$\newcommand\restr[2]{{\left.#1\right|_{#2}}}$ I'm fairly new to this whole area, so correct me if there are any technical errors in any of this.
The base category for a classical sheaf is the category of open sets of a topological space. If $\mathcal U$ is such a category, then a presheaf on $\mathcal U$ is nothing but a functor $\mathcal F\colon\mathcal U^{op}\to\textbf{Set}$ (say). We say that $\mathcal F$ is also a sheaf if whenever $U\in \mathcal U$ is covered by open sets $U_\alpha \in \mathcal U$ the following conditions hold:
- If $s,t\in\mathcal F(U)$ and $\restr{s}{U_\alpha}=\restr{t}{U_\alpha}$ for all $\alpha$, then $s=t$.
- If there exist $s_\alpha\in\mathcal F(U_\alpha)$ that are compatible in the sense that for all $\alpha,\beta$ we have $\restr{s_\alpha}{U_\alpha\cap U_\beta}=\restr{s_\beta}{U_\alpha\cap U_\beta}$, then there is some $s\in\mathcal F(U)$ with $\restr{s}{U_\alpha}=s_\alpha$ for each $\alpha$.
In other words, we could define a presheaf on any category, but when it comes to defining a sheaf, we need two pieces of topological information: what it means for a collection of objects in $\mathcal U$ to cover a given object, and what the intersection $\mathcal U_\alpha\cap U_\beta$ means when $U_\alpha,U_\beta$ are objects of $\mathcal U$.
In the case where $\mathcal U$ is the category of open sets of some topological space $X$, this information can all be recovered from the category $\mathcal U$ itself: unions (which are, after all, the same thing as covers in this context) are given by coproducts and finite intersections are given by products, since we know that the category $\mathcal U$ is closed under these operations.
The modern definition of a site extends this idea. As remarked above, a presheaf on a category $\mathcal C^{op}$ is just a functor $\mathcal C^{op}\to\textbf{Set}$. If we want to define a sheaf on $\mathcal C$ we need to make sense of the notions of covering and finite intersections on $\mathcal C$. One way of doing this is by defining a Grothendieck pretopology on the category $\mathcal C$.
In a Grothendieck pretopology, we axiomatize the definition of a covering by declaring certain sets of morphisms $\{U_\alpha\to U\}$ to be covering families (by analogy with the inclusion maps $\{U_\alpha\to U\}$ where $(U_\alpha)$ is a cover by open sets) and require that they satisfy some natural axioms. We deal with intersections categorically by requiring that certain fibre products exist; specifically:
For all objects $U$ of $\mathcal C$, and for all morphisms $U_\alpha\to U$ which appear in some covering family of $U$, and for all morphisms $V\to U$, the fibred product $U_\alpha\times_U V$ exists.
This fibred product corresponds exactly to the categorical intersection $U\cap V$ in our category $\mathcal U$ of open sets (OK, we took the product rather than the fibre product, but the product in $\mathcal U$ is just the fibre product over the whole space $X$, so we're fine). Incidentally, I suspect that the reason we decide to axiomatize covers and define intersections categorically as opposed to some other combination is that covers are what allow us to patch things together in interesting ways, while intersections are more of a utility that makes the gluing work.
The idea of a Grothendieck pretopology is supposed to generalize the classical category $\mathcal U$ of open sets in a topological space, and indeed if we define the covering families in $\mathcal U$ to be the families $\{U_\alpha\to U\}$ such that $\bigcup_\alpha U_\alpha=U$ then we get a Grothendieck pretopology on $\mathcal U$.
But it's also well known that we can define a sheaf on a basis $\mathcal B$ of open sets for the topology $\mathcal U$. Indeed, if $\mathcal B$ is closed under finite intersections, and we $\mathcal F\colon\mathcal B^{op}\to\textbf{Set}$ satisfies the sheaf axioms within $\mathcal B$, then it extends to a unique sheaf on the whole of $\mathcal U$.
But here's the point - it's more work, but $\mathcal B$ doesn't have to be closed under finite intersections. The details are given on page 3 of these notes - if $\mathcal B$ is a basis of open sets for $\mathcal U$, we say that a functor $\mathcal F:\mathcal B^{op}\to\textbf{Set}$ is a sheaf if:
Whenever $(U_\alpha)$ is a cover of $U$, we can choose a cover $(U_{\alpha\beta\gamma})$ of each intersection $U_\alpha\cap U_\beta$ such that the object $\mathcal F(U)$ is the limit of the diagram formed by the $\mathcal F(U_\alpha)$, the $\mathcal F(U_{\alpha\beta\gamma})$ and the restriction arrows.
Then any sheaf on $\mathcal B$ extends uniquely to a sheaf on the whole of $\mathcal U$.
This is much less convenient than the case where $\mathcal B$ is closed under finite intersections, but still seems like a natural notion. In this case, though, the category $\mathcal B$ does not have fibre products (since it's not closed under finite intersections). Yet fibre products are required in the definition of a Grothendieck pretopology.
So my question is this: is it just convenience that makes us require that fibre products exist in a Grothendieck pretopology (it would certainly be inconvenient to have to deal with covers of finite intersections all the time rather than the intersections themselves)? In other words, is it possible to relax the fibre product criterion in order to get a consistent (yet less clean) theory that generalizes the situation of sheaves on a basis of open sets that's not closed under finite intersections? Or is there some reason why that's impossible?