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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJul 14th 2015
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   Author: Mike Shulman
   Format: MarkdownItexI feel like I should know the answer to this. Let $A\to B$ be relatively codiscrete, i.e. the naturality square $$\array{ A & \to & \sharp A\\ \downarrow && \downarrow\\ B & \to & \sharp B}$$ is a pullback. Now factor it as an effective epi followed by a mono, $A\to M\to B$. Is the mono $M\to B$ still relatively codiscrete?

   I feel like I should know the answer to this. Let $A\to B$ be relatively codiscrete, i.e. the naturality square

   $$\array{ A & \to & \sharp A\\ \downarrow && \downarrow\\ B & \to & \sharp B}$$

   is a pullback. Now factor it as an effective epi followed by a mono, $A\to M\to B$. Is the mono $M\to B$ still relatively codiscrete?

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 15th 2015
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   Author: DavidRoberts
   Format: MarkdownItexIn what sort of category (I only ask because you say _effective_ epi)

   In what sort of category (I only ask because you say effective epi)

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 15th 2015
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   Author: Mike Shulman
   Format: MarkdownItexWell, I meant it in a cohesive (∞,1)-topos. But an answer in a cohesive 1-topos (where all epis are effective) would already be interesting.

   Well, I meant it in a cohesive (∞,1)-topos. But an answer in a cohesive 1-topos (where all epis are effective) would already be interesting.

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeJul 15th 2015
   • (edited Jul 15th 2015)
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   Author: Zhen Lin
   Format: MarkdownItexIf we knew that the reflector preserves the relevant factorisations, then it would just be the pullback pasting lemma + the fact that we have a stable orthogonal factorisation system. Curiously, the axiom of choice implies this is the case for simplicial sets (as a cohesive 1-topos).

   If we knew that the reflector preserves the relevant factorisations, then it would just be the pullback pasting lemma + the fact that we have a stable orthogonal factorisation system. Curiously, the axiom of choice implies this is the case for simplicial sets (as a cohesive 1-topos).

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 15th 2015
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   Author: Mike Shulman
   Format: MarkdownItexRight (I don't even see where the pullback pasting lemma is needed). Since the reflector is left exact, it always preserves monos, so the question is, does it preserve effective epis?

   Right (I don’t even see where the pullback pasting lemma is needed). Since the reflector is left exact, it always preserves monos, so the question is, does it preserve effective epis?

6. • CommentRowNumber6.
   • CommentAuthorZhen Lin
   • CommentTimeJul 15th 2015
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   Author: Zhen Lin
   Format: MarkdownItexI thought we needed the pullback pasting lemma in order to relate the pullbacks of the reflected factorisation to the original factorisation? Anyway. I've just realised that the axiom of choice is not needed for simplicial sets: the reflector has an explicit construction in terms of finite products, and it preserves (effective) epimorphisms because finite products of (effective) epimorphisms are (effective) epimorphisms in a topos. So the same argument would work for simplicial objects in any topos. Another easy observation: if the inclusion of the subcategory of codiscrete objects preserves (effective) epimorphisms, then the reflector also preserves (effective) epimorphisms. Is the converse true? The usual argument about the creation of colimits in categories of algebras for a monad doesn't quite apply here, I think.

   I thought we needed the pullback pasting lemma in order to relate the pullbacks of the reflected factorisation to the original factorisation?

   Anyway. I’ve just realised that the axiom of choice is not needed for simplicial sets: the reflector has an explicit construction in terms of finite products, and it preserves (effective) epimorphisms because finite products of (effective) epimorphisms are (effective) epimorphisms in a topos. So the same argument would work for simplicial objects in any topos.

   Another easy observation: if the inclusion of the subcategory of codiscrete objects preserves (effective) epimorphisms, then the reflector also preserves (effective) epimorphisms. Is the converse true? The usual argument about the creation of colimits in categories of algebras for a monad doesn’t quite apply here, I think.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJul 16th 2015
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   Author: Mike Shulman
   Format: MarkdownItexSo, the outer rectangle is a pullback by assumption. If the reflector preserves both classes, then the reflected factorization is again a factorization. Therefore, since the factorization system is stable, the pullback of the reflected factorization is again a factorization of the original map. Hence, by uniqueness of factorizations, it is the original factorization. Am I missing something? I also think that since colimits in a reflective subcategory are in general obtained by reflecting colimits in the ambient category, if the reflector preserves a particular colimit, then the subcategory is closed under that colimit, i.e. its inclusion preserves them.

   So, the outer rectangle is a pullback by assumption. If the reflector preserves both classes, then the reflected factorization is again a factorization. Therefore, since the factorization system is stable, the pullback of the reflected factorization is again a factorization of the original map. Hence, by uniqueness of factorizations, it is the original factorization. Am I missing something?

   I also think that since colimits in a reflective subcategory are in general obtained by reflecting colimits in the ambient category, if the reflector preserves a particular colimit, then the subcategory is closed under that colimit, i.e. its inclusion preserves them.

8. • CommentRowNumber8.
   • CommentAuthorZhen Lin
   • CommentTimeJul 16th 2015
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   Author: Zhen Lin
   Format: MarkdownItexYour argument is the same as mine, but I think you are using the pullback pasting lemma in the sentence beginning with "therefore". There is no issue with colimits in reflective subcategories – but the issue when it comes to (effective) epimorphisms is, I think, a bit more subtle. For instance, assuming the axiom of choice, any endofunctor whatsoever on $\mathbf{Set}$ must preserve epimorphisms, but it is not obvious to me whether reflective subcategories of $\mathbf{Set}$ must have the same epimorphisms. (If we knew that the reflector preserves pushouts, then the reflective subcategory would have the same epimorphisms, but that appears to be a strictly stronger assumption.)

   Your argument is the same as mine, but I think you are using the pullback pasting lemma in the sentence beginning with “therefore”.

   There is no issue with colimits in reflective subcategories – but the issue when it comes to (effective) epimorphisms is, I think, a bit more subtle. For instance, assuming the axiom of choice, any endofunctor whatsoever on $\mathbf{Set}$ must preserve epimorphisms, but it is not obvious to me whether reflective subcategories of $\mathbf{Set}$ must have the same epimorphisms. (If we knew that the reflector preserves pushouts, then the reflective subcategory would have the same epimorphisms, but that appears to be a strictly stronger assumption.)

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJul 17th 2015
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   Author: Mike Shulman
   Format: MarkdownItex> I think you are using the pullback pasting lemma in the sentence beginning with “therefore”. Oh... I suppose if you use the definition of [[stable factorization system]] that just says the left class is closed under pullback. I was thinking of it as by definition meaning that the whole factorization is of a morphism is closed under pullback. > If we knew that the reflector preserves pushouts, then the reflective subcategory would have the same epimorphisms, but that appears to be a strictly stronger assumption. Preserving all pushouts is, yes, but I believe the pushouts that exhibit a split epi as an epi are [[absolute colimit|absolute pushouts]], so they should also be preserved by any functor. For effective epis, we may need to require the reflector to be left exact, so that it also preserves the construction of the kernel (but $\sharp$ in my original question is left exact).

   I think you are using the pullback pasting lemma in the sentence beginning with “therefore”.

   Oh… I suppose if you use the definition of stable factorization system that just says the left class is closed under pullback. I was thinking of it as by definition meaning that the whole factorization is of a morphism is closed under pullback.

   If we knew that the reflector preserves pushouts, then the reflective subcategory would have the same epimorphisms, but that appears to be a strictly stronger assumption.

   Preserving all pushouts is, yes, but I believe the pushouts that exhibit a split epi as an epi are absolute pushouts, so they should also be preserved by any functor.

   For effective epis, we may need to require the reflector to be left exact, so that it also preserves the construction of the kernel (but $\sharp$ in my original question is left exact).

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 31st 2015
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    Author: Mike Shulman
    Format: MarkdownItexJust a remark: if the base topos satisfies AC, and $B$ is a set, then the answer is yes. For $\sharp = \nabla \Gamma$, and $\Gamma$ preserves all epis since it is a left adjoint, and sets since it is a right adjoint; thus it preserves (effective) epis with set codomain, hence (by AC) takes them to split epis, and split epis are preserved by any functor (including $\nabla$).

    Just a remark: if the base topos satisfies AC, and $B$ is a set, then the answer is yes. For $\sharp = \nabla \Gamma$, and $\Gamma$ preserves all epis since it is a left adjoint, and sets since it is a right adjoint; thus it preserves (effective) epis with set codomain, hence (by AC) takes them to split epis, and split epis are preserved by any functor (including $\nabla$).

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJan 1st 2016
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    Author: Mike Shulman
    Format: MarkdownItexAnd if the base topos satisfies AC and the cohesive $(\infty,1)$-topos is sheaves on some ∞-cohesive 1-site, then the answer is again yes: since $\nabla$ is given by mapping out of the underlying ∞-groupoids of the objects of the site, so if these ∞-groupoids are sets then (by AC) mapping out of them preserves (effective) epis. This is good enough for what I want right now (namely, Euclidean-topological ∞-groupoids), but it would still be nice to answer the question in general.

    And if the base topos satisfies AC and the cohesive $(\infty,1)$-topos is sheaves on some ∞-cohesive 1-site, then the answer is again yes: since $\nabla$ is given by mapping out of the underlying ∞-groupoids of the objects of the site, so if these ∞-groupoids are sets then (by AC) mapping out of them preserves (effective) epis. This is good enough for what I want right now (namely, Euclidean-topological ∞-groupoids), but it would still be nice to answer the question in general.