nLab
canonical topology

Definition

The canonical topology on a category $C$ is the Grothendieck topology on $C$ which is the largest subcanonical topology. More explicitly, a sieve $R$ is a covering for the canonical topology iff every representable functor is a sheaf for every pullback of $R$. Such sieves are called universally effective-epimorphic.

Examples

On a Grothendieck topos

If $C$ 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 $C$ has a small generating set. The canonical topology of a Grothendieck topos is also special in that every sheaf is representable; that is, $C\simeq {\mathrm{Sh}}_{\mathrm{canonical}}(C)$.

Notice that if $(D,J)$ is a site of definition for the topos $C$, then this says that

$${\mathrm{Sh}}_J(D)\simeq {\mathrm{Sh}}_{\mathrm{canonical}}({\mathrm{Sh}}_J(D)).$$Sh_J(D) \simeq Sh_{canonical}(Sh_J(D)) \,.