Sets as a category of sheaves on the category of finite sets

Question

Can one give a Grothendieck topology $J$ on $\mathbb F$, the category of finite sets such that $Sh(\mathbb F,J)\simeq \mathcal Set$?

I have the following observations:

1. $[\mathbb F^\text{op},\mathcal Set]_{\text{finite product preserving}}\simeq \mathcal Set$
2. If we choose $J_{\text{triv}}$ to be the trivial topology, then $Sh(\mathbb F,J_\text{triv})\simeq [\mathbb F^\text{op},\mathcal Set]$, which is too large.
3. If we choose $J_\text{can}$ to be the canonical topology, then $Sh(\mathbb F,J_\text{can})\simeq \mathbb F$, which is too small.

Alternatively, how might one define a sheafification functor $a:[\mathbb F^\text{op},\mathcal Set]\to \mathcal Set$?

Answer 1

You can take $J$ to be the canonical topology on $\mathbb{F}$. Indeed, if $F$ is any sheaf with respect to the canonical topology, then $F$ is naturally isomorphic to the functor $A\mapsto X^A$ where $X$ is the value of $F$ on a singleton, since any finite set $A$ is covered by maps from a singleton to each of its points. Conversely, for any set $X$, the functor $A\mapsto X^A$ is easily seen to be a sheaf in the canonical topology (since every covering in the canonical topology is jointly surjective). It is then easy to check that the category of sheaves in the canonical topology is equivalent to $\mathcal{Set}$ by taking each sheaf to its value on a singleton.