Editing Grothendieck topology (section)

=== Grothendieck topology ===
A '''Grothendieck topology''' ''J'' on a category ''C'' is a collection, ''for each object c of C'', of distinguished sieves on ''c'', denoted by ''J''(''c'') and called '''covering sieves''' of ''c''.  This selection will be subject to certain axioms, stated below.   Continuing the previous example, a sieve ''S'' on an open set ''U'' in ''O''(''X'') will be a covering sieve if and only if the union of all the open sets ''V'' for which ''S''(''V'') is nonempty equals ''U''; in other words, if and only if ''S'' gives us a collection of open sets that [[Cover (topology)|cover]] ''U'' in the classical sense.

==== Axioms ====
The conditions we impose on a Grothendieck topology are:

* (T 1) (Base change) If ''S'' is a covering sieve on ''X'', and ''f'': ''Y'' → ''X'' is a morphism, then the pullback ''f''<sup><math>\ast</math></sup>''S'' is a covering sieve on ''Y''.
* (T 2) (Local character)  Let ''S'' be a covering sieve on ''X'', and let ''T'' be any sieve on ''X''.  Suppose that for each object ''Y'' of ''C'' and each arrow ''f'': ''Y'' → ''X'' in ''S''(''X''), the pullback sieve ''f''<sup><math>\ast</math></sup>''T'' is a covering sieve on ''Y''.  Then ''T'' is a covering sieve on ''X''.
* (T 3) (Identity) Hom(&minus;, ''X'') is a covering sieve on ''X'' for any object ''X'' in ''C''.

The base change axiom corresponds to the idea that if {''U<sub>i</sub>''} covers ''U'', then {''U<sub>i</sub>'' ∩ ''V''} should cover ''U'' ∩ ''V''.  The local character axiom corresponds to the idea that if {''U<sub>i</sub>''} covers ''U'' and {''V<sub>ij</sub>''}<sub>''j <math>\in</math>J<sub>i</sub>''</sub> covers ''U<sub>i</sub>'' for each ''i'', then the collection {''V<sub>ij</sub>''} for all ''i'' and ''j'' should cover ''U''.  Lastly, the identity axiom corresponds to the idea that any set is covered by itself via the identity map.

==== Grothendieck pretopologies ====
In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category ''C'' contains certain fibered products. In this case, instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain.  These collections are called '''covering families'''.  If the collection of all covering families satisfies certain axioms, then we say that they form a '''Grothendieck pretopology'''.  These axioms are:

* (PT 0) (Existence of fibered products) For all objects ''X'' of ''C'', and for all morphisms ''X''<sub>0</sub> → ''X'' that appear in some covering family of ''X'', and for all morphisms ''Y'' → ''X'', the fibered product ''X''<sub>0</sub>&nbsp;&times;<sub>''X''</sub>&nbsp;''Y'' exists.
* (PT 1) (Stability under base change) For all objects ''X'' of ''C'', all morphisms ''Y'' → ''X'', and all covering families {''X''<sub>''α''</sub> → ''X''}, the family {''X''<sub>''α''</sub> &times;<sub>''X''</sub> ''Y'' → ''Y''} is a covering family.
* (PT 2) (Local character) If {''X''<sub>''α''</sub> → ''X''} is a covering family, and if for all α, {''X''<sub>''βα''</sub> → ''X''<sub>''α''</sub>} is a covering family, then the family of composites {''X''<sub>''βα''</sub> → ''X''<sub>''α''</sub> → ''X''} is a covering family.
* (PT 3) (Isomorphisms) If ''f'': ''Y'' → ''X'' is an isomorphism, then {''f''} is a covering family.

For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology.

For categories with fibered products, there is a converse.  Given a collection of arrows {''X''<sub>''α''</sub> → ''X''}, we construct a sieve ''S'' by letting ''S''(''Y'') be the set of all morphisms ''Y'' → ''X'' that factor through some arrow ''X''<sub>''α''</sub> → ''X''.  This is called the sieve '''generated by''' {''X''<sub>''α''</sub> → ''X''}.  Now choose a topology.  Say that {''X''<sub>''α''</sub> → ''X''} is a covering family if and only if the sieve that it generates is a covering sieve for the given topology.  It is easy to check that this defines a pretopology.

(PT 3) is sometimes replaced by a weaker axiom:

* (PT 3') (Identity) If 1<sub>''X''</sub> : ''X'' → ''X'' is the identity arrow, then {1<sub>''X''</sub>} is a covering family.

(PT 3) implies (PT 3'), but not conversely.  However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3).  These families generate a pretopology.  The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism ''Y'' → ''X'' is Hom(&minus;, ''X'').  Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.