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I have incorporated Jonas’ comment into the text at pretopos, changing the definition to “a category that is both exact and extensive”, as this is sufficient to imply that it is both regular and coherent.
Just a terminological comment for the benefit of future readers:
in the Elephant, (pretopos)$=$(coherent,effective,positive). In pretopos, (pretopos)$=$(exact,extensive)$\Leftrightarrow$(ditto, with “coherent” added).
The discrepancy between “effective” and “exact” is terminological: “effective” is just Johnstone’s synonym for “exact” (cf. Elephant p.24)
This leaves the question whether (coherent,exact,positive)$\Leftrightarrow$(coherent,exact,extensive); this is true for the maximally-strong, terminological reason that (positive)$=$definitionally$=$(extensive). (cf e.g. here for more)
[ I shied away from making a footnote on this in pretopos, since to achieve a maximally-clear comment like “This is sometimes synonymously stated as “effective and positive” one has to know whether it is true in general that (effective,positive)$\Rightarrow$(coherent), as is true when “effective,positive” is replaced with “exact,extensive”; the latter I did not stop to to try to ascertain, in particular since “positive” seems to usually only be discussed for categories which are assumed to be coherent in the first place ]
It’s a bug we know about, see here. Hopefully Richard Williamson will be able to fix it.
Thank you very much for reporting this, Kevin. I have now fixed $\Pi W$-pretopos and hopefully $\Pi$-pretopos (if there were any occurrences of ’$\infty$-pretopos’ which genuinely referred to $\infty$-categories, these will now wrongly be $\Pi$-pretoposes as well, but I could not find any such occurrences). Please report any other occurrences of question marks and seemingly erroneous $\infty$-symbols that you come across!
Thanks for doing these fixes Richard!
The term “$\infty$-pretopos” is also used in the literature (at least, in the Elephant) for what we on the nLab call an infinitary (1-)pretopos. Apparently we didn’t mention that anywhere though! I’ve now added a note to pretopos.
Added conceptual completeness to Related concepts.
At infinitary coherent category we have remarked on the $\kappa$-ary version. Maybe we should use our terminology, and move the remark about the Elephant’s terminology to that page?
Sure, that’s fine. I feel that $\sigma$-topos is still a better name, though, than $\aleph_1$-ary pretopos (or $\aleph_1$-ary regular pretopos), even though it’s not systematic. It’s a special enough case that a special name (with echoes of $\sigma$-algebras) for it is useful.
We might have to agree to disagree on that one. (-: Are there interesting examples of $\sigma$-pretoposes that are not infinitary pretoposes?
The category of countable sets (in the presence of AC)? Probably something like countably-presented sets without AC
Is that category interesting for a reason other than being a $\sigma$-pretopos?
Not sure (-: I was wondering if there was some kind of computability-related category, or one that a particular type of quasi-finitist might be interested in. Or an arithmetic universe?
I note that Simpson and Streicher, Constructive toposes with countable sums as models of constructive set theory, https://doi.org/10.1016/j.apal.2012.01.013 prove a nontrivial result about $\sigma$-$\Pi$-pretoposes, relating them closely to a variant of CZF.
In particular they give the example of the ex/reg completion of the category of modest sets over the second Kleene algebra $K_2$ as having countable but not small sums (and in fact, is essentially small).
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