Either this is horribly wrong, or it must be well-known. So I guess Iโm asking for either a rebuttal or a reference.
Take a โsmallishโ category . By this I mean that for every object the collection of all maps ending in must be a set. On this set, letโs call it for Yonedaโs sake, we can define a pre-order if there is a commuting diagram
Misplaced &
A sieve on is the same thing as a downset in with respect to this pre-order. Composition with gives a map such that is a downset (or, sieve) in whenever is a downset in .
A Grothendieck topology on is a function which assigns to every object a collection of sieves on satisfying:
,
if then for every morphism ,
a sieve on is in if there is a sieve such that for all morphisms in .
From this it follows for all downsets and in that if and then and if both then also .
In other words, the collection defines an ordinary topology on , and the second condition implies that we have a covariant functor
sending
That is, one can view a Grothendieck topology as a functor to ordinary topological spaces.
Furher, the topos of sheaves on the site seems to fit in nicely. To a sheaf
one associates a functor of flabby sheaves on having as stalks
for all points in
and as sections on the open set all functions of the form