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For a category $C$ it is required that the morphisms of any two objects of $C$ form a set (c.f. Lang: Algebra, or Weibel: An introduction to homological algebra).

What's the point about this requirement ? Would there be any disadvantages / logical deficiencies if one allows the morphisms to form a proper class ?

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What I am saying is this. Take any mathematical object, defined by a list of axioms. Choose of the axioms and then ask on MO "what is the point of this axiom anyway? Why can't I drop it?". I am not so sure that this is a very good way of generating MO questions! But of course some of these questions are good, the answer being "if you drop this axiom then you recover the notion of a (blah), and these are widely used in (blah)". Is this question really one of the good ones? If so then there will probably be some natural situations where the axiom fails. Where are these situations?? –  Kevin Buzzard Dec 9 '10 at 20:07
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Here's a "category" with proper-class hom-sets. The objects are all sets; for any two sets A and B, Hom(A,B) is the class of all sets C together with a surjection to A and a map to B. In other words, multifunctions from A to B. I don't have a use for it but I can't say confidently that it is useless and it is certainly not that artificial. –  Ryan Reich Dec 9 '10 at 20:11
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@Ryan, David, Qiaochu: together you have convinced me that this is a genuine question. Nice one :-) –  Kevin Buzzard Dec 9 '10 at 20:54
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@Kevin Perhaps another example that will seem less artificial is the localisation of a non-finitely complete, non-small category $C$ at a class $W$ of arrows not having a category of fractions a la Gabriel-Zisman. A priori this is not locally small, since the arrows of $W^{-1}C$ are (classes of) zig-zags of arbitrary length, with the wrong-way arrows in $W$. I can't quite recall if Quillen uses it for non-small categories, but this is how he defines the fundamental groupoid of a category (taking $W = Arr(C)$), and this is certainly an interesting question for me. –  David Roberts Dec 9 '10 at 20:55
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Dear Kevin: this issue really arises when defining derived categories via localization as in David Roberts' comment just above. See 10.4.4 in Weibel's book, for instance. (In fact, Weibel should have "locally small" hypotheses in many places where it is omitted.) Also, when Grothendieck proves his criterion for an abelian category with a "generating object" to have enough injectives (via various axioms called things like AB1, AB2*, etc.), he really uses transfinite induction on Hom sets in a clever way. So there are reasons other than Yoneda. –  BCnrd Dec 9 '10 at 21:44
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4 Answers 4

If you are doing serious category theory, then at some point you will come across what are affectionately known as 'size considerations' or similar. In particular, any presheaf category $Cat(C,Set)$ and the subcategory of sheaves is not locally small (homs are sets) when $C$ is not a small category (set of objects). For example, you might want to consider the category of sheaves on the category of spaces, or schemes, or on a topos (these are not usually small). Then the Yoneda embedding, as Ryan points out, will not work, which is a bit of a problem.

One workaround is the axiom of universes, say with two universes $U \in V$. Then you can talk about locally small categories in $U$ - homs are elements of $U$ whereas the objects form a subset of $U$ (so these categories are '$U$-large'). Then the presheaf category consists of functors to the category of sets which are (isomorphic to) elements of $U$. The (pre)sheaf category is then locally small in $V$, and the Yoneda embedding for this category is taken into presheaves with values in the category of sets which are (isomorphic to) elements of $V$.

Whenever you see the phrase 'locally small', you can be sure someone is using some sort of foundations that distinguishes between large and small - Universes, GBN class-set theory or similar - to get around the issue.

Edit: Actually if one wants to think of schemes as sheaves on CRing, then you need to think hard about local smallness, otherwise the category of schemes will not be category under the naive definition that the homs are sets.

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This is great---thanks. I think this answer more than justifies the existence of the question. –  Kevin Buzzard Dec 9 '10 at 20:53
    
Kevin, whenever we form Ext-sheaves, those Hom's had better be sets or else it is really confusing what is going on. –  BCnrd Dec 9 '10 at 21:46
    
To quote from Clark Barwick's arXiv paper math/1012.1889 out today, "We use the Universe Axiom of [SGA4 I, Exp I, §0], and we fix two universes U ∈ V. All rings, modules, schemes, etc., will be U-small. We employ the following notational conventions for categories or ∞-categories: (N.1) Roman characters A, B, C, . . . , etc., will denote categories and ∞-categories that are essentially V-small and locally U-small. (N.2) Bold characters A, B, C, . . . , etc., will denote categories and ∞-categories that are locally V-small." –  David Roberts Dec 10 '10 at 1:31
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I have to admit I had never heard of the distinction that Thierry Zell and Adam Hughes point out in their answers; I had always learned that a "small" category as opposed to a "large" one was merely the stipulation that the objects form a set, but that hom-sets were always sets.

That said, in hindsight the reason this is correct is Yoneda's Lemma: that there exists a fully faithful embedding of any category in its functor category to Sets. Of course, if hom-sets are classes then one cannot do this, since there is no such thing as the class of all classes but there is a class of all sets. Since Yoneda's Lemma is arguably the most important single fact about category theory, it is worth preserving.

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I quite agree. Yoneda is certainly the most useful tool in category, at least in all the things I care about. –  Adam Hughes Dec 9 '10 at 19:50
    
Why is there "no such thing as the class of all classes" ? In some comments above the category of all categories is mentioned. So naively it's not hard to consider a category with objects all classes and morphism all mappings between two classes. And in fact this construct appears in 6 iv) (page 10) of the article: math.stanford.edu/~feferman/papers/ess.pdf There is also mentioned a Yoneda lemma for arbitrary (i.e. not necessarily locally small) categories. –  Ralph Dec 10 '10 at 16:07
    
@Ralph: It depends what you mean by "class". I am not a set theorist, and I know of only two theories which treat them: NBG (von Neumann-Bernays-Goedel) and Grothendieck universes. Neither one allows a class to contain itself. The ultimate reason is that, without some form of restriction on the axiom of comprehension, doing so would allow Russell's paradox. In that Feferman paper, it appears (though I don't know because, not being a set theorist, I didn't read it carefully) that he is using a set theory with such a restriction, based on types. –  Ryan Reich Dec 12 '10 at 19:59
    
@Ralph: also, in the comments mentioning the category of all categories, I think they mean small categories. Look at the third answer to mathoverflow.net/questions/3278/… for a discussion of this issue. –  Ryan Reich Dec 12 '10 at 19:59
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Ryan, thanks for the reply. I'm neither a set theorist and don't know under which restrictions one can build a "category of classes". In any case I agree that in practise Yondea's lemma is most reasonable in conjunction with (locally) small categories. –  Ralph Dec 14 '10 at 1:20
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If you look at Steve Awodey's book Category Theory (Oxford Logic Guides * 49) you'll see that on p. 22, Definition 1.12 is that a category in which $\hom_\mathbf{C}(A,B)$ is a set for every pair of objects $A$ and $B$ is called locally small.

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And I would add that since it warrants an adjective, there might be cases when this condition doesn't hold. –  David Roberts Dec 9 '10 at 20:56
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As David mentioned, category theorists generally tend to shy away from such `size issues', because in a sense they do not touch the heart of the matter, as the original question rightfully suggests. Apart from such foundational issues, I can think of two practical reasons where it is important that homs be sets.

First, there is Freyd's celebrated Adjoint functor theorem. It gives conditions that characterize precisely when a given functor has an adjoint. Crucially, one of the conditions, called the `solution set condition', is that a certain class is in fact a set. This shows that size issues do play a fundamental role in category theory, which came as quite a surprise to most people.

Second, one can think of Enriched categories. It turns out that a lot of category theory goes through if homs are not necessarily objects of the category Set of sets and functions, but objects in an arbitrary monoidal category, with composition being a morphism of that category. For example, relating to Kevin and Qiaochu's comments above, a 2-category can be seen as a category enriched in Cat, the category of categories. But this also gives some surprising examples. For example, a metric space can be seen as a category enriched in $\mathbb{R}^+$, i.e. the poset $[0,\infty]$ with monoidal structure given by addition. And of course a locally small category is just a Set-enriched category. This is not an argument against `large categories' per se, but does indicate that a lot of murky waters can be avoided by only considering locally small categories.

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Re your first point: perhaps it was a surprise that size issues matter, but it should not have been. This answer (mathoverflow.net/questions/33385/preservation-of-limits/…) gives a convincing, pretty basic argument that large colimits should not be considered merely "large" colimits but actually something entirely different from small ones. It doesn't deal with morphisms, though. –  Ryan Reich Dec 10 '10 at 8:44
    
@Ryan: I’m not sure I’d read that answer as showing a fundamental difference between large and small colimits; everything it says holds with “small” interpreted as “strictly smaller than κ”, for any regular cardinal κ. –  Peter LeFanu Lumsdaine Dec 10 '10 at 15:22
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