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I'm puzzled by this comment "The objects (or arrows) of a category need not constitute a set. If they do, the category is said to be small. If they don't, the category is large." in the definition of the axioms of category theory here ( http://en.wikibooks.org/wiki/Category_Theory/Categories ).

If this is in the axioms, it is, presumably important, and the distinction between a large and small category something one needs to understand.

This page doesn't define 'set', but, if a set is a collection of discrete objects, then I'd see the arrows of a category being defined as, just that, 'the set of arrows of a category'.

Is the point being made that the arrows (or objects) need not be discrete?

Or is there quite a different definition of a 'set' being appealed to? If so, what is it?

Or, perhaps more pertinently, what is an example of a large category?

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    $\begingroup$ Well, not any collection of objects form a set - for example, the set of all sets is excluded axiomatically. Hence my guess would be that if one took the category of all sets, it would be large. But I will let someone wiser than I answer properly. $\endgroup$
    – Joe Tait
    Feb 16 '14 at 10:25
  • $\begingroup$ Besides the category of all sets (or categories), also Russel's Paradox comes to mind. $\endgroup$
    – Roland
    Feb 16 '14 at 10:27
  • $\begingroup$ Russell's paradox disappears if you drop the unnecessary axiom of non-contradiction and admit dialetheisms. $\endgroup$ Feb 17 '14 at 3:30
  • $\begingroup$ As an aside, sets aren't exclusively collections of discrete objects. Sets can be continuous, e.g. the set of real numbers. $\endgroup$
    – Vaelus
    Jul 24 '20 at 13:45
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Examples of large categories are the category of sets, of groups, of topological spaces, of rings, of vector spaces, of modules, of ....., also, any proper class gives rise to two large categories, the discrete one and the indiscrete one on the set.

The importance of size in category theory is an issue of the existence of certain constructions. For instance, it is useful to know if a category admits certain limits or colimits, since if it does, then you know you can always construct a certain object as the (co)limit of some diagram in the category. If a category is a complete lattice, then it has all limits and colimits. All here means no restriction at all on the diagram. In the other direction, if a category has all limits or all colimits, then it actually must be a poset (and thus a complete lattice).

This shows that there is some mutual exclusivity between having all (co)limits and having more than two arrows between objects. Since we are quite often interested in categories that are not posets, and we are also interested in having (co)limits, we must set some size constraints on the diagrams for the (co)limits. Typically, one defines a category to be small (co)complete, if it has all small (co)limits. That means that any diagram indexed by a small category will have a (co)limit. Large diagrams may or may not have limits.

Also, without the restriction of size, the difference between limits and colimits becomes blurred. It is possible to exhibit any limit as a (potentially) large colimit, and vice versa. Same holds for left and right Kan extensions.

Size issues also play a role in knowing that functor categories exist and also in constructions of left/right adjoints, but I won't get into that since my answer is long enough, and I hope it answers your question sufficiently, at least for now. And see Importance of 'smallness' in a category, and functor categories.

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  • $\begingroup$ I don't see why any of those are 'large'. The definition I've seen says that a 'large category' is one where the objects or arrows do not constitute a set. All the examples you give seem to be small by that definition. $\endgroup$ Feb 17 '14 at 3:28
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    $\begingroup$ @PeterBrooks the objects of the category of sets are all sets. All sets do not form a set. The objects of the category of groups are all groups. All groups do not constitute a set. Etc. $\endgroup$ Feb 17 '14 at 3:37
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    $\begingroup$ I can't agree. All sets do form a set. Sets are objects. All groups form a set, the set of all groups, because groups are objects - ∀x, x is a set, x ∊ S. Defines S, the set of all sets. $\endgroup$ Feb 17 '14 at 5:35
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    $\begingroup$ @PeterBrooks please read a bit about Russell's paradox. $\endgroup$ Feb 17 '14 at 8:05
  • $\begingroup$ I'm familiar with Russell's paradox - I've not looked at this site for a while. You might find it useful to look into dialetheism. dialetheism.org/dialetheism/… $\endgroup$ Oct 15 '14 at 16:35
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A category of sets $C$ consists of a class $\DeclareMathOperator{\ob}{ob}\ob(C)$ of objects and a class $\hom(C)$ of morphisms. Every set $A$ is an object in $\ob(C)$. ِAssume (temporarily), that $\ob(C)$ is a set, then by Cantor's theorem $|\mathscr{P}(\ob(C))| > |\ob(C)|$. On the other hand, by our assumption, all elements of $\mathscr{P}(\ob(C))$ are sets, and thus contained in $\ob(C)$, therefore $|\mathscr{P}(ob(C))| ≤ |\ob(C)|$. Contradiction. So the class of all sets is not a set, and the category of all sets is large (not small).

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As mentioned here in the wiki article on categories, the category of all sets is large category. The "set of all sets" is not a set, as it is excluded axiomatically.

Also, see the question Category of all categories vs. Set of all sets.

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  • $\begingroup$ By what axiom? Why is this axiom needed? $\endgroup$ Feb 17 '14 at 6:50
  • $\begingroup$ The 'set of all sets' is a set, and is not excluded axiomatically, unless you happen to include the axiom of non-contradiction, which is not necessary for logic. Russell's paradox disappears if you recognise that it's a dialetheia. $\endgroup$ Oct 23 '14 at 22:43
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As stated in other answers, the category of all sets is a large category. This is because considering the existence of the set of all sets leads to contradictions. Assuming any class of objects (or morphisms) is a set leads to, for example, Russel's paradox. This is the reason for the distinction between large and small sets; small categories have more structure and properties that can be used for proofs about their objects and hom-classes.

You indicated that this distinction may be irrelevant if the existence of statements which are both true and false (dialetheia) is assumed. This may be true, and it may be possible to come up with a consistent category theory in which there is no distinction between large and small sets. It's possible constructions from this version of category theory could give insight which could be translated back to conventional category theory for everyone else to understand.

However the creation of this alternate version of category theory would be quite arduous. Conventional systems of logic become trivial if statements can be both true and false, so to discern anything useful, this alternate category theory would have to be built from scratch in another, possibly unintuitive system of logic. Remember that there isn't any existing body of work to build upon for this alternate category theory; there are no existing theorems to cite which have been proved by someone else.

Ultimately, mathematicians choose to make the distinction between large and small categories because it allows category theory to be consistent with a large body of existing work which can be built upon.

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