topos theory

Contents

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, $C\simeq {\mathrm{Sh}}_{\mathrm{canonical}}\left(C\right)$.

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

${\mathrm{Sh}}_{J}\left(D\right)\simeq {\mathrm{Sh}}_{\mathrm{canonical}}\left({\mathrm{Sh}}_{J}\left(D\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$Sh_J(D) \simeq Sh_{canonical}(Sh_J(D)) \,.

Revised on April 15, 2012 13:02:13 by Urs Schreiber (82.113.98.59)