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  1. The observation that the preorder sub(x)sub(x) of subobjects of some object xx in an elementary topos happens to be a Heyting algebra is the starting point of categorical semantics.

    The dual situation where the preorder sup(x)sup(x) consisting of epimorphisms out of the object is considered (which should give an Esakia space in an elementary topos, I guess), I have never seen mentioned in context of categorical logic. Is this really dual to the sub(x)sub(x)-case in that we obtain a supobject classifier (respectively: an object co-classifier if we are in (,1)(\infty,1)-categories) satisfying dual statements etc. ?

    What are the reasons for this not being mentioned? Maybe I just have to look under a different keyword to find some material.

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeJul 14th 2012
    • (edited Jul 14th 2012)

    It is not possible to have a “supobject” classifier \mho in an elementary topos: the universal epimorphism would be an arrow 0\mho \to 0, but the initial object of a cartesian closed category is strict, so 0\mho \cong 0.

  2. Is this idea becoming more interesting if we consider it in an ”elementary cotopos” (finite colimits, cocartesian co-closed, and ”sup object classifier”)?

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeJul 14th 2012

    Epimorphisms out of an object are quotient objects, not superobjects. A superobject (following the term ‘superset’) would be a monomorphism out of the object.

    I wouldn’t expect quotients objects of an object in a topos to form an Esakia space; Esakia duality says that Heyting algebras and Esakia spaces are essentially the same thing, except that the morphisms go in opposite directions. The quotient objects (of a fixed object in a topos) will still form a lattice, albeit not a Heyting algebra but some other kind of lattice.

    If you dualise everything and look at quotient objects in a ‘cotopos’, then this is really just the same as looking at subobjects in the opposite category (which is a topos), so we’re back to Heyting algebras.