Vtyjkyuk

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)?

In general "the category $mathbfSet$" 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 $mathbfSet$ are all pure sets. But if your foundational axioms are something like ZFA (ZF with atoms = urelements), then the objects of $mathbfSet$ include sets of atoms. The presence or absence of atoms is generally speaking irrelevant for the categorical properties of the category $mathbfSet$; in both cases it is a cocomplete well-pointed elementary topos, etc. (Other choices of set-theoretic axioms do affect the properties of $mathbfSet$, 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 $mathbfSet$ of all sets we can consider the category $mathbfSet_mathrmpure$ of pure sets (those which hereditarily contain no atoms). Both of these will have all the same basic categorical properties, and the inclusion $mathbfSet_mathrmpurehookrightarrowmathbfSet$ 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 $mathbfSet$ 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.

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.