Let $\mathcal{J}$ be a Grothendieck topology on a small category $\mathbb{C}$. If $C$ is an object of $\mathbb{C}$ for which $\mathcal{J}(C)$ contains the empty cover/sieve, then if $F : \mathbb{C}^{\mathsf{op}} \to \mathsf{Set}$ is any $\mathcal{J}$-sheaf, it is my understanding that $F(C)$ must be a singleton, since any element of $F(C)$ will be an amalgamation for the (empty) matching family for the empty cover of $C$, and such amalgamations are unique. Is it a 'common' situation for an object to have an empty cover in a Grothendieck topology? Is it possible to assume without loss of generality that no object is covered by the empty sieve, in the sense that there is another site $(\mathbb{D}, \mathcal{K})$ with this property for which the topos of $\mathcal{J}$-sheaves is equivalent to the topos of $\mathcal{K}$-sheaves?
Amalgamation of matching family for empty cover
category-theorytopos-theory
Best Answer
It's extremely common for there to be objects $C$ for which $\mathcal{J}(C)$ contains the empty sieve. Take the Grothendieck topology arising from any topological space in the standard way: then the empty set has an empty cover.
You have the same situation with the join Grothendieck topology on a distributive lattice with all joins and meets - a partial order with all joins will have a least element, and this element will have an "empty cover" since it is the join of the empty set. Not surprising, given that this is a generalisation of the above example.
I'm not sure whether it is always possible to assume WLOG that there is no empty covering of any element.