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Grothendieck topologies as functors to Top

Either this is horribly wrong, or it must be well-known. So I guess Iโ€™m asking for either a rebuttal or a reference.

Take a โ€˜smallishโ€™ category C. By this I mean that for every object C the collection of all maps ending in C must be a set. On this set, letโ€™s call it y(C) for Yonedaโ€™s sake, we can define a pre-order fโ‰คg if there is a commuting diagram

Misplaced &

A sieve S on C is the same thing as a downset in y(C) with respect to this pre-order. Composition with h:Dโ†’C gives a map h:y(D)โ†’y(C) such that hโˆ’1(S) is a downset (or, sieve) in y(D) whenever S is a downset in y(C).

A Grothendieck topology on C is a function J which assigns to every object C a collection J(C) of sieves on C satisfying:

  • y(C)โˆˆJ(C),
  • if SโˆˆJ(C) then hโˆ’1(S)โˆˆJ(D) for every morphism h:Dโ†’C,
  • a sieve R on C is in J(C) if there is a sieve SโˆˆJ(C) such that hโˆ’1(R)โˆˆJ(D) for all morphisms h:Dโ†’C in S.

From this it follows for all downsets S and T in y(C) that if SโŠ‚T and SโˆˆJ(C) then TโˆˆJ(C) and if both S,TโˆˆJ(C) then also SโˆฉTโˆˆJ(C).

In other words, the collection JC={โˆ…}โˆชJ(C) defines an ordinary topology on y(C), and the second condition implies that we have a covariant functor

J:Cโ†’Top sending Cโ†ฆ(y(C),JC)

That is, one can view a Grothendieck topology as a functor to ordinary topological spaces.

Furher, the topos of sheaves on the site (C,J) seems to fit in nicely. To a sheaf

A:Copโ†’Sets

one associates a functor of flabby sheaves A(C) on (y(C),JC) having as stalks

A(C)h=Im(A(h)) for all points h:Dโ†’C in y(C)

and as sections on the open set SโŠ‚y(C) all functions of the form

sa:Sโ†’โจ†hโˆˆSA(C)h where sa(h)=A(h)(a) for some aโˆˆA(C).

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