Sieves over covering families? Topology or pretopology? 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?
 A: 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...
