Atomic topology ($J(U)=\{S\subseteq yU:S\ne \varnothing \}$)

The atomic topology declares exactly the nonempty sieves covering. It exists precisely when arrows with a common codomain admit a common refinement. On nonempty finite sets and surjections this follows from the fiber product. Its J-closed sieves are only the empty and maximal sieves.

Use a small skeleton of nonempty finite sets and surjections, with every nonempty sieve covering. Fiber-product projections are surjections, so the atomic coverage exists. Surjections are effective quotients of their kernel pairs, making the site subcanonical. Its sheaves admit the primitive decomposition of an atomic finite-surjection sheaf.

An element $x\in F(n)$ is primitive if it does not descend along any surjection $n\to n-1$. All elements at cardinality one are primitive. Repeated descent reaches a primitive ancestor in finitely many steps, and the primitive-element kernel rigidity lemma makes the ancestor unique up to bijection.

Partition the elements of a sheaf by their primitive-ancestor equivalence classes under bijection. Each class gives a sheaf subfunctor, because its membership is preserved and reflected along covering surjections. A representative primitive element names an epic representable map onto its class component. Thus every sheaf is the coproduct of these components, each an atom in a topos.

If primitive elements have equal restrictions along $\alpha :P\to m$ and $\beta :P\to n$, then these surjections have equal kernels. To rule out $\alpha (a)=\alpha (b)$ with unequal $\beta$ images, identify those two images by $q:n\to n-1$. The set of pairs equal under $\alpha$ and $q\beta$ maps surjectively both to $P$ by projection and to the kernel pair of $q$ by applying $\beta$. Injective restrictions force the kernel-pair matching condition on the primitive element, contradicting descent along $q$.

For any sheaf $F$ and surjection $q:n\to k$, $F(k)\to F(n)\rightrightarrows F(n{\times }_{k}n)$ is an equalizer. In particular every $F(q)$ is injective. These follow from the sheaf condition on the sieve generated by the covering arrow $q$ and control primitive elements.

For each pair $f:V\to U$, $g:W\to U$, require $fh=gk$ for some arrows $h:T\to V$, $k:T\to W$. This is necessary and sufficient for stability of nonempty sieves under pullback; the other topology axioms then follow. Nonempty finite sets with arbitrary functions fail it at distinct singleton-to-two-point maps.