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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 11th 2012
• (edited Sep 11th 2012)

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.

• CommentRowNumber2.
• CommentAuthorR.Arthur
• CommentTimeNov 25th 2021

Changed the array to a proper tikz-cd

1. fix an mistake (capital or small)

Linuxmetel

2. fix an mistake (“of of” to “of”)

Linuxmetel

• CommentRowNumber5.
• CommentAuthorjulia9367
• CommentTimeJul 5th 2023
• (edited Jul 5th 2023)

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.