🚀 go-pugleaf

I've been wondering about some aspects of the theory of sheaves on a
topological space and more generally on a site. In the case of
sheaves on a topological space X, there are two approaches to the
study of sheaves:
(1) view a sheaf as a special kind of presheaf, a presheaf being a 
    contravariant functor from the topology to the category of sets.
(2) view a sheaf as a local homeomorphism from a space Y to X. The space
    Y is called an espace etal\'e over X.

To go from (1) to (2), let F be the sheaf and consider the disjoint union
of all of the spaces U x F(U), where U runs over all of the
open subsets of X and where F(U) is given the discrete topology. Then
form the quotient space of this disjoint union with respect to the
equivalence relation generated by identifying points of V x F(U)
with their images in V x F(V) whenever V is an open subset of U.
On the other hand, given an espace etal\'e Y over X, the associated
functor associates to each open subset U of X the set of continuous
sections of Y over U.

Thus these notions are equivalent. The functorial approach in (1) is
in many ways preferable, it seems, because it generalizes to the setting
of sheaves on sites (i.e. on categories with Grothendieck topologies).
On the other hand, the espace etale has the advantage of concreteness.

If one gets down to specific advantages, the following come immediately
to mind:
(i) it is very convenient to form the inverse image sheaf using the espace
    etal\'e approach: if X'--> X is a continous function and if Y is an
    espace etal\'e over X, then the inverse image sheaf on X' arises from
    the espace etal\'e on X' which is the fibre product of X' and Y over X.
    The inverse image sheaf regarded as a functor is much more complicated,
    being defined as the sheaf associated to a presheaf defined using
    direct limits.

(ii) For the direct image sheaf, on the other hand, it seems that it is
     more convenient to use the functorial approach: if f:X' --> X is  
     continuous and F is a sheaf on X', then the direct image   sheaf is simply
     the composition of the sheaf on X' with the functor from the topology
     of X to the topology of X' induced by f. That is conceptually quite
     simple, whereas I am not aware of a direct topological construction
     of the espace etal\'e on X associated to this sheaf (if you are aware
     of one, please let me know).

This suggests that it is desirable to keep both of these approaches 
available and to be able to carry out constructions using either one.
If that is the case, then it is also natural to ask whether the
espace etal\'e can be generalized to the setting of Grothendieck
topologies. Note that the construction of the espace etal\'e for
a sheaf on a topological space (i.e. (1) => (2)) was carried
out entirely in the setting of topological spaces and the sheaf
was concretely realized as a topological space over X using only
topological constructions. It is natural then to wonder whether,
given a sheaf F on a site E, there is a natural construction of a
site E' and a morphism of sites E' --> E which generalizes the
construction of an espace etal\'e, the sheaf being recovered as some
kind of sheaf of sections. I think that if one works with Grothendieck
topological spaces (i.e. sets equipped with a Grothendieck topology of subsets,
one can also get
an espace etal\'e (with a Grothendieck topology of subsets) by the
same construction, but I haven't really checked it. So that tends to
support the idea of a general construction with sites.

If you know how to do this for sites in general, please let me know. I have 
had some interesting and imaginative suggestions from some people I have 
asked privately, but I have not seen a complete solution.

Allan Adler
ara@altdorf.ai.mit.edu