ag.algebraic geometry - Why are non-singleton covering families often ignored?

In applications I've seen, what matters is the topos, not the site. If this is true for your applications, you should feel free to replace your site by any site that produces the same topos. I think you can always make the following conventions without changing the topos (edit: not true, you need some hypothesis on the site; see comments):

1. If ${U_ito X}_{iin I}$ is a covering, then so is the singleton $bigsqcup_I U_ito X$.


2. ${U_ito bigsqcup_{jin I} U_j}_{iin I}$ is always a covering.

After that, you can use convention 1 to replace any non-singleton covering by a singleton covering, which is easier to think about (at least easier to symbolically manipulate). But you do have to keep 2 around to be able to prove, for example, that $F(bigsqcup_{I} U_i)=prod_{I} F(U_i)$ for any sheaf $F$. As far as I can tell, everybody I know adopts these two conventions and then just talks about singleton covers.

An alternative (probably better) explanation comes from the sieve-theoretic formulation of Grothendieck topologies. All that matters about a covering is the sieve that it generates, and $bigsqcup_I U_ito X$ generates the same sieve as ${U_ito X}_{iin I}$ (edit: also not true; perhaps someone who understands the sieve approach could say what the right statement is). I think the canonical reference for this approach is SGA 4 ("sieve"="crible").