TITLE: Meaning of Grothendieck quote: recover a topological space by its category of sheaves
QUESTION [7 upvotes]: In "Récoltes et Semailles"(- Grothendieck), there is a moment when the author talks about the idea of sheaves of sets over a topological space, then taking the category of sheaves (of sets over a topological space):

It [the category of sheaves] functions as a kind of "superstructure of measurement", called the
  "Category of Sheaves" ( over the given space), which henceforth shall be
  taken to incoorporate all that is most essential about that space. This
  is in all respects a lawful procedure, ( in terms of "mathematical common 
  sense") because it turns out that one can "reconstitute" in all respects,
  the topological space by means of the associated "category of sheaves"
  ( or "arsenal" of measuring instruments)
  (For the mathematical reader) Strictly speaking, this is only true
  for so-called "tame" spaces. However these include virtually all of the
  spaces one has to deal with, notably the "separable spaces" so dear to
  functional analysts.
( The verification of this is a simple exercise- once someone thinks to
  pose the question, naturally) One needs nothing more ( if one feels the
  need for one reason or another), henceforth one can drop the initial space
  and only hold onto its associated "category" ( or its "arsenal"), which
  ought to be considered as the most complete incarnation of the
  "topological (or spatial) structure" which it exemplifies

(English translation by Roy Lisker)
Now, my question is: what is the meaning of this part, especially the part when he says that one can "reconstitute" in all respects the topological space by means of the associated "category of sheaves"?

REPLY [13 votes]: Grothendieck means you can recover a topological space $X$ from the category of sheaves $\text{Sh}(X)$, under mild hypotheses. It goes like this. $\text{Sh}(X)$ has a terminal object $1$, and the lattice of subobjects of $1$ (isomorphism classes of monomorphisms into $1$) turns out to be precisely the lattice of open subsets of $X$. Hence from $\text{Sh}(X)$ one can recover the locale determined by $X$. And sober topological spaces are determined by their locales, or equivalently by their lattices of open subsets.