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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 4th 2014
   • (edited Sep 4th 2014)
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   Author: DavidRoberts
   Format: MarkdownItexGiven a [[locally connected topos]] $\Gamma\colon E \leftrightarrows Set : \Delta$, we have the extra left adjoint $\pi_0 \dashv \Delta$. $\pi_0$ preserves epis, as any left adjoint does. If $E$ is boolean I'm sure that $\pi_0$ also reflects epis, at least assuming classical logic in the background. And I can't think of a non-boolean example where $\pi_0$ doesn't reflect epis (just my own ignorance, rather than any compelling mathematical reason). Alternatively, I'm suspicious of $\pi_0$ reflecting epis in the case of boolean $E$ without classical logic. To be concrete, say I've got an epi $T \to V$ in boolean locally connected $E$ where $V$ is connected (throw in enough colimits if you like). $T$ has a component (as $\pi_0(T)$ is inhabited assuming $Set$ is constructively well-pointed), say $T_0 \subset T$. Then $f\colon T_0 \to V$ induces an iso on applying $\pi_0$, but there's no immediately compelling reason I can see for it to be epi. Given that $E$ is boolean, I can see that $f$ is _not not_ epi, since if it were not epi, $V$ would not be connected since $V$ would split as the image of $T_0$ and its complement. Presumably the topos of directed graphs would be good to consider. Hmmm...

   Given a locally connected topos , we have the extra left adjoint . preserves epis, as any left adjoint does. If is boolean I’m sure that also reflects epis, at least assuming classical logic in the background. And I can’t think of a non-boolean example where doesn’t reflect epis (just my own ignorance, rather than any compelling mathematical reason).

   Alternatively, I’m suspicious of reflecting epis in the case of boolean without classical logic.

   To be concrete, say I’ve got an epi in boolean locally connected where is connected (throw in enough colimits if you like). has a component (as is inhabited assuming is constructively well-pointed), say . Then induces an iso on applying , but there’s no immediately compelling reason I can see for it to be epi. Given that is boolean, I can see that is not not epi, since if it were not epi, would not be connected since would split as the image of and its complement.

   Presumably the topos of directed graphs would be good to consider. Hmmm…

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 4th 2014
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   Author: DavidRoberts
   Format: MarkdownItexIn the last comment that should be _reflexive_ directed graphs (see [[cohesive topos#Graphs]], and indeed, $\pi_0$ (there called $\Pi_0$) does _not_ reflect epis. I've probably implicitly used the atomicity in the specific topos I have at hand :-S

   In the last comment that should be reflexive directed graphs (see cohesive topos#Graphs, and indeed, (there called ) does not reflect epis.

   I’ve probably implicitly used the atomicity in the specific topos I have at hand :-S

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeSep 4th 2014
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   Author: Zhen Lin
   Format: MarkdownItexThe topos of (reflexive directed) graphs is not boolean, though?

   The topos of (reflexive directed) graphs is not boolean, though?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 4th 2014
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   Author: DavidRoberts
   Format: MarkdownItexYeah, but that addresses my first question as to a counterexample I was seeking.

   Yeah, but that addresses my first question as to a counterexample I was seeking.

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeSep 4th 2014
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   Author: Zhen Lin
   Format: MarkdownItexReturning the penultimate paragraph of #1, it seems you have assumed that $\pi_0$ reflects initial objects. That is a weak form of Nullstellensatz, and I'm not sure it holds non-classically. Here's the classical proof: take a subcanonical site for the topos consisting of connected objects, which we may do because it is locally connected; then constant presheaves are sheaves, so $\pi_0$ can be computed as the colimit of the underlying presheaf and we are done. Of course, if you assume $\pi_0$ reflects epimorphisms, then it also reflects initial objects: indeed, in a topos, any epimorphism whose codomain is an initial object must be an isomorphism.

   Returning the penultimate paragraph of #1, it seems you have assumed that reflects initial objects. That is a weak form of Nullstellensatz, and I’m not sure it holds non-classically.

   Here’s the classical proof: take a subcanonical site for the topos consisting of connected objects, which we may do because it is locally connected; then constant presheaves are sheaves, so can be computed as the colimit of the underlying presheaf and we are done.

   Of course, if you assume reflects epimorphisms, then it also reflects initial objects: indeed, in a topos, any epimorphism whose codomain is an initial object must be an isomorphism.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 4th 2014
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   Author: DavidRoberts
   Format: MarkdownItexHmm, thanks. I've just taken $\pi_0$ reflecting epis as a hypothesis now. It will hold in the final construction I do, so even if it is restrictive it makes the hypotheses cleaner.

   Hmm, thanks. I’ve just taken reflecting epis as a hypothesis now. It will hold in the final construction I do, so even if it is restrictive it makes the hypotheses cleaner.