# Is there a Category Set with ur-elements?

The question says it all. Below is some context.

For the RDF Semantics to be useful in the real world, the objects one talks about have to be things people care about. A quick summary of RDF and how it is used is presented in the Stack Exchange (failed) Question How should one model RDF semantics in terms of category theory?. It was shown in the answer to another question What kind of Categorical object is an RDF Model? what such a category of models is. But I am getting questions from well known semanticists like the following:

I know what a set is, and I know what it means to talk of, say, a set of numbers or a set of spiral galaxies, or the set of oak trees growing in Escambia county in 2018. But I have absolutely no idea what a free category of a quiver of oak trees would be.

The category Set contains all sets of /pure/ set theory, right? The sets described by ZFC /without/ ur-elements? Because those are not the slightest interest to me, speaking as a semanticist. My sets (and Tarski's sets) have real stuff in them.

If there is such a category how is it related to Set? (Is there a literature on it, I can refer my friend to)?

• What is a ur-element? – Taroccoesbrocco Sep 8 '18 at 8:48
• @Taroccoesbrocco: It's an atom, a non-set object which can be an element of sets, but has no elements on its own. Kinda like the empty set, but not the empty set. – Asaf Karagila Sep 8 '18 at 8:49
• I added a link to the wikipedia entry on ur-element. If there is a better link feel free to change it, or tell me. – Henry Story Sep 8 '18 at 8:53
• I would be very surprised if the category of sets with urelemente looked different from the usual category of sets. It seems like if Choice holds then each set of (sets and) urelemente is going to be isomorphic to a pure set (e.g. some ordinal). – Malice Vidrine Sep 8 '18 at 9:56
• "My sets (and Tarski's sets) have real stuff in them." I'm curious about this claim about Tarski's view of sets. I understand that it's a fictional well-known semanticist, not you, who is making the claim. But do you have a reference? – Alex Kruckman Sep 8 '18 at 16:03

In general "the category $\mathbf{Set}$" refers to the category whose objects are all sets in whatever your chosen foundational system is. (Or perhaps only those sets that belong to some Grothendieck universe, but that's irrelevant to the point here.) If your foundational axioms are ZFC, in which all sets are pure (sets of sets of sets of ...), then the objects of $\mathbf{Set}$ are all pure sets. But if your foundational axioms are something like ZFA (ZF with atoms = urelements), then the objects of $\mathbf{Set}$ include sets of atoms. The presence or absence of atoms is generally speaking irrelevant for the categorical properties of the category $\mathbf{Set}$; in both cases it is a cocomplete well-pointed elementary topos, etc. (Other choices of set-theoretic axioms do affect the properties of $\mathbf{Set}$, but generally speaking the presence or absence of atoms does not.)

One way to make precise the question "how is the category of sets-with-atoms related to the category of pure-sets" is to work in a theory such as ZFA, where in addition to the category $\mathbf{Set}$ of all sets we can consider the category $\mathbf{Set}_{\mathrm{pure}}$ of pure sets (those which hereditarily contain no atoms). Both of these will have all the same basic categorical properties, and the inclusion $\mathbf{Set}_{\mathrm{pure}}\hookrightarrow\mathbf{Set}$ is fully faithful (since the notion of "function" is the same in all cases). If we assume the axiom of choice, then this inclusion is moreover essentially surjective, and hence an equivalence of categories; this is because under AC any set (even a set of atoms) can be well-ordered and is therefore isomorphic to a von Neumann ordinal, which is a pure set. (It's generally not sensible to ask whether two large categories are isomorphic, only whether they're equivalent.) But in the absence of the axiom of choice, there might in principle be sets containing atoms that are not isomorphic to any pure set. (For instance, I expect that probably a permutation model of ZFA contains such sets, although I'm not conversant enough with such things to give a specific example.)

With all that said, I think that your friend is asking the wrong question. There is no sense in which a mathematical set can contain real oak trees. The elements of a mathematical set, be they other sets or atoms, are other mathematical objects, not real objects that you can see, pick up, or touch. (Even a mathematical Platonist, I think, would agree that the Platonic mathematical objects exist in another Platonic world and are not things like trees that we can see and touch.) The way we use mathematics to model the real world is by choosing to represent certain real objects by certain mathematical ones. So the presence or absence of urelements in the objects of the category $\mathbf{Set}$ is irrelevant to our ability to use it to talk about oak trees; in either case we have to use some mathematical set to "label" the oak trees and thereby transfer theorems proven about the former to real facts about the latter.

• Very helpful answer. This reminds me somewhat to the David Lewis' work on Mereology for which I found the following article online by John P. Burgess princeton.edu/~jburgess/Lewis.doc . If I remember correctly Lewis thought of sets in a way as referring to the things they contained. In the end all there was are the singletons. Then he asked what the empty set referred to and his genial answer was the mereological fusion of all possibilities, which gives maths its necessary character while also allowing us to speak of burgers... – Henry Story Sep 8 '18 at 16:13
• I think the idea is that one wants to say that a set of one burger is different from the set of one tank, even if they are isomorphic. But that is the question I asked "Isomorphism in the Real World" math.stackexchange.com/questions/2494830/… (that was closed but the linked question gave some interesting answers, but in the pure math space). It kind of seems to raise the question if sets are abstract, how can one do semantics with sets, if sets themselves need semantics. – Henry Story Sep 8 '18 at 16:49
• I'm not quite sure what that question means, but my guess is that the answer is just that you have to stop somewhere, otherwise it's "turtles all the way down". Mathematical logicians study models of abstract theories, and they construct those models inside mathematics. That model-constructing mathematics can in turn be formalized inside a set theory like ZFC, while on the other hand the formal system of ZFC is one of the abstract theories of which they can consider models. There is no contradiction here, it's just that sometimes we are using sets and other times we are studying them. – Mike Shulman Sep 8 '18 at 18:05
• @HenryStory Sets don't need semantics. ZFC, say, is a formal theory. You can use and understand ZFC via proof theory. Manipulation of proofs can be formalized in very weak systems like PRA which is not a set theory nor does it require sets. That said, on the one hand manipulating PRA formulas requires something similar/comparable to PRA, on the other hand PRA (modulo ultrafinitist concerns) is closely related to computation as we'd do it with real, physical computers. – Derek Elkins Sep 8 '18 at 19:15
• For my conversation with my friend who accepts ZFA, all I needed was to show him that there is a category of ZFA and it's relation to Set. This is so I can then move on to bring in Category Theoretic tools if applicable, in terms of Category of Sets with urelements. – Henry Story Sep 9 '18 at 6:28

For the title question at face value: Of course, one can consider the category whose class of objects is the class of sets with urelements (i.e., the things a suitable theory of sets with urelements considers), whose morphisms are the maps between such sets, and where the compositoin of morphisms is the composition of maps. See, I just defined it and the defining properties of a category are immediately verified.

• A simple answer such as yours is great. Sometimes people stumble on questions like this. Perhaps I should add to the question if this is a well known category with some literature about it, or if it is isomorphic to the category Set. That would be helpful. – Henry Story Sep 8 '18 at 9:20