nLab
big and little toposes

Context

Topos Theory

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Big and little toposes

Idea

There are two different (related) relationships between Grothendieck topoi and a notion of generalized space. (Recall that a Grothendieck topos TT is a category of sheaves T=Sh(S)T = Sh(S) on some site SS.)

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 SS is the category of open subsets Op(X)Op(X) of a topological space XX (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 TT 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 TT as a well-behaved category whose objects are generalized spaces. This tends to be a useful point of view when the site SS is a category of all test spaces in some sense, such as Top, Diff, or CartSp. In this case, we refer to TT 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.

Relationships

Objects in a big topos Sh(S)Sh(S) may be thought of as spaces modeled on SS, 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)Sh(X), can also be regarded as a kind of generalized spaces, but generalized spaces over XX on which the rigid structure of morphisms in Op(X)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))Sh(Op(X)) is equivalent to the category of etale spaces over XX—i.e. spaces “modeled on XX” in a certain sense. More generally, for any topos EE, the objects of EE can be identified with local homeomorphisms of toposes into EE.

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.)

The big and little topos of an object

If XX is a topological space, then the canonical little topos associated to XX is the sheaf topos Sh(X)Sh(X). On the other hand, if SS is a site of probes enabling us to regard XX as an object of a big topos H=Sh(S)H = Sh(S), then we can also consider the topos H/XH/X as a representative of XX. These two toposes are often called the little topos of XX (or petit topos of XX) and the big topos of XX (or gros topos of XX) respectively.

There might be some debate about whether H/XH/X is, itself, “a little topos” or “a big topos.” While it certainly contains information about the space XX specifically, its objects are not “spaces locally modeled on XX” but rather spaces locally modeled on the big site SS which happen to have a map to XX. The standard phrase “the big topos of XX” is the most descriptive.

Note that if XX is actually an object of the site SS, then H/XH/X can be identified with the topos of sheaves on the slice site S/XS/X (and otherwise, it can be identified with the topos of sheaves on the category of elements of XSh(S)X\in Sh(S)). This site S/XS/X is often referred to as the big site of XX, as compared to the little site, which is Op(X)Op(X) (or appropriate replacement). The topos Sh(S/X)Sh(S/X) can thus be viewed as spaces modelled on SS, but parameterised by the representable sheaf XX.

Note that when S=TopS=Top with its local-homeomorphism topology, there is a canonical functor Op(X)S/XOp(X) \to S/X which preserves finite limits and both preserves and reflects? covering families. Therefore, it induces both a geometric morphism H/XSh(X)H/X \to Sh(X) and one Sh(X)H/XSh(X) \to H/X, of which the latter is the left adjoint of the former in Topos. In other words, the geometric morphism H/XSh(X)H/X \to Sh(X) is local, and in particular a homotopy equivalence of toposes. This fact relating the big and little toposes of XX also holds in other cases.

Axiomatizations

  • If a site SS is given by a Grothendieck pretopology, then one can define an associated notion of a little site associated to any object of SS, 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.

Examples

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

On the other hand, a big topos in which XX is incarnated is a category of sheaves on a site of test spaces with which XX may be probed. For instance for C=C = Top, or Diff or CartSp with their standard coverages, Sh(C)Sh(C) is such a big topos.

In good cases, the intrinsic properties of XX 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 XX with constant coefficients gives the same answer in each case.

References

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

  • Bill Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7.(pdf)

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

  • Bill Lawvere, Categories of space and quantity in: J. Echeverria et al (eds.), The Space of mathematics , de Gruyter, Berlin, New York (1992)

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

  • Bill Lawvere Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)

on page 14.

Under the term categories of cohesion these axioms are discussed in

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

Another definition of gros vs petit toposes and remarks on applications in Galois theory is in

  • Nick Duncan, Gros and petit toposes (pdf)

There is also something relevant in this article:

A discussion and comparison of big vs little approaches to (,1)(\infty,1)-topos theory began at these blog entries:

Revised on October 10, 2014 10:09:01 by David Corfield (129.12.18.116)