improve a link
Linuxmetel
]]>Clarified the opening sentence to state
A category is extensive if it has coproducts that interact well with a certain class of pullbacks.
Since not all pullbacks are assumed to exist.
]]>Added fact: extensive categories with finite products are distributive. This is already mentioned in distributive category but I wanted a link going from this page to that one.
]]>I have re-organized the section outline for readability:
Merged the remarks in a section titled “Remarks” with other scattered remarks into a list of Remark-environments below the definition.
Moved all further definitions (super-extensive, pre-extensive) as subsections into the Definition section below that.
Added a Properties-section and added there the characterizan of connected objects in extensive categories (copied over from the entry connected object, where I just edited that proof as announced here).
]]>added (here) pointers to where in the literature the various equivalent conditions are discussed (also added DOI links for these).
added mentioning of the example of free coproduct completion (here)
]]>I wonder if it would be too presumptuous to appropriate the name “disjunctive category” for what the page currently calls “pre-lextensive categories”. It seems that some people have used “disjunctive category” to mean “extensive category”, but it seems a waste of good terminology to have two names for the same thing, and since pre-lextensivity suffices to interpret disjunctive logic it seems a natural back-formation.
]]>Urs, please, I trust you can make the reasonable assumption I know the mathematics. I’m just editing the phrasing to make it read more smoothly. My reason above was not phrased the best, maybe.
A smooth map has pullbacks along maps transversal to it. But “ has pullbacks along transversal maps” makes it sound to my ears like “transversal maps” is a class of morphisms of along which an arbitrary map can be pulled back, like the class of submersions.
That’s all. :-)
]]>It’s not important, but it was okay the way it was:
Pullback is along a map. Fiber product is of a cospan. Pullback along a transversal map, transversal to the map being pulled back, is what the text naturally needed here.
Just saying. But let’s please not discuss this further. :-)
]]>Edited it to
The category Diff of smooth manifolds is infinitary extensive, though it does not have all pullbacks (only those involving a cospan of transversal maps).
to make the case match. The link to transversal maps (plural) before was referring to a single map along which one is pulling back.
]]>Thomas has alerted me that this page had claimed that smooth manifolds “lack all pullbacks”. I have changed it to: “… does not have all pullbacks (only those along transversal maps)”.
]]>Corrected a mistake in the definition of an infinitary extensive category.
]]>Re #26: I overlooked this at the time, but have merged the two threads now.
]]>Removed the discussion.
]]>Here is the discussion:
While creating this page, we had the following discussion regarding “finitely” versus “infinitary.”
+–{.query} Can we say ’small-extensive’? Or even redefine ’extensive’ to have this meaning, using ’finitely extensive’ for the first version? —Toby
I think “extensive” is pretty well established for the finite version, and I would be reluctant to try to change it. I wouldn’t object too much to “small-extensive” for the infinitary version in principle, but -positive is used in the Elephant and possibly elsewhere. I think the topos theorists think by analogy with -pretopos, which I don’t think we have much hope of changing, despite the unfortunate clash with “-topos.” But you can use “finitary disjunctive” and “disjunctive” in the lex case, which most examples are. -Mike
Mike: Okay, I just ran across one paper that uses “(infinitary) extensive” for the infinitary version the first time it was introduced, and then dropped the parenthetical for the rest of the paper. I also recall seeing “extensive fibration” used for a fibration having disjoint and stable indexed coproducts, which is certainly a (potentially) infinitary notion. So perhaps there is no real consensus on whether “extensive” definitely implies the finite version or the infinitary one.
Toby: It would be nice to not overload the prefix ’-’ so much. It's like ’continuous’; default to small.
Mike: I agree that it would be nice to avoid -. What if we do what we did for omega-category? That is, if you want to be unambiguous, say either “finitary extensive” or “infinitary extensive,” and in any particular context you are allowed to define “extensive” at the beginning to be one of the two and use it without prefix in what follows.
Toby: Sure. Of course, the general concept is -extensive, where is any cardinal (which we may assume to be regular).
=–
]]>Added some links and references. I would also suggest to move the terminological discussion at the end into the forum thread in line with the current approach to query boxes.
]]>For the sake of minimizing confusion, add a parenthetical mentioning the finite case implies the infinite case.
]]>For the sake of minimizing confusion, add a parenthetical mentioning the finite case implies the infinite case.
]]>I’ve made the analogous change to the infinitary version, on the assumption it’s required there too.
]]>This thread needs to be merged with the other discussion thread.
]]>I agree – corrected point 3 having checked Carboni-Lack-Walters
]]>I’m looking at the equivalent characterizations of extensive. Am I missing something or is condition #3 not actually equivalent to the others? In particular, given condition #3, it’s not clear how you would show that a span has a pullback.
]]>The étale and fppf topologies satisfy WISC, at least over a locally noetherian base scheme. This is basically because they are defined by adjoining to the Zariski topology certain (finite) families of the form where is finitely presented over . The fpqc topology has no such finiteness requirement.
]]>Conversely, the fpqc site doesn’t. The question is, where is the transition?
]]>Yes, that looks right. Every Zariski-covering sieve contains one that is generated by a finite set of principal affine open subschemes.
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