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comment Why do filtered colimits commute with finite limits?
Thanks, Dylan. I agree that J being filtered will have to be used. But what properties of Set should be used?
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answered Topos associated to a category
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comment Finitary mono preserving functors on varieties that fail to preserve intersections.
An empty intersection of subobjects is the whole object, so is always preserved. The example I gave involved failure to preserve a binary intersection. Perhaps it would be helpful if you explained what you were really after.
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comment Finitary mono preserving functors on varieties that fail to preserve intersections.
Rob, I'm not sure quite what this means. I take it that the modified version sends everything to 1? In that case the empty set will no longer possess an algebra structure.
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comment Finitary mono preserving functors on varieties that fail to preserve intersections.
The endofunctor T of Set sending a set X to a singleton 1 if it is non-empty, and to the empty set 0 if it is empty preserves monomorphisms, since it sends every map to a monomorphism. But it does not preserve the intersection of the two maps from a singleton 1 to a two-element set 2.
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comment Classification of Quasi-topoi
it extends to Grothendieck toposes; not sure about more generally than that.