Added that example.

]]>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 $0$.

]]>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.

]]>examples of disjoint 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 says on that p. 34:

]]>We call a coherent category

positiveif it has disjoint finite coproducts

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…

]]>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.

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

]]>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.