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One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a thing.

Models of NF are permutation-invariant in the sense that given an $\in$-automorphism $\gamma$ on a model $\mathcal{M}$ of NF, the model with the same underlying set and new membership relation $x\in_\gamma y:=x\in\gamma(y)$ is also a model of exactly the same stratified sentences (and hence all the axioms of NF).

I'm interested in getting a look at this in the language of category theory (because). I know the category theory that one would know from working Awodey's Category Theory cover-to-cover, but I'm working my way through Mac Lane & Moerdijk, and revisiting Goldblatt's Topoi, to try and brush up on categorical model theory. I'm really not sure what I should be looking into that might offer more insight than, say, rephrasing the whole problem in ETCS. So far I've not recognized anything as clearly the right way to address the question.

So: Are there any areas of category theory (hopefully a little narrower than "topos theory" or the like), or specific types of constructions, that would be especially handy for getting a fresh look at what's really happening in this kind of permutation-invariant theory? Would an answer to this question depend on what kinds of constructions give us models of NF? Am I barking up the wrong tree entirely?

Sorry I can't be more specific in the question, but the whole thing is in the pre-exploratory phase...

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You're not going to have a lot of luck using topos theory to tackle problems in NF: a general model of NF can't be a topos because it's not Cartesian closed! –  Clive Newstead Jan 18 at 6:45
Well aware! The topos theory isn't to understand NF as a topos, which it clearly isn't. It's to have a setting to categorically understand NF's models (or models of related systems that have known interpretations, since nobody has an example of a model of NF yet...). –  Malice Vidrine Jan 18 at 6:50
My hope is that there's a more informative route than "assume you have a model of NF in a topos and see what it looks like". –  Malice Vidrine Jan 18 at 6:54

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