Schanuel topos: singleton coverage vs atomic topology

Question

I have seen two different definitions of the Schanuel topos:

1. Topos of sheaves on $\mathbf{FinSet}_{\mathsf{mono}}^{\mathsf{op}}$ with the atomic topology (all non-empty sieves cover)

2. Topos of sheaves on $\mathbf{FinSet}_{\mathsf{mono}}^{\mathsf{op}}$ with the singleton coverage

Are these definitions somehow equivalent?

Answer 1

The atomic topology is generated by the singleton coverage in the sense that a sieve is non-empty iff it contains a subsieve that is generated by a single arrow (take any arrow f in S, and close {f} under precomposition).

In general, given a coverage (=set of covering families) on a category, the generated Grothendieck topology is the smallest topology containing all sieves generated by the covering families. Since topologies are upwards closed under inclusion, the generated topology contains in particular all sieves that are supersets of covering families. Thus, the atomic topology is generated by the singleton coverage simply because it contains precisely all sieves that are supersieves of singletons.