Olivia Caramello's website
Unifying theory
Concrete examples
Stone-type dualities
In preparation... (in the meantime you can read the preliminaries below and this paper).
Morphisms of sites
In this section we briefly recall the theory of morphisms of sites.
For any small categories 
and
, any functor 
induces an
essential geometric morphism
, whose
inverse image is the functor given by composition with
; the left
adjoint to this functor will be denoted by
, from which
can be
recovered as its restriction to the representables:
A morphism of sites
is a
functor
such that the functor
restricts
to a functor
; as shown
in
this work, any morphism of sites
induces a
geometric morphism
, whose
inverse image can be identified with such restriction. If we denote by
and
t the
functors obtained by composition of the Yoneda embedding with the
associated sheaf functor, we have that the following diagram commutes:
commutes, for a functor
and a
geometric morphism
, then is
cover-preserving, that is it sends any 
-covering sieve
to (a family generating a)
-c-covering sieve.
Indeed, it is easy to prove that for any site
and any
sieve 
on an
object 
of
,
is
-covering if and
only if the family
of arrows
in 
with
codomain 
is epimorphic.
If
and
are cartesian
categories then a functor
is a
morphism of sites
if and
only if it is cartesian (i.e., finite-limit preserving) and
cover-preserving.