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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 $\phi$.

  1. 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.

  1. Why is $\coprod_{j\in J}R_j$ the pullback of $\phi$ and $f$?
  2. 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).

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    $\begingroup$ The translation to singleton coverages does not work well unless you assume your site is extensive. The pretopology you start with should also be superextensive. $\endgroup$
    – Zhen Lin
    Jan 18, 2023 at 6:24
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    $\begingroup$ I think that some authors, in their haste to simplify to singleton coverages, neglect to state some "obvious" assumptions. For example, that coproducts should be pullback stable and that every coproduct cocone is a covering family. In other words, the category should be extensive and the coverage should be superextensive. $\endgroup$
    – Zhen Lin
    Jan 18, 2023 at 11:10

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