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Stream: learning: questions

Topic: Isbell envelope and bicompletion

Nathanael Arkor (Sep 12 2020 at 16:52):

The Isbell envelope of a small category $\mathbf C$ naturally contains both the free cocompletion and free completion of $\mathbf C$. Is the Isbell envelope the free bicompletion, or if not, is there any relationship between the two? It seems like a natural question, but I didn't spot any discussion in the original papers (though perhaps I was confused by the older terminology there).

fosco (Sep 13 2020 at 15:01):

(not an answer, but I was trying to see what is the embedding you're talking about)

This is actually nice: $(-)^\sharp : P \mapsto (P, \mathcal O P, \xi)$ where $\xi$ is the natural transformation image of the identity of ${\cal O}P$ under the natural isomorphism ${\sf Cat}({\cal C}^\text{op}\times {\cal C},{\sf Set})(P\times {\cal O}P, \hom)\cong{\sf Cat}({\cal C}^\text{op}\times {\cal C},{\sf Set})({\cal O}P,{\cal O}P)$ is a fully faithful embedding of ${\sf Cat}({\cal C}^\text{op},{\sf Set})$ into ${\cal E}nv({\cal C})$. (Dually for copresheaves, you use $\rm Spec$.)

Is there a slick way to prove that it's fully faithful without getting dirty?

John Baez (Sep 14 2020 at 15:30):

The concept of "free bicompletion" makes me nervous - does anyone discuss it? The reason it makes me nervous is that if we restrict to posets, there's a concept of complete lattice which really means "complete and cocomplete poset"... but there's no free complete lattice on a set of 3 or more elements.

Martti Karvonen (Sep 14 2020 at 18:25):

Joyal has a paper titled "Free bicompletions of categories" discussing bicompletions. The main result is that for any category $C$ there is a bicomplete category $\lambda C$ and an inclusion $i\colon C\to\lambda C$ that is universal in that any functor from $C$ to a bicomplete category factors uniquely (up to iso) via $i$ and a bicontinuous functor. However, the paper leaves most of the proofs out (e.g. one comment says that something "can be proved using standard categorical methods that we shall not discuss"), so you might have to reverse-engineer what Joyal had in mind. I'm not sure how the Isbell envelope is related. I'm also left wondering how this squares with the claim above about complete lattices, as it sounds like there is some tension there.

Nathanael Arkor (Sep 14 2020 at 18:38):

Yes, Joyal characterises the free bicompletion, but doesn't give any kind of explicit construction as far as I remember, which makes it harder to compare. I'm also curious about the complete lattice…

Martti Karvonen (Sep 14 2020 at 19:34):

I think that the apparent tension between non-existence of free complete lattices on sets and the existence of free bicompletions of cats comes down to size issues: for a category (large or small) its bicompletion exists but is in general large. The point about lattices is that one would've hoped that the "free complete lattice" on a set would be a small lattice, but alas that can't be achieved in general.

Dan Doel (Sep 14 2020 at 19:37):

Oh, that makes sense, I guess. Even the free (co)completion construction usually given (presheaves) is large.

Nathanael Arkor (Sep 14 2020 at 19:42):

Ah, that's true. In a sense, categories are more intuitive than lattices, in that it's a little odd that free join semilattices do exist.

Dan Doel (Sep 14 2020 at 19:43):

That's why predicative mathematics makes the most sense. :)