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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 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.

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4 Answers

up vote 9 down vote accepted
  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 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).

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There are many other reasons to want $\mathcal{C}$ to be small. One is that we want $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ to have a subobject classifier. (This implies that for any object $c$ there is only a set of sieves on $c$; though the sieves themselves may be proper classes.) Another is that we want $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ to be bounded over $\textbf{Set}$ (and so a Grothendieck topos). –  Zhen Lin Jan 24 '13 at 0:29
    
@Zhen - good points. –  David Roberts Jan 24 '13 at 1:17
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Since the question is about topology I suppose the OP is interested in forming sheaves. Then the thing is: over a large site there is in general no sheafification functor ncatlab.org/nlab/show/sheafification. For how to get around this, the keyword is dense subsite ncatlab.org/nlab/show/dense+sub-site –  Urs Schreiber Jan 24 '13 at 13:11
    
Thank you for detailed answer. I am reading your and other answers to understand. Regarding Question 2, I understand that the category of Hausdorff and 2nd countable manifolds is small. The question is: why can we "pass" the small category equivalent to the original category we want to consider? Metzler says "the base category should be thought of as the category of local models", but I can't understand this explanation at all. –  H. Shindoh Jan 24 '13 at 20:34
    
We can pass to a small equivalent subcategory because everything you want to do is invariant under equivalence of categories. Or, bringing in the pov that Urs mentioned, as long as you get a category of sheaves which is equivalent to the one-universe-up category of sheaves on the whole site, that is what counts. As David C mentioned, all you need really is to consider the site with objects $\mathbb{R}^n$ and all smooth functions between them (these are the local models for manifolds, in the hopefully obvious sense), and you recover manifolds as non-representable sheaves on this site. –  David Roberts Jan 24 '13 at 23:10
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Re 2:

In my opinion, none of the answers just yet have hit the nail on the head about "why this trick works". The real reason is Urs' comment about dense subsites. Whether one takes manifolds to mean 2nd countable + Hausdorff, or whether one removes these conditions and considers all topological spaces with a smooth atlas, the topos of sheaves over their corresponding sites are equivalent. This is why the trick works; it's not simply that you don't care about pathological manifolds (it's actually very convenient to have them around when you're talking about differentiable stacks), but that you don't need to include them in your site of definition, because they are faithfully represented by the sheaves they induce over any full subcategory of manifolds, containing at least one manifold of each dimension. E.g., one can consider the full subcategory of manifolds spanned by only those of the form $\mathbb{R}^n,$ and takes sheaves on this site, and this is equivalent to taking sheaves on the site of all manifolds. The latter site is NOT essentially small, but it doesn't matter. In full generality, you need a site with a "small set of topological generators": see SGA4 (in particular expose ii, theorem 3.4) and the link Urs posted http://ncatlab.org/nlab/show/dense+sub-site

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+1. Very nice answer –  David White Jan 24 '13 at 20:38
    
"topological spaces with a smooth atlas"? Do you mean any topological space which is the orbit space of some groupoid in $Cart$ or $Diff$? –  David Roberts Jan 24 '13 at 23:06
    
@David Roberts: No. I just meant to say, any smooth manifold, without the requirement that it is 2nd countable, Hausdorff, paracompact... just a bare atlas. –  David Carchedi Jan 25 '13 at 0:03
    
OK, that's what I thought originally, but then came up with the 'explanation' I wrote. –  David Roberts Jan 25 '13 at 0:59
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Ad 2: The chapter on set theory in the Stacks Project explains in detail how to construct small categories of schemes, closed under various operations, and then the chapter on topologies constructs the usual topologies (Zariski, fpqc, fppf, étale, etc.) on these categories - without running into the set-theoretic difficulties which all(?) the other sources ignore. The rough idea is to use the reflection principle from mathematical logic in order to get a large enough ordinal number $\alpha$ which bounds the schemes under consideration. In practice, I think that you can ignore $\alpha$ because you can always choose it large enough, at least as for the proofs of specific theorems for example about cohomology groups etc.

As David has explained, we don't have to do all this in the case of manifolds, which probably was the case the question was aiming at. Therefore this answer here is just additional info for the interested reader...

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More precisely, the stacks project constructs, for any given set of schemes, a category of schemes closed under all the 'usual operations', so any given problem or theorem, which only uses a set of schemes, takes place inside one of these small categories. Off-topic, this is one of the ideas in McLarty's program to show modern arithmetic geometry only needs very weak foundations: any given concrete problem only needs a countable site, so we can work down in arithmetic :) –  David Roberts Jan 24 '13 at 2:56
    
Interesting. The reflection principle is equivalent to the axiom of replacement, so this is a natural example of the axiom being used outside of set theory. They also give a direct proof, but the proof itself uses replacement. –  arsmath Jan 24 '13 at 23:04
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The other answers are all good, but I thought I would also point out that one doesn't have to require that sites be small, or have small dense sub-sites, or satisfy WISC. I think one does generally want to assume that each covering sieve contains one that is generated by a small family, and also an additional "solution-set condition" that is vacuous for small sites, which I studied in this paper.

In particular, I showed there that there is a reasonable notion of "the category of (small) sheaves" on such a large site, and in most cases it does admit a "sheafification" functor from the category of (small) presheaves. What's different is that sheafification doesn't necessarily have a right adjoint, so that "sheaves" can't necessarily be identified with particular presheaves. The "category of small sheaves" is generally not a Grothendieck topos, nor an elementary one, but it satisfies all of Giraud's axioms except for the existence of a small generating set, and it has "the same" universal property (in a certain sense) as the topos of sheaves on a small site.

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Interesting. So if sheaves aren't a special type of presheaf, then what are they? –  David Carchedi Jan 25 '13 at 14:50
    
Things formally glued together from objects of the site. For instance, schemes can be identified with "small sheaves" on the large site of all rings. –  Mike Shulman Jan 26 '13 at 2:53
    
I think small sheaves are really great. In fact, in my heart I think that Grothendieck got the definition of sheaves on a site wrong. What he wanted was small sheaves, although for probably all practical purposes the only advantage to using small sheaves is that you never have to say the word 'universe'. But I'm a little confused by something you write here, Mike. I think of small sheaves as sheaves that can be expressed as small colimits of representable sheaves. In particular, small sheaves are a full subcategory of sheaves and hence of presheaves. –  JBorger Jan 27 '13 at 0:06
    
If you define 'small' with reference to a universe, then yes, small sheaves are a full subcategory of large sheaves and also of large presheaves. My point is that a small sheaf may not necessarily be a small presheaf, neither in the sense of "being a small colimit of representables (as a presheaf)" nor in the sense of "taking values in small sets". –  Mike Shulman Jan 27 '13 at 1:03
    
Moreover, you can define the category of small sheaves without needing to assume a universe, giving 'small' the other meaning of 'a set' (rather than a proper class). In this case, there is generally no way to define the category of large sheaves, or that of large presheaves. That's the context in which I meant my statement that small sheaves are not particular (small) presheaves. –  Mike Shulman Jan 27 '13 at 1:05
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