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1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeMar 19th 2019
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   Author: Karol Szumiło
   Format: MarkdownItexConsider a morphism $A \to C$ that is a coproduct inclusion, i.e., there is another morphism $B \to C$ that together make $C$ into a coproduct of $A$ and $B$. I believe that it is not always the case that morphisms of this form are closed under retracts. Basically, one can construct a minimal counterexample category that has just this one coproduct with an extra morphism $A \to B$ and a few more necessary to make it a retract of $A \to C$. My question: what are some reasonably lightweight conditions on the ambient category for coproduct inclusions to be closed under retracts?

   Consider a morphism $A \to C$ that is a coproduct inclusion, i.e., there is another morphism $B \to C$ that together make $C$ into a coproduct of $A$ and $B$. I believe that it is not always the case that morphisms of this form are closed under retracts. Basically, one can construct a minimal counterexample category that has just this one coproduct with an extra morphism $A \to B$ and a few more necessary to make it a retract of $A \to C$.

   My question: what are some reasonably lightweight conditions on the ambient category for coproduct inclusions to be closed under retracts?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2019
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   Author: Urs
   Format: MarkdownItexSorry for flooding your question with trivia. This here to bump it back up.

   Sorry for flooding your question with trivia. This here to bump it back up.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMar 19th 2019
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   Author: Mike Shulman
   Format: MarkdownItexI would expect it to be true in any [[extensive category]].

   I would expect it to be true in any extensive category.

4. • CommentRowNumber4.
   • CommentAuthorKarol Szumiło
   • CommentTimeMar 20th 2019
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   Author: Karol Szumiło
   Format: MarkdownItexThanks, this works indeed. In an extensive category coproduct inclusions are closed under pullback. If we have a retract of a coproduct inclusion, then one of the squares that exhibit it can be shown to be pullback by hand. We just need to know that coproduct inclusions are monic which is Proposition 2.6 in the Carboni--Lack--Walters paper linked from nLab

   Thanks, this works indeed. In an extensive category coproduct inclusions are closed under pullback. If we have a retract of a coproduct inclusion, then one of the squares that exhibit it can be shown to be pullback by hand. We just need to know that coproduct inclusions are monic which is Proposition 2.6 in the Carboni–Lack–Walters paper linked from nLab