Note that when $S=Top$ with its local-homeomorphism topology, there is a canonical functor $Op(X) \to S/X$ which preserves finite limits and both preserves and reflects? covering families. Therefore, it induces both a geometric morphism $H/X \to Sh(X)$ and one $Sh(X) \to H/X$, of which the latter is the left adjoint of the former in Topos. In other words, the geometric morphism $H/X \to Sh(X)$ is local, and in particular a homotopy equivalence of toposes. This fact relating the big and little toposes of $X$ also holds in other cases.
If a site $S$ is given by a Grothendieck pretopology, then one can define an associated notion of a little site associated to any object of $S$, and hence both a little topos and a big topos, which are related as above.
One proposed axiomatization of the notion of big topos is that of a cohesive topos.
For $X$ a topological space, the little topos that it defines is the category of sheaves $Sh(X) := Sh(Op(X))$ on the category of open subsets of $X$. A general object in this topos can be regarded as an etale space over $X$. The space $X$ itself is incarnated as the terminal object $X = * \in Sh(X)$.
On the other hand, a big 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, $Sh(C)$ is such a big topos.
In good cases, the intrinsic properties of $X$ do not depend on whether one regards it as a little topos or as an object of 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.
The suggestion that a notion of gros topos is needed goes back to some remarks in Pursuing Stacks . A precise axiom system capturing the notion is first proposed in
The axioms 0 and 1 for toposes of generalized spaces given there later became called the axioms for a cohesive topos
Further discussion of this axiomatics for gros toposes is in
where a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”) as covariant and contravariant functors out of a distributive category are considered.
The left and right adjoint to the global section functor as a means to identify discrete spaces and codiscrete space is also mentioned in
on page 14.
Under the term categories of cohesion these axioms are discussed in
Another definition of gros vs petit toposes and remarks on applications in Galois theory is in
There is also something relevant in this article:
Mathieu Anel, Grothendieck topologies from unique factorization systems (arXiv:0902.1130)
Mamuka Jibladze, Homotopy types for “gros” toposes, thesis, pdf
A discussion and comparison of big vs little approaches to $(\infty,1)$-topos theory began at these blog entries: