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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.
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