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Oct 23, 2016 at 14:43 comment added Arrow Sorry for bugging you again. If you have time, could you take a look at this question?
Oct 23, 2016 at 14:25 comment added Arrow I do mean finite conjunctions. I think I'll start calling these things sub-posets with meets. Thanks again for the great answer.
Oct 23, 2016 at 14:22 comment added Simon Henry @Arrow : if by "inf lattice" you mean with finite conjunctions the yes. For infinite conjunctions then No, but the converse is true.
Oct 20, 2016 at 18:54 comment added Arrow Thank you for the very complete and interesting answer! Forgive me but much of it is beyond me for now (though I definitely intend to study this material and the entirety of your answer soon!); for now I would like to make sure I understand correctly - with the identification you mention in your first point, it is true that a subobject of $\Omega$ which is object-wise a sub-inf-lattice is internally a sub-inf-lattice basically because the internal logic of a topos does not see conjunctions locally?
Oct 20, 2016 at 14:14 vote accept Arrow
Oct 19, 2016 at 11:30 history answered Simon Henry CC BY-SA 3.0