Could not include topos theory - contents
The canonical topology on a category is the Grothendieck topology on which is the largest subcanonical topology. More explicitly, a sieve is a covering for the canonical topology iff every representable functor is a sheaf for every pullback of . Such sieves are called universally effective-epimorphic.
If is a Grothendieck topos, then the canonical covering sieves are those that are jointly epimorphic. Moreover, in this case the canonical topology is generated by small jointly epimorphic families, since has a small generating set.
Notice that if is a site of definition for the topos , then this says that
For simplicity, assume is a small subcanonical site. The quasi-inverse of the Yoneda embedding has a very simple description: it is the functor that sends a sheaf to its restriction along the Yoneda embedding .
Indeed, suppose is a sheaf. We claim that is determined up to unique isomorphism by its restriction along the embedding . Indeed, let be a -sheaf. Then is the colimit of a canonical small diagram of representable sheaves on in a canonical way. Consider the colimiting cocone on : it is a universal effective epimorphic family and is therefore a covering family in the canonical topology on . Thus, is indeed determined up to unique isomorphism by the restriction of to . We must also show that the restriction is actually a sheaf on ; but this is true because -covering sieves in become universal effective epimorphic families in .
Thus we obtain a functor that is left quasi-inverse to the embedding , and the argument above shows that it is also a right quasi-inverse.
A textbook account in topos theory is in
Discussion in the refined context of higher topos theory is in