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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeJul 14th 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexThe observation that the preorder $sub(x)$ of subobjects of some object $x$ in an elementary topos happens to be a Heyting algebra is the starting point of [[categorical semantics]]. The dual situation where the preorder $sup(x)$ consisting of epimorphisms out of the object is considered (which should give an [Esakia space](http://en.wikipedia.org/wiki/Esakia_duality) in an elementary topos, I guess), I have never seen mentioned in context of categorical logic. Is this really dual to the $sub(x)$-case in that we obtain a supobject classifier (respectively: an object co-classifier if we are in $(\infty,1)$-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.

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

   The dual situation where the preorder $sup(x)$ 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 $sub(x)$-case in that we obtain a supobject classifier (respectively: an object co-classifier if we are in $(\infty,1)$-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.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeJul 14th 2012
   • (edited Jul 14th 2012)
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   Author: Zhen Lin
   Format: MarkdownItexIt is not possible to have a "supobject" classifier $\mho$ in an elementary topos: the universal epimorphism would be an arrow $\mho \to 0$, but the initial object of a cartesian closed category is strict, so $\mho \cong 0$.

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

3. • CommentRowNumber3.
   • CommentAuthorStephan A Spahn
   • CommentTimeJul 14th 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexIs this idea becoming more interesting if we consider it in an ''elementary cotopos'' (finite colimits, cocartesian co-closed, and ''sup object classifier'')?

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

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJul 14th 2012
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   Author: TobyBartels
   Format: MarkdownItexEpimorphisms 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.

   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.