3 typo

A) Let $\tau$ be a Grothendieck pretopology on any category with fiber products. Define a new Grothendieck pretopology $\tau'$ where a cover $\{U_i \to X\}$ is a $\tau'$ cover if and only if there exists a refinement $\{V_{ij} \to X\}$ (i.e. there exists for each $ij$ an $X$-morphism $V_{ij} \to U_i$) such that $\{V_{ij} \to X\}$ is a $\tau$ cover. Then $\tau$ and $\tau'$ generate the same topologies.

B) The operation (lets say $\Phi$) that takes topologies to pretopologies composed with the operation (say $\Psi$) that takes pretopologies to topologies gives back the same topology that you started with (i.e. $\Psi \Phi = id$). So for any pretopology $\tau$, the pretopology $\Phi\Psi \tau$ gives the same topology as $\tau$.

C) Related to (A), on the category of affine schemes, we can define an fppf cover as a family $\{U_i \to X\}$ which is jointly surjective, and such that each morphism is flat, and finitely presentedand quasi-finite. We can define an fpqc cover (still using only affine schemes) in could also include the same way but ommiting requirement that in addition the quasi-finiteness hypothesismorphisms be quasi-finite. Corollary 17.16.2 in EGA IV implies that these two pretopologies gives rise to the same topology.

In response to the first question, there are surely people more expert than me, but it seems to me that Grothendieck topologies in algebraic geometry are almost always defined via Grothendieck pretopologies. I think topologies are the nicer concept, but while the idea is suited to proving very general things about topoi, if you have a specific pretopology such as the étale, Zariski, Nisnevich, flat, cdh, envelopes (also called proper cdh - see 18.3 in Fulton's "Intersection theory" and the Mazza, Voevodsky, Weibel book), etc having actual scheme morphisms in your hands allows you to apply the strong algebr-geometric results which are largely responsible for making topologies such a powerful tool in algebraic geometry. Of course, I'm sure someone in logic would have a very different point of view.

In relation to this line of thought, Voevodsky - while working with the Nisnevich and cdh topologies - found it useful to further simplify the data leading to a topology in the notion of a cd structure (see the papers "Unstable motivic homotopy categories in Nisnevich and cdh-topologies" and "Homotopy theory of simplicial sheaves in completely decomposable topologies").

While I'm talking about Voevodsky's work, he also uses from time to time an idea which he calls "covers of normal form" in "Homology of schemes I", but the same idea is used implicitely in the appendix to "Singular homology of abstract algebraic varieties". The way I understand this phenomenon is the following. If you have two pretopologies $\sigma$ and $\rho$ on a category with fiber products, then the covers of the pretopology generated by $\sigma$ and $\rho$ are finite compositions of covers which are either a $\sigma$ cover or a $\rho$ cover. Lets denote the new pretopology by $\langle \sigma, \rho \rangle$. Many pretopologies $\tau$ in common use, are actually generated by two other pretopologies in the sense that $\langle \sigma, \rho \rangle$ and $\tau$ give rise to the same topology.

For example, the cdh pretopology is generated in this way (by definition) by the Nisnevich pretopology and the pretopology of envelopes. Voevodsky shows that the h-pretopology is generated like this Zariski and the pretopology whose covers are jointly surjective families of proper morphisms. The qfh pretopology is generated like this by étale and the pretopology whose covers are jointly surjective families of finite morphisms.

2 added a response to question 1

C) Related to (1), A), on the category of affine schemes, we can define an fppf cover as a family $\{U_i \to X\}$ which is jointly surjective, and such that each morphism is flat, finitely presented and quasi-finite. We can define an fpqc cover (still using only affine schemes) in the same way but ommiting the quasi-finiteness hypothesis. Corollary 17.16.2 in EGA IV implies that these two pretopologies gives rise to the same topology.

In response to the first question, there are surely people more expert than me, but it seems to me that Grothendieck topologies in algebraic geometry are almost always defined via Grothendieck pretopologies. I think topologies are the nicer concept, but while the idea is suited to proving very general things about topoi, if you have a specific pretopology such as the étale, Zariski, Nisnevich, flat, cdh, envelopes (also called proper cdh - see 18.3 in Fulton's "Intersection theory" and the Mazza, Voevodsky, Weibel book), etc having actual scheme morphisms in your hands allows you to apply the strong algebr-geometric results which are largely responsible for making topologies such a powerful tool in algebraic geometry. Of course, I'm sure someone in logic would have a very different point of view.

In relation to this line of thought, Voevodsky - while working with the Nisnevich and cdh topologies - found it useful to further simplify the data leading to a topology in the notion of a cd structure (see the papers "Unstable motivic homotopy categories inNisnevich and cdh-topologies" and "Homotopy theory of simplicial sheaves in completely decomposable topologies").

While I'm talking about Voevodsky's work, he also uses from time to time an idea which he calls "covers of normal form" in "Homology of schemes I", but the same idea is used implicitely in the appendix to "Singular homology of abstract algebraic varieties". The way I understand this phenomenon is the following. If you have two pretopologies $\sigma$ and $\rho$ on a category with fiber products, then the covers of the pretopology generated by $\sigma$ and $\rho$ are finite compositions of covers which are either a $\sigma$ cover or a $\rho$ cover. Lets denote the new pretopology by $\langle \sigma, \rho \rangle$. Many pretopologies $\tau$ in common use, are actually generated by two other pretopologies in the sense that $\langle \sigma, \rho \rangle$ and $\tau$ give rise to the same topology.

For example, the cdh pretopology is generated in this way (by definition) by the Nisnevich pretopology and the pretopology of envelopes. Voevodsky shows that the h-pretopology is generated like this Zariski and the pretopology whose covers are jointly surjective families of proper morphisms. The qfh pretopology is generated like this by étale and the pretopology whose covers are jointly surjective families of finite morphisms.

1

A) Let $\tau$ be a Grothendieck pretopology on any category with fiber products. Define a new Grothendieck pretopology $\tau'$ where a cover $\{U_i \to X\}$ is a $\tau'$ cover if and only if there exists a refinement $\{V_{ij} \to X\}$ (i.e. there exists for each $ij$ an $X$-morphism $V_{ij} \to U_i$) such that $\{V_{ij} \to X\}$ is a $\tau$ cover. Then $\tau$ and $\tau'$ generate the same topologies.

B) The operation (lets say $\Phi$) that takes topologies to pretopologies composed with the operation (say $\Psi$) that takes pretopologies to topologies gives back the same topology that you started with (i.e. $\Psi \Phi = id$). So for any pretopology $\tau$, the pretopology $\Phi\Psi \tau$ gives the same topology as $\tau$.

C) Related to (1), on the category of affine schemes, we can define an fppf cover as a family $\{U_i \to X\}$ which is jointly surjective, and such that each morphism is flat, finitely presented and quasi-finite. We can define an fpqc cover (still using only affine schemes) in the same way but ommiting the quasi-finiteness hypothesis. Corollary 17.16.2 in EGA IV implies that these two pretopologies gives rise to the same topology.