Editing Grothendieck topology (section)

== Continuous and cocontinuous functors ==
There are two natural types of functors between sites.  They are given by functors that are compatible with the topology in a certain sense.

=== Continuous functors ===
If (''C'', ''J'') and (''D'', ''K'') are sites and ''u'' : ''C'' → ''D'' is a functor, then ''u'' is '''continuous''' if for every sheaf ''F'' on ''D'' with respect to the topology ''K'', the presheaf ''Fu'' is a sheaf with respect to the topology ''J''. Continuous functors induce functors between the corresponding topoi by sending a sheaf ''F'' to ''Fu''.  These functors are called '''pushforwards'''.  If <math>\tilde C</math> and <math>\tilde D</math> denote the topoi associated to ''C'' and ''D'', then the pushforward functor is <math>u_s : \tilde D \to \tilde C</math>.

''u''<sub>''s''</sub> admits a left adjoint ''u''<sup>''s''</sup> called the '''pullback'''. ''u''<sup>''s''</sup> need not preserve limits, even finite limits.

In the same way, ''u'' sends a sieve on an object ''X'' of ''C'' to a sieve on the object ''uX'' of ''D''.  A continuous functor sends covering sieves to covering sieves.  If ''J'' is the topology defined by a pretopology, and if ''u'' commutes with fibered products, then ''u'' is continuous if and only if it sends covering sieves to covering sieves and if and only if it sends covering families to covering families.  In general, it is ''not'' sufficient for ''u'' to send covering sieves to covering sieves (see SGA IV 3, {{lang|fr|Exemple}} 1.9.3).

=== Cocontinuous functors ===
Again, let (''C'', ''J'') and (''D'', ''K'') be sites and ''v'' : ''C'' → ''D'' be a functor.  If ''X'' is an object of ''C'' and ''R'' is a sieve on ''vX'', then ''R'' can be pulled back to a sieve ''S'' as follows: A morphism ''f'' : ''Z'' → ''X'' is in ''S'' if and only if ''v''(''f'') : ''vZ'' → ''vX'' is in ''R''.  This defines a sieve.  ''v'' is '''cocontinuous''' if and only if for every object ''X'' of ''C'' and every covering sieve ''R'' of ''vX'', the pullback ''S'' of ''R'' is a covering sieve on ''X''.

Composition with ''v'' sends a presheaf ''F'' on ''D'' to a presheaf ''Fv'' on ''C'', but if ''v'' is cocontinuous, this need not send sheaves to sheaves.  However, this functor on presheaf categories, usually denoted <math>\hat v^*</math>, admits a right adjoint <math>\hat v_*</math>.  Then ''v'' is cocontinuous if and only if <math>\hat v_*</math> sends sheaves to sheaves, that is, if and only if it restricts to a functor <math>v_* : \tilde C \to \tilde D</math>.  In this case, the composite of <math>\hat v^*</math> with the associated sheaf functor is a left adjoint of ''v''<sub>*</sub> denoted ''v''<sup>*</sup>. Furthermore, ''v''<sup>*</sup> preserves finite limits, so the adjoint functors ''v''<sub>*</sub> and ''v''<sup>*</sup> determine a [[geometric morphism]] of topoi <math>\tilde C \to \tilde D</math>.

=== Morphisms of sites ===
A continuous functor ''u'' : ''C'' → ''D'' is a '''morphism of sites''' ''D'' → ''C'' (''not'' ''C'' → ''D'') if ''u''<sup>''s''</sup> preserves finite limits.  In this case, ''u''<sup>''s''</sup> and ''u''<sub>''s''</sub> determine a geometric morphism of topoi <math>\tilde C \to \tilde D</math>. The reasoning behind the convention that a continuous functor ''C'' → ''D'' is said to determine a morphism of sites in the opposite direction is that this agrees with the intuition coming from the case of topological spaces.  A continuous map of topological spaces ''X'' → ''Y'' determines a continuous functor ''O''(''Y'') → ''O''(''X'').  Since the original map on topological spaces is said to send ''X'' to ''Y'', the morphism of sites is said to as well.

A particular case of this happens when a continuous functor admits a left adjoint. Suppose that ''u'' : ''C'' → ''D'' and ''v'' : ''D'' → ''C'' are functors with ''u'' right adjoint to ''v''. Then ''u'' is continuous if and only if ''v'' is cocontinuous, and when this happens, ''u''<sup>''s''</sup> is naturally isomorphic to ''v''<sup>*</sup> and ''u''<sub>''s''</sub> is naturally isomorphic to ''v''<sub>*</sub>.  In particular, ''u'' is a morphism of sites.