Can someone please say, or give a reference which discusses, why we should bother with covering sieves/topologies over covering families/pretopologies? Pretopologies seem to be sufficient for defining sheaves and feel more (geometrically) intuitive, to me.
It seems Jean Giraud introduced the idea of sieves after Grothendieck's first definition of a topology on a category. I would like to know why Giraud did that?
For reference: I am talking about the definitions as given here https://en.wikipedia.org/wiki/Grothendieck_topology
I have heard (i) the indicies in the definition of a covering family (pretopology) are "annoying" (ii) the requirement of pull backs is also "less than desirable" ... Are these the only reasons?
In my understanding, the difference between Grothendieck topologies and pretopologies is the same as the difference between (usual) topologies and basis of the topology. You can define a usual topology say on $\mathbb{R}^n$ by requiring that open balls of radius $\frac{1}{n}$ whose center have rational coordinates are open. From this, there is a way to recover the full topology.
Say two basis are equivalent if they define the same topologies. There is a quick way to see if two basis $\mathfrak{B}_1,\mathfrak{B}_2$ are equivalent : just check if for any $U_1\in\mathfrak{B}_1$ and any $x\in U_1$, there exists $U_2\in\mathfrak{B}_2$ such that $x\in B_2$ and $B_2\subset B_1$ and the other way around.
Now the problem with Grothendieck topologies is that this quick way does not exists. More precisely, given two Grothendieck pretopologies $Cov_1, Cov_2$, say they are equivalent if they define the same Grothendieck topology. Now there is in general no way to quickly compare them with the following lines : if $\{U_i\to X\}$ is a cover in $Cov_1$, then there exists $\{V_j\to X\}$ in $Cov_2$ such that...
Well, in fact the condition for comparing them leaves the domain of coverings : we need sieves.
In a similar vein, if we have two pretopologies, $Cov_1,Cov_2$ how to define the pretopology defined by their union ?
(In fact given a pretopology $Cov$, there is a way to saturate it which makes possible the comparison, but this saturation process uses sieves, or even if it avoid sieves, it is not easier than them...)
I never heard that the indices may be annoying, though I understand why (if we need to compare objects, like Cech nerves, associated to two coverings which differ by their order, though this is trivial, a rigorous proof becomes more technical...), the thing is, in most Grothendieck topology (the exact name is superextensive topology), $\{U_i\to X\}$ is a covering iff $\{\coprod_i U_i\to X\}$ is a covering, so we can replace any covering by a single object $\{U\to X\}$. Hence, the indices are not a problem anymore...
Now, for many purposes, pretopologies are sufficient ! In fact, I never used sieves though I used several Grothendieck topologies. In algebraic geometry most topologies are given by covering and they are already comparable : a Zariski covering is in particular an étale covering which is also an $fppf$-covering...