Let $(C,\mathcal{T})$ be an arbitrary site with a pretopology $\mathcal{T}$. The category $C$ has coproducts.
As a pretopology I mean the definition 2.24 of Grothendieck topology in Angelo Vistoli’s 2007 Notes on Grothendieck topologies, fibered categories and descent theory.
Let $\{U_i\rightarrow U\}_{i\in I}\in\mathcal{T}$ be a covering, then we get a singleton covering (a covering with just a single morphism) by forming the single map $\coprod_{i\in I}U_i\rightarrow U$. With that, we create a family $\mathcal{T_S}$ of singleton coverings. My aim is to proof the pretopology properties on $\mathcal{T_S}$. I got, that $\mathcal{T_S}$ contains all isomorphisms and is closed under composition. So the last part is to show, that $\mathcal{T_S}$ is closed under pullbacks along arbitrary morphisms. So let $\phi:T\rightarrow U\in\mathcal{T_S}$ be a singleton covering and $f:S\rightarrow U$ be an arbitrary morphism. I saw on nLab (Relationship to singleton covers), that there is a singleton covering $\psi:R\rightarrow S$ such that $f\psi$ factors through $\phi$.
- Why do we know that factorisation?
For by the pullback condition on a Grothendieck topology, there is a covering family $\{\psi_j:R_j\rightarrow S\}$ such that each $f\psi_j$ factors through $\phi$. Then nLab says, that $\coprod_{j\in J}R_j\rightarrow S$ is the required singleton covering. I understand, that $\coprod_{j\in J}R_j\rightarrow S\in\mathcal{T_S}$. Unfortunately, I still have questions for which I cannot find an answer.
- Why is $\coprod_{j\in J}R_j$ the pullback of $\phi$ and $f$?
- Do I need for question (2.) more conditions on $C$?
Why is $C$ in my question not extensive (resp. superextensive)? In Ralf Meyer's and Chengchang Zhu's Groupoids in categories with pretopology, there is the translation to singleton pretopologies with a category $C$, which has only coproducts and is not necessarily extensive (resp. superextensive). So, on the one hand some says $C$ has to be extensive (resp. superestensive), like nLab, and on the other hand the paper "Groupoids in categories with pretopology" says we just need coproducts. Is there a mistake in "Groupoids in categories with pretopology"? I couldn't find a good counterexample, where $T_S$ is not a pretopology, if $C$ is not extensive (resp. superextensive).