Editing Grothendieck topology (section)

=== Sieves ===
In a Grothendieck topology, the notion of a collection of open subsets of ''U'' stable under inclusion is replaced by the notion of a [[sieve (category theory)|sieve]].  If ''c'' is any given object in ''C'', a '''sieve''' on ''c'' is a [[subfunctor]] of the functor Hom(&minus;, ''c''); (this is the [[Yoneda embedding]] applied to ''c'').  In the case of ''O''(''X''), a sieve ''S'' on an open set ''U'' selects a collection of open subsets of ''U'' that is stable under inclusion.  More precisely, consider that for any open subset ''V'' of ''U'', ''S''(''V'') will be a subset of Hom(''V'', ''U''), which has only one element, the open immersion ''V'' → ''U''.  Then ''V'' will be considered "selected" by ''S'' if and only if ''S''(''V'') is nonempty. If ''W'' is a subset of ''V'', then there is a morphism ''S''(''V'') → ''S''(''W'') given by composition with the inclusion ''W'' → ''V''.  If ''S''(''V'') is non-empty, it follows that ''S''(''W'') is also non-empty.

If ''S'' is a sieve on ''X'', and ''f'': ''Y'' → ''X'' is a morphism, then left composition by ''f'' gives a sieve on ''Y'' called the '''[[pullback]] of''' ''S'' '''along''' ''f'', denoted by ''f''<sup><math>^\ast</math></sup>''S''.  It is defined as the [[fibered product]] ''S''&nbsp;&times;<sub>Hom(&minus;, ''X'')</sub>&nbsp;Hom(&minus;, ''Y'') together with its natural embedding in Hom(&minus;, ''Y'').  More concretely, for each object ''Z'' of ''C'', ''f''<sup><math>^\ast</math></sup>''S''(''Z'') = { ''g'': ''Z'' → ''Y'' | ''fg'' <math>\in</math>''S''(''Z'') }, and ''f''<sup><math>^\ast</math></sup>''S'' inherits its action on morphisms by being a subfunctor of Hom(&minus;, ''Y'').  In the classical example, the pullback of a collection {''V''<sub>i</sub>} of subsets of ''U'' along an inclusion ''W'' → ''U'' is the collection {''V''<sub>i</sub>∩W}.