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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 25th 2021
• (edited Aug 25th 2021)

added pointer (below a new Proposition-environment here) to where Paré states/proves the characterizations of L-finite limits (namely on his p. 740, specifically in his Prop. 7 – this also makes clear which typo the footnote is about).

Similarly I added pointer (below a new Remark-environment here) to where Paré discusses the relation to K-finitness (namely the next page), together with attribution to Richard Wood.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2021
• (edited Aug 25th 2021)

I suspect that ’$L$-finite’ is the sort of thing that comes into its own in a constructive setting. Just as K-finite=finite=D-finite in classical mathematics (if I’m not mistaken), but these are different in the internal logic of a topos, one might need a special notion of finiteness of diagrams (added: invariant under suitable equivalence) that captures what is meant when doing internal category theory in a general topos.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 25th 2021

Incidentally, what’s an example of an L-finite category/limit that’s not finite?

I don’t see that Paré’s article gives any such example. But we need to give one for the claim at finite limit that finite limits are not the saturation class of pullbacks+terminal object.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2021

Surely a non-finite category with an initial object is L-finite?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 25th 2021

Oh, I see, sure. I’ll add this as a remark now.

But do we also have an interesting example?

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2021

A finite coproduct of such is more non-trivial. Or take a finite category with some objects with no non-identity arrows coming out of them. Then attach a copy of the poset $\omega$ to each one.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 25th 2021

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeAug 25th 2021

K-finite sets are defined on this page in Remark 1.2, while at the same time the link K-finite set redirects to finite set where ’K-finiteness’ is given as

admits a surjection from some finite set [n]; that is, it is a quotient set of a finite set.

Is that obvious that these definitions are equivalent?

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeAug 25th 2021

This is also asserted in the section “Finiteness without infinity” on the page finite set. I don’t remember the proof offhand, but it’s probably in the Elephant.

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