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I call the Vopěnka's principle:

Every subfunctor of an accessible functor is accessible

but other formulations (which may lose equivalence in weak contexts?) are also interesting to me.

If this is true, then I wonder how many axioms in the external theory can be removed (moving towards the free topos with NNO) so that Vopěnka's principle is still inherited by topoi.

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  • $\begingroup$ It seems unlikely to me that anything like Vopěnka's principle would ever hold in the free topos with NNO. Maybe constructively it becomes a lot weaker, but since it's classically a large cardinal principle it feels like it ought to entail the existence of far larger objects than those that live in a free topos (since the objects thereof all live in $V_{\omega +\omega}$ roughly speaking). $\endgroup$
    – James E Hanson
    Commented Oct 15, 2023 at 2:00
  • $\begingroup$ Thanks for your comment! But I don’t quite understand in what context you are talking. “If the external theory is ZFC + Vopěnka's principle, then it is unlikely to hold in a free topos?” But this topos is not a Grothendieck topos and therefore does not relate to my question. I ask: “If the external theory is the free topos + some axioms + Vopěnka's principle, then what axioms should be so that Grothendieck’s topoi inherits Vopěnka's principle.” $\endgroup$
    – Arshak Aivazian
    Commented Oct 15, 2023 at 2:44
  • $\begingroup$ Sorry, I misread your comment earlier. My point is that Vopěnka's principle is such a large scale set-theoretic phenomenon that it doesn't really feel like it will 'fit' in a topos whose only objects are those generated freely by an NNO. For starters it's a statement about proper classes, which you can't even really natively talk about in toposes. $\endgroup$
    – James E Hanson
    Commented Oct 15, 2023 at 4:15

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