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So I have an introductory knowledge of category theory but there is one concept I can't get my head around and would like some help:

When my class had categories defined we said a category $\mathcal{A}$ is a collection of objects (not necessarily a set) and for every two objects $A, B$ there is a set Hom($A$,$B$) that are all the morphisms (arrows) between the two objects.

I was getting along fine with this until my professor mentioned that we cannot have a category of categories. He said we could certainly look at the collection of all categories but given two categories the functors between them may not form a set thus violating our definition. So this is where my confusion is. I understand that not all collections can be sets, i.e. under general set theory we cannot have a set of all sets that don't contain themselves (Russell's Paradox). But I don't quite see what is wrong with having a set of all functors between categories.

I'm well aware that there are things called Higher categories that allow my Hom space be a category on its own right and this would allow me to form a higher category of categories or some such (I'm not very familiar with this area, but it's also not quite what I'm asking at the moment).

So if possible I would love an example (or an explanation of) two categories where the collection of functors between them cannot be a set. I spoke to my professor about it but he pretty much just said it can lead to paradoxes, but I don't see how.

Any help is appreciated, Thanks

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  • $\begingroup$ Well, you describe a "locally-small" category, where the $\hom(-,-)$ are sets. They can be "bigger" collections (classes), which would give us a "large" category (example of a large cat: Set). THEN the question is: are you talking about the category of small categories, or of large categories? (Or both?) $\endgroup$
    – Alex Nelson
    Sep 28, 2013 at 19:33
  • $\begingroup$ Daniel Rust' answer is clear and simple and adequately answers your question. I just would like to add that the definition you have been given is not the most general one. What you are working with are generally called locally small categories. In any case, it is pretty obvious that you cannot make the category of all categories, for exactly the same reason that you cannot have the set of all sets or the mother of all mothers for that matter. The reason is simply that such things would have to contain themselves and this is a little awkward, and is considered unacceptable. .... $\endgroup$
    – magma
    Sep 29, 2013 at 23:36
  • $\begingroup$ ....So you cannot have a giant category that contains itself (since it contains all categories). The way out is to have hierarchies of collections: sets, classes, conglomerates. You can have the class of all sets and the conglomerate of all classes, but not the set of all sets or the class of all classes. See The Joy of Cats katmat.math.uni-bremen.de/acc/acc.pdf for an eye-opening view on the quasicategory of all categories. $\endgroup$
    – magma
    Sep 29, 2013 at 23:52

2 Answers 2

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The class of functors from the one object category to the category of sets (which is determined by the image of the single object) is not a set because there is no 'set of all sets'.

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It's only possible to say that there is no 'set of sets' if you believe that Russell's paradox rules it out.

Russell's paradox disappears if you accept dialetheias. If the idea of a 'large category' being one that contains things such as a 'set of all sets' there is no real problem because the 'set of all sets' is perfectly coherent if you omit the otiose axiom of non-contradiction.

If this is what is being meant - and I'm really not sure that it is - then I'd be interested to know the implications of dialetheism for category theory.

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  • $\begingroup$ If I understand your idea correctly, you say that we have a category of sets, then we consider a large category whose objects are categories themselves, one of which is the category of sets we previously had. But now you effectively changed the definition of the word "set". Previously you considered sets as objects of one category, and now you consider them as objects of a different category. The idea is clearer with universes, when we move to a larger universe, the previous universe is now set, but we also change the meaning of the word "set" to include new objects. $\endgroup$
    – Asaf Karagila ♦
    Feb 16, 2014 at 6:57
  • $\begingroup$ I'm not sure why you see that as different definitions of 'set'. It doesn't matter what the elements of a set might be, elephants, categories, sets, sets of sets, those are all sets. $\endgroup$
    – Peter Brooks
    Feb 16, 2014 at 7:47
  • $\begingroup$ Because there is no "absolute" notion of set (and if there is, then the collection of all sets is certainly not a set). Instead we have models of set theory, and their elements are called sets. Once you change the model of set theory that you are working with, you change the definition of what it means to be a set. If we say that numbers are elements of $\Bbb N$, then $-1$ is not a number, but we then decide that "number" means a real number, then now $-1$ is a number. We changed the context, we changed the definition of "number". Same goes here. $\endgroup$
    – Asaf Karagila ♦
    Feb 16, 2014 at 7:54
  • $\begingroup$ I understand your point. A set contains objects of any category. That's the point, really, and that's why this was called naive set theory and there's Russell's paradox. With that definition, then there is no change in the definition of 'set' - any new category can, automatically be a member of a set. That there is no universal set is proved by contradiction, a proof that relies upon the law of non-contradiction, so is not safe. $\endgroup$
    – Peter Brooks
    Feb 16, 2014 at 10:20

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