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nLab > Latest Changes: disjoint coproduct

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2012
    • (edited Nov 8th 2012)

    I have expanded various sections at disjoint coproduct. In particular towards the end is now a mentioning of the fact that in a positive category morphisms into a disjoint coproduct are given by factoring disjoint summands of the domain through the canonical inclusions.

    Also,I made positive category and variants redirect to extensive category.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeNov 8th 2012

    Is the phrase “positive category” intended to include coherent-ness, or not?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2012

    Is the phrase “positive category” intended to include coherent-ness, or not?

    Johnstone in the Elephant on p. 34 says “positive” for “coherent + disjoint coproducts”, as you will know. You once wrote “Extensive categories are also called positive categories, especially if they are also coherent.”

    I am agnostic about it.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 8th 2012

    I thought he only said “positive coherent category”, never “positive category” without the adjective “coherent”. But I don’t have the Elephant in front of me right now…

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2012
    • (edited Nov 9th 2012)

    It says on that p. 34:

    We call a coherent category positive if it has disjoint finite coproducts

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeNov 9th 2012

    That could equally well be read in either way. I guess I read [past tense] it my way and you read it your way. (-: I guess I was assuming that he was using it by analogy with the adjective “effective” for regular categories. I’m pretty sure he does say “effective regular category”, not ever just “effective category”.

    • CommentRowNumber7.
    • CommentAuthorSam Staton
    • CommentTimeOct 5th 2020

    examples of disjoint coproducts.

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorSam Staton
    • CommentTimeOct 5th 2020
    I put some examples and non-examples of disjoint coproducts. But just now I can't think of a naturally occuring category that has all coproducts disjoint but which isn't extensive.
    • CommentRowNumber9.
    • CommentAuthorThomas Holder
    • CommentTimeDec 12th 2020

    Added the category PfnPfn of sets and partial functions as an example that is not extensive. I guess the category VectVect of vector spaces would be another example of this kind.

    diff, v12, current

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 17th 2020

    It’s clear that although coproducts are disjoint in VectVect, they are not stable under pullback. Consider pulling back the coproduct ℝ⊕ℝ\mathbb{R} \oplus \mathbb{R} with its coproduct inclusions i 1,i 2:ℝ↪ℝ⊕ℝi_1, i_2: \mathbb{R} \hookrightarrow \mathbb{R} \oplus \mathbb{R} along the diagonal inclusion Δ:ℝ↪ℝ⊕ℝ\Delta: \mathbb{R} \hookrightarrow \mathbb{R} \oplus \mathbb{R}. This pulling back is given by intersecting subspaces. But 1 ℝ⊕ℝ∩Δ1_{\mathbb{R} \oplus \mathbb{R}} \cap \Delta, with codomain ℝ\mathbb{R}, cannot be the coproduct of i 1∩Δi_1 \cap \Delta and i 2∩Δi_2 \cap \Delta, which are both 00.

    • CommentRowNumber11.
    • CommentAuthorThomas Holder
    • CommentTimeDec 21st 2020

    Added that example.

    diff, v14, current

  12. fix a link

    Linuxmetel

    diff, v19, current

  13. improve a link

    Linuxmetel

    diff, v19, current

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJul 26th 2023

    Added a constructive phrasing of disjointness for arbitrary coproducts.

    diff, v20, current

1 to 14 of 14

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