To define a Grothendieck topology of a category, we usually require that the category is small. 


**Question 1: Why do we need to require the category to be small?**


I thought that the problem was that we wanted a sieve to be a set. (In the category of manifolds, the maximal sieve of an arbitrary manifold $X$ is not a set.) But my thought must be wrong because of the following remark.


When we want to deal with the category of manifolds (schemes, or topological spaces), [Metzler](http://arxiv.org/abs/math/0306176) says that we can avoid this problem by choosing a subcategory which is small. For example, the category $\mathbf{M}$ of smooth manifolds which are second countable and Hausdorff.


**Question 2: Why does this trick work?**


Metzler explains the reason in Remark 13 in the linked paper, but I can not understand it (because I do not know what is the problem.)


**Question 3: If we define a Grothendieck topology as an equivalence class of basis of Grothendieck topology, does this definition work?**


In the definition of a basis of Grothendieck topology (See Def 5 in the paper by Metzler for the definition), we seem to deal with only sets. So I think that the definition is well-defined for any category.



Grothendieck topology for a non-small category
1. We need the category $C$ to be small so as to define the functor category $Cat(C^{op},Set)$, i.e. the category of presheaves, so that it is locally small.
2. The trick works with the category of manifolds because the category of smooth Hausdorff second countable manifolds is equivalent to a small category (use the embedding theorem, so that every such manifold is embedded in a high-dimensional $\mathbb{R}^n$ - there are only a set of such submanifolds if we take them to be literal subsets of Euclidean spaces). People generally don't want to deal with non-Hausdorff manifolds (some do) and almost never non-second countable ones (only show up as pathological counterexamples, for example the long ray). So one can repeat this trick for any large category which is equivalent to a small category.
3. There might be a proper class of bases for any given topology on a large site.
It is entirely equivalent to work with a pretopology/basis rather than a topology, as one doesn't need the full topology to define sheaves (or possibly stacks etc), which is the whole point. Once you have the topos of sheaves, the choice of basis is immaterial, though sometimes convenient for calculations.


The real thing you want is that the axiom [WISC](http://ncatlab.org/nlab/show/WISC) is satisfied, especially when the definition of a pretopology (=basis) doesn't demand that each object only has a set's worth of covering families. Consider for example the category of groups with the pretopology consisting of single epimorphisms as covering families. This is a large category such that each object has a proper class of covering families. But one can find a set of epimorphisms such that any epimorphism is refined by one in that set, and this is the statement of WISC. For a small category, WISC is automatically satisfied, and also when the definition of basis is such that only a set of covering families is associated to any object.


A non-example is the category of schemes with the fpqc topology, and so there we cannot sheafify an arbitrary presheaf. Another non-example can be the category of sets if one assumes the negation of the axiom of choice (independent work of van den Berg, Karagila and myself).


