Added a constructive phrasing of disjointness for arbitrary coproducts.
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Linuxmetel
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Linuxmetel
]]>Added that example.
]]>It’s clear that although coproducts are disjoint in , they are not stable under pullback. Consider pulling back the coproduct with its coproduct inclusions along the diagonal inclusion . This pulling back is given by intersecting subspaces. But , with codomain , cannot be the coproduct of and , which are both .
]]>Added the category of sets and partial functions as an example that is not extensive. I guess the category 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 positive if 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.
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