nLab
subcanonical coverage

A coverage, Grothendieck topology, or Grothendieck pretopology (all of which are different ways of presenting a site) is said to be subcanonical if all representable functors 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 $X:C^{\mathrm{op}}\to \mathrm{Set}$ 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 $X(g)(x_f)=x_{fg}$ 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 $h_{f}g=h_{fg}$ for any $g:V\prime \to V$, there exists a unique $k:U\to Z$ such that $h_f=kf$.

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 a regular epimorphism.

Discussion

This is from the old subcanonical pretopology.

Mike: Does this page deserve to coexist with subcanonical coverage or should it redirect?

Zoran Skoda: I did not see the other page. I looked for subcanonical topology, subcanonical site and did not find anything. Coverage is far less standard term which will be hence overlooked by a casual and external user of nlab and human google seeker for wiki pages. As you see I did not find it. Now I put the link, let it be the way it is: for example subcanonical coverage does not have the term subcanonical site explained so the search does not find that term.

Toby: In principle this could exist separately, since coverage and Grothendieck pretopology are not defined the same way, but in practice I would redirect unless something is written that really makes use of the description as a pretopology rather than as a coverage.