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 definitions 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 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.
   • CommentAuthorUrs
   • CommentTimeSep 11th 2012
   • (edited Sep 11th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have touched the formatting at _[[direct sum]]_ and then expanded a little: 1. Added a paragraph to the Idea-section such that something familiar is mentioned right at the beginning; 1. Expanded on the example of direct sums in $Ab$ by drawing the cocone diagrams and explicitly mentioning the universal property. 1. Mentioned the relation to formal linear combinations. 1. Mentioned the examples of direct sums of modules.

   I have touched the formatting at direct sum and then expanded a little:

   1. Added a paragraph to the Idea-section such that something familiar is mentioned right at the beginning;

   2. Expanded on the example of direct sums in $Ab$ by drawing the cocone diagrams and explicitly mentioning the universal property.

   3. Mentioned the relation to formal linear combinations.

   4. Mentioned the examples of direct sums of modules.

2. • CommentRowNumber2.
   • CommentAuthorR.Arthur
   • CommentTimeNov 25th 2021
   • PermaLink
   Author: R.Arthur
   Format: MarkdownItexChanged the array to a proper tikz-cd <a href="https://ncatlab.org/nlab/revision/diff/direct+sum/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/direct+sum/30">v30</a>, <a href="https://ncatlab.org/nlab/show/direct+sum">current</a>

   Changed the array to a proper tikz-cd

   diff, v30, current

3. • CommentRowNumber3.
   • CommentAuthornLab edit announcer
   • CommentTimeAug 15th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexfix an mistake (capital or small) Linuxmetel <a href="https://ncatlab.org/nlab/revision/diff/direct+sum/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/direct+sum/31">v31</a>, <a href="https://ncatlab.org/nlab/show/direct+sum">current</a>

   fix an mistake (capital or small)

   Linuxmetel

   diff, v31, current

4. • CommentRowNumber4.
   • CommentAuthornLab edit announcer
   • CommentTimeAug 15th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexfix an mistake ("of of" to "of") Linuxmetel <a href="https://ncatlab.org/nlab/revision/diff/direct+sum/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/direct+sum/31">v31</a>, <a href="https://ncatlab.org/nlab/show/direct+sum">current</a>

   fix an mistake (“of of” to “of”)

   Linuxmetel

   diff, v31, current

5. • CommentRowNumber5.
   • CommentAuthorjulia9367
   • CommentTimeJul 5th 2023
   • (edited Jul 5th 2023)
   • PermaLink
   Author: julia9367
   Format: MarkdownItexHere are some things I think are true relating these notions to the finite biproducts and infinite coproducts in additive categories. 1. Let $\mathcal{A}$ be an additive category. The weak direct product agrees with the coproduct. For the finite case it's just the ordinary products, which are biproducts and thus also the coproducts. In the infinite case it becomes the directed colimit of the finite subbiproducts which is equal the full infinite coproduct. 2. Let $\mathcal{A}$ further be regular. Then the finite direct sums agree with the finite biproducts. This is because in this case the canonical morphism from the coproduct to the product is an iso. We only need the regular, so that the image and thus the direct sum is even defined. Any other reasonable image notion would work too. 3. Let further directed colimits in $\mathcal{A}$ be exact, then the infinite direct sum agrees with the coproduct. Exact implies that taking colimits commutes with taking the image, this is really all we care about here. We consider for any finite subbiproduct of the coproduct its inclusion as a bifactor into the full infinite product. The bifactor should be isomorphic to its image under this inclusion. Now since the directed colimit commutes with the image, the full coproduct is also isomorphic to its image in the full product.

   Here are some things I think are true relating these notions to the finite biproducts and infinite coproducts in additive categories.

   1. Let $\mathcal{A}$ be an additive category. The weak direct product agrees with the coproduct. For the finite case it’s just the ordinary products, which are biproducts and thus also the coproducts. In the infinite case it becomes the directed colimit of the finite subbiproducts which is equal the full infinite coproduct.
   2. Let $\mathcal{A}$ further be regular. Then the finite direct sums agree with the finite biproducts. This is because in this case the canonical morphism from the coproduct to the product is an iso. We only need the regular, so that the image and thus the direct sum is even defined. Any other reasonable image notion would work too.
   3. Let further directed colimits in $\mathcal{A}$ be exact, then the infinite direct sum agrees with the coproduct. Exact implies that taking colimits commutes with taking the image, this is really all we care about here. We consider for any finite subbiproduct of the coproduct its inclusion as a bifactor into the full infinite product. The bifactor should be isomorphic to its image under this inclusion. Now since the directed colimit commutes with the image, the full coproduct is also isomorphic to its image in the full product.