Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeMar 19th 2019
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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