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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.”

• 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.

• 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 $Pfn$ of sets and partial functions as an example that is not extensive. I guess the category $Vect$ of vector spaces would be another example of this kind.

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeDec 16th 2020

It’s clear that although coproducts are disjoint in $Vect$, they are not stable under pullback. Consider pulling back the coproduct $\mathbb{R} \oplus \mathbb{R}$ with its coproduct inclusions $i_1, i_2: \mathbb{R} \hookrightarrow \mathbb{R} \oplus \mathbb{R}$ along the diagonal inclusion $\Delta: \mathbb{R} \hookrightarrow \mathbb{R} \oplus \mathbb{R}$. This pulling back is given by intersecting subspaces. But $1_{\mathbb{R} \oplus \mathbb{R}} \cap \Delta$, with codomain $\mathbb{R}$, cannot be the coproduct of $i_1 \cap \Delta$ and $i_2 \cap \Delta$, which are both $0$.

• CommentRowNumber11.
• CommentAuthorThomas Holder
• CommentTimeDec 21st 2020