nLab subcanonical coverage

Definition

A coverage (resp. Grothendieck topology, resp. Grothendieck pretopology) defining a site is called subcanonical if all representable presheaves on this site are sheaves. Of course, a subcanonical site is one whose coverage is subcanonical.

The term “subcanonical” comes about because the largest coverage for which the representables are sheaves is called the canonical coverage, and the subcanonical coverages are precisely the “sub-coverages” of the canonical one.

Effective-epimorphic sieves

An alternate definition is that a Grothendieck coverage is subcanonical if and only if all of its covering sieves $R\hookrightarrow C(-,U)$ are effective-epimorphic, meaning that the morphisms $f:V\to U$ in $R$ form a colimit cone under the diagram consisting of all morphisms between them over $U$. To see this, first recall that if $R\hookrightarrow C(-,U)$ is a sieve, then a functor $X:C^{op}\to Set$ satisfies the sheaf axiom for $R$ if and only if

• for every family $(x_f)_{f\in R}$ which is compatible, in the sense that $X(g)(x_f) = x_{f g}$ whenever this makes sense, there exists a unique $x\in X(U)$ such that $x_f = X(f)(x)$.

Interpreting this when $X$ is a representable functor $C(-,Z)$, we obtain

• for every family of maps $(h_f:V\to Z)$, where $f:V\to U$ is in $R$, such that $h_f g = h_{f g}$ for any $g:V'\to V$, there exists a unique $k:U\to Z$ such that $h_f = k f$.

But this says precisely that $R$ is effective-epimorphic, as defined above.

In fact, since the covering sieves in a subcanonical coverage must also satisfy pullback-stability, they must be not only effective-epimorphic but universally effective-epimorphic (meaning that any pullback of them is effective-epimorphic). It is then easy to see that the canonical coverage consists precisely of all the universally effective-epimorphic sieves.

Note also that if $f:V\to U$ is a single morphism having a kernel pair $p,q:V\times_U V \;\rightrightarrows\; V$, then the sieve generated by $f$ is effective-epimorphic iff $f$ is the coequalizer of its kernel pair, and thus iff $f$ is an effective epimorphism.

Examples

See at coverage.

Related concepts

• canonical topology

• trivial topology