We all know that we quickly encounter issues of size when we start category theory. Categories, we are told, are collections of objects and morphisms satisfying certain axioms. But we very soon meet, among our very first examples of categories, categories like Set whose objects are too many for the collection of them to itself be a set. So what kind of ‘collection’ do we have here?
But here’s another issue about that initial example of the category Set which is ignored by many writers of introductory texts.
The issue can be posed, bluntly, like this: when the category Set is mentioned early on in an introductory text what category is that?
Typically no explanation is given at this stage. But of course which category we are dealing with depends on our set theory. For an NF-iste, the category of NFsets has very different properties from what is intended (for a start NFsets is not cartesian closed and later we learn that Set is). But fair enough, in an intro book you aren’t going to mention that in the opening pages! Rather the authors are, surely, intending to point to stuff that their beginning readers can be assumed to know about, and are saying, “Hey, you in fact already know about some categories, for example …” The charitable reading, then, is that authors are relying on their readers to think of Set as comprising the sets they already know and love.
But what sets are those?
Well, for many readers (if they have done any more than had a bit of set-notation waved at them) the sets they know and love are the pure sets of the cumulative hierarchy — pure in that there are no urlements, no memberless entities in the universe of sets other than the empty set. If set theories with urelements are mentioned in passing in an intro set-theory course, it is usually only to be dismissed and forgotten about. So: in the absence of special explicit signals to the contrary, we might well reasonably take the category Set mentioned in the very early pages to be a category of pure sets of the usual hierarchy (or at least the hierarchy up to some inaccessible, or whatever). Just sometimes this is made explicit. Thus, Horst Schubert in his terse but very good and clear Categories (§3.1) writes “One has to be aware that the set theory used here has no “primitive (ur-) elements”; elements of sets, or classes …, are always themselves sets.”
But then what are we to make, a bit later in our introductory books, of e.g. the usual presentation of the Yoneda embedding as $latex \mathcal{Y}\colon \mathscr{C} \to [\mathscr{C}^{op}, \mathbf{Set}]$. Putting it this way, if you look at the details, assumes that hom-collections $latex \mathscr{C}(A, B)$ for $latex A, B \in \mathscr{C}$ actually live in $latex \mathbf{Set}$. And since such a hom-collection is a set of $latex \mathscr{C}$-morphisms, that assumes that the $latex \mathscr{C}$-morphisms — irrespective of what the [small] category $\mathscr{C}$ is — must live in the world of pure sets too. [Sure, we may want the relevant hom-collections to be set-sized in the Yoneda embedding case — but being no bigger than set-sized is one thing, living in the universe of pure sets is something else!]
But do we really want to assume that morphisms are always pure sets?
Might we not be looking to category theory for a story about how different bits of the mathematical universe hang together which need not presuppose some over-arching, all-in, set-theoretic reductionism, and so in particular doesn’t presuppose from day one that all morphisms are pure sets?
Now, as I noted, the foundational sections you often meet early in category theory books worry away about questions of size (sets vs classes etc.). But the present worry is orthogonal to all that, and is in a way more basic. If we want to make no assumption that the denizens of different bits of the mathematical universe are all cut from the same cloth, we won’t want to slip into assuming that sets of these denizens are all pure sets. So in particular, do we really want to assume that a collection of morphisms (hom-set) must actually live in Set, if that’s the category mentioned back almost on p.1 of the book which is naturally read (and sometimes explicitly said to be) a category of pure sets? It may be that Set eventually gets to play a special role in the mathematical universe, with other most other categories being representable inside it: but surely this should be something to be shown later, not an assumption to be built in from the start.
In Chapter 3 of his fine Basic Category Theory, Tom Leinster makes the right moves here. Back track. Ask again: what are the sets that, before we start into category theory, we know and love and which are pointed to as constituting an exemplar of a category? Not the sets of pure ZFC (which mathematicians other than set theorists could even spell out the axioms which are supposed to tell you what these are?) but rather what we might call the sets of the working mathematician — sets of naturals, perhaps, or sets of reals, or sets of points in a space, or a set of continuous functions, or … And these are sets with ur-elements, elements that are not themselves presumed to be sets. (Or if we ascend to talk of sets of sets of naturals, for example, it remains the case that when we descend again through members of members … everything typically bottoms out with primitive non-sets.)
We assume various operations are permissible on these sets, and we can codify the permissible operations (for the purposes of the non-set-theorist, falling short of ZFC). Now Leinster associates this with Lawvere’s codification of the Elementary Theory of the Category of Sets (ETCS). But of course, Zermelo was in the same game, aiming to codify the principles of set-building that the working mathematician actually needs, aiming to provide enough to do the usual constructions but not enough to threaten paradox, and Zermelo’s original set theory too allowed for ur-elements. But then, if Set is to be a category of sets-with-urelements, built up using the principles working mathematicians actually need (whatever they are) — which indeed makes some of appeals to Set in category theory much more natural — then this really needs to be said (as Leinster does, and others don’t, and some actually deny).
Of course, we could go Schubert’s way and take hom-sets to be pure sets, so that morphisms and the other apparatus of category theory are all pure sets. But seeing category theory in this way as simply in the service of an across-the-board set-theoretic reductionism would surely be an unhappy way of looking at things: category theory seemed to promise more than that!