Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
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.
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.
Surely a non-finite category with an initial object is L-finite?
Oh, I see, sure. I’ll add this as a remark now.
But do we also have an interesting example?
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.
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?
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.
1 to 9 of 9