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In SGA4, the first axiom of a Grothendieck pretopology is given as:

PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$ sont quarrables. (Rappelons qu’un morphisme $Y \to X$ est dit quarrable si pour tout morphisme $Z \to X$ le produit fibré $Z \times_X Y$ est représentable).

A translation of this is :

If $\{U_i \to U \}$ is a covering and $f : V \to U$ any morphism, then all $U_i Ă—_U V$ exist.

I'm wondering how the quantifiers are to be parsed in this. There's two ways I can see reading this axiom:

  1. For each morphism $g_i : U_i \to U $ in a covering family $F \in Cov(U)$, and for each $f : V \to U$, the pullback of $g_i$ and $f$ exists
  2. For every covering family $F \in Cov(U)$ and morphism $f : V \to U$, there is a pullback between $f$ and $F$, i.e. a single fiber-product of $f$ and all $(f_i : U_i \to U) \in F$

i.e. Is there a pullback for each morphism in a covering family, or is there one pullback for the whole family?

(1) seems to go against my intuition for covering: the entire family taken together covers $U$, so why should a single morphism in the family have to have a pullback with an arbitrary morphism?

But (2) leaves me unclear about what it means to have a pullback between a morphism and a whole family of morphisms.

Which is the intended reading of the definition? Is there a reference that spells this out explicitly? I've had trouble, since most references assume you're working in a category with all pullbacks and hence don't need this axiom.

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    $\begingroup$ It seems to me that it says "the morphisms in the family of morphisms are squarable". That is, it is stating a condition on morphisms, which is required to apply for each morphism in a family, rather than stating a condition directly on families of morphisms. That would be reading (1). As to whether it fits with the intuition or the intention, I couldn't say. $\endgroup$
    – LSpice
    Commented Jul 7, 2023 at 12:59
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    $\begingroup$ It is the first one. The French text has no ambiguity (it defines that a map is quarrable if the pullback exists along any "base change", and the condition is that every map in every covering family is quarrable). This is also the definition in stacks.math.columbia.edu/tag/03NH $\endgroup$
    – Z. M
    Commented Jul 7, 2023 at 13:09
  • $\begingroup$ Without looking at SGA4, I think the point is that there should be another axiom saying that the induced family $\{U_i\times_U V\to V\}$ is a covering of $V$ but you first need this family to exist. $\endgroup$
    – Simon Wadsley
    Commented Jul 7, 2023 at 18:59
  • $\begingroup$ In other words you shouldn't necessarily try to isolate the intuitive meaning of a single part of a definition outside the context of the whole. $\endgroup$
    – Simon Wadsley
    Commented Jul 7, 2023 at 19:01

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