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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.
Is the phrase “positive category” intended to include coherent-ness, or not?
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.
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…
It says on that p. 34:
We call a coherent category positive if it has disjoint finite coproducts
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”.
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 $0$.
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