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I have created exhaustive category — not just the page, but the terminology. No one at MO seemed to know a name for this exactness property, so I made one up. The adjective “exhaustive” seems harmonious with “extensive” and “adhesive”, and expresses the idea that the subobjects in a transfinite union “exhaust” the colimit. But I would welcome other opinions and suggestions.
I have added an alternative characterization of exhaustive categories, analogous to similar ones for adhesive and extensive categories. There should also be a third characterization involving embeddings into an infinitary pretopos preserving transfinite unions.
That’s nicer! Maybe it’s just me, but that second characterization flows more smoothly: a category is exhaustive if transfinite composites of monos exist, are monos and are preserved by pullback. I would make that the default characterization.
I had that reaction when I first encountered the versions for extensive and adhesive categories: coproducts are disjoint and stable, or pushouts of monos are monos, stable, and are pullbacks. In fact when I first made the page for adhesive categories I left out the “cube lemma” characterization entirely. However, recently these sorts of definitions have been growing on me, not just because they are more uniform across different exactness properties, but because they directly express stack conditions, and because they are surprisingly useful. Cf. HTT 6.1.3.9(4).
However, for uniformity with extensive category and adhesive category I agree that both definitions should at least be presented on an equal footing; I’ve so rearranged the page.
Does the characterization of HTT 6.1.3.9(4) exist anywhere on the nLab?
Does the characterization of HTT 6.1.3.9(4) exist anywhere on the nLab?
No, I think that is not on the $n$Lab. Most of the surrounding statements are neither, I think.
More properties added to exhaustive category: transfinite unions preserve finite limits. Is the fact that the diagonal functor of a filtered category is final anywhere on the nLab?
Is the fact that the diagonal functor of a filtered category is final anywhere on the nLab?
Not quite explicitly yet, but at sifted (infinity,1)-category it has the general statement.
I am adding the remark about filtered categories now…
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