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For the purposes of this question, let me fix a "true" universe of sets, which I will call the "true sets". Recall that a category is locally presentable if it is cocomplete and accessible. Both these notions involve sets --- I will interpret them in terms of the true sets. Indeed, the word "category" involves sets (there should be a set of morphisms between any two objects) and again here I mean true sets.

Now choose some other model of set theory, which I will call the "fake sets". I presume that the fake sets assemble into a category, at least if the fake model is built "internally" to the true model.

Under what circumstances is the category of fake sets locally presentable?

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  • $\begingroup$ Is there a purely set-theoretic manner of asking the question? $\endgroup$ – Joel David Hamkins Feb 15 '16 at 20:21
  • $\begingroup$ @JoelDavidHamkins No, because it's not a perfectly well-posed question, because I don't really know what a "model of sets" is. But here's a related question. If I'm not mistaken, among the universe U of sets, there some but not all of the sets are "constructible", which I think is called V. Is there a functor from U to V that is the identity on finite sets and takes colimits to colimits? It should be a sort of "rounding up" functor. $\endgroup$ – Theo Johnson-Freyd Feb 16 '16 at 2:18
  • $\begingroup$ Gödel's constructible universe is usually denoted by L, and the full set-theoretic universe is usually denoted by V. If these are different, then you can't map V to L with a functor that is the identity on finite sets, since not all finite sets will be in L. If x is not in L, then {x} is a finite set that is not in L. $\endgroup$ – Joel David Hamkins Feb 16 '16 at 2:21
  • $\begingroup$ If what you want is an $\in$-embedding from $V$ to $L$, this is related to my question mathoverflow.net/q/101821/1946, which is still open, but in joint work with Woodin, Magidor and others, we have a bunch of partial results. $\endgroup$ – Joel David Hamkins Feb 16 '16 at 2:25
  • $\begingroup$ Ah, sorry for the notation! By "is the identity on finite sets" I mean something less strict than it sounds --- perhaps I should have said "is the identity on finite cardinalities". $\endgroup$ – Theo Johnson-Freyd Feb 16 '16 at 22:30
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Your category of fake set is going to be in particular a boolean elementary topos. By Giraud's theorem, It is locally presentable if and only if it is a Grothendieck topos, so a category of sheaves over a site...

If you additionally formulate the axiom of choice so that it corresponds to external axiom of choice then the only possibility is a boolean valued model (i.e. a category of sheaves over a complete boolean algebra).

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  • $\begingroup$ Perhaps you can clarify your remark, "the only possibility is a Boolean-valued model," since of course every model of set theory can be seen as a Boolean-valued model, for any desired Boolean algebra. You mean something more specific. $\endgroup$ – Joel David Hamkins Feb 15 '16 at 22:18
  • $\begingroup$ @JoelDavidHamkins I think he means the Boolean-valued model constructed from a Boolean algebra as at for instance en.wikipedia.org/wiki/…. (I thought this was fairly standard terminology?) $\endgroup$ – Mike Shulman Feb 16 '16 at 5:03
  • $\begingroup$ that is indeed what I mean, (I also thought it was a standard terminology but I am not a model theorist at all).In fact I realized that there is several results in topos theory that says that any boolean grothendieck topos can be written as localic over an atomic topos, which maybe can be translated into a statement saying that any such model (even not satisfying choice) can be obtained by taking a permutation model inside a boolean valued model... but there is a few details that need to be clarified and I can't assert just now... $\endgroup$ – Simon Henry Feb 16 '16 at 8:12
  • $\begingroup$ I suppose it was clear what you had meant, although I find that usage to be sloppy. If $B$ is a Boolean algebra, then a $B$-valued model (in a language) is a structure with a domain of names, and an assignment of instances of those names under the fundamental relations to truth values in $B$. If $V$ is the set-theoretic universe, then $V^B$ is the class of $B$-names in the forcing sense, and this is what you meant. But there are many other $B$-valued models, for example, $W^B$ for various inner models $W$, or $V^C$ for any complete subalgebra $C$ of $B$, or inner models of $V^B$, etc. $\endgroup$ – Joel David Hamkins Feb 16 '16 at 13:45

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