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Stream: learning: questions

Topic: Cauchy complete linear categories

John Baez (Jun 20 2023 at 20:06):

Does anyone know a reference for the fact that the Cauchy completion of a category $\mathsf{C}$ enriched over vector spaces is formed by taking the full subcategory of $\mathsf{Vect}$-valued presheaves on $\mathsf{C}$ consisting of retracts of finite coproducts of representables?

John Baez (Jun 20 2023 at 20:07):

If worst comes to worst, I'll settle for a reference for the analogous fact with $\mathsf{Vect}$ replaced by $\mathsf{Ab}$. Even the oldest sources I've looked at seem to take this for granted.

Mike Shulman (Jun 20 2023 at 23:12):

As far as I know (and you probably know this too), this notion of Cauchy completion for enriched categories was first defined by Lawvere in his 1973 paper "Metric spaces, generalized logic, and closed categories", where he claims that "it can be shown" that the characterization you mention holds for $\mathsf{Ab}$, making it the bad sort of [[folklore]]. It seems like someone in the intervening 50 years ought to have written out a proof, but I don't know of one.

The closest I can find right now is in Cauchy completeness for DG-categories by Nikolić, Street, and Tendas (2021), they show (Prop 6.1) that the Cauchy completion of a category enriched over $\mathbb{Z}$-graded abelian groups is formed by taking the full subcategory of enriched presheaves ("right modules") consisting of retracts of finite coproducts of suspensions of representables. Inspecting the proof, it's clear that the grading could be by any abelian group instead of $\mathbb{Z}$, and taking it to be the trivial group yields the result for $\mathsf{Ab}$. But surely there must be an earlier and better reference than that.

Giacomo (Jun 21 2023 at 08:59):

I do not know any earlier references, but that result could be seen as a consequence of Prop 4.23 (see also 4.22) of my paper Flat vs. filtered colimits in the enriched context, joint with Lack. There, among other things, we characterize Cauchy completions for $\mathcal V$-categories enriched over, what we call, a "locally dualizable" base; the main point being that such a base has a strong generator $\mathcal G$ made of dualizable objects.
$\mathbf{Vect}$ is an example of these, and the strong generator $\mathcal G$ is given by the unit; therefore the $\mathcal G$-copowers used in Prop. 4.23 are trivial (i.e. $P\cdot \mathcal C(-,C)\cong \mathcal C(-,C)$ in the statement). Using that the absolute=Cauchy weights are the objects of the Cauchy completion, this gives the result you wanted.

John Baez (Jun 21 2023 at 14:33):

Mike Shulman said:

As far as I know (and you probably know this too), this notion of Cauchy completion for enriched categories was first defined by Lawvere in his 1973 paper "Metric spaces, generalized logic, and closed categories", where he claims that "it can be shown" that the characterization you mention holds for $\mathsf{Ab}$, making it the bad sort of [[folklore]]. It seems like someone in the intervening 50 years ought to have written out a proof, but I don't know of one.

Yes, I tracked the thread of references back and reached this point! It was sort of like pulling on a long fishing line and after a long struggle finding I'd caught nothing.

John Baez (Jun 21 2023 at 14:37):

Giacomo said:

I do not know any earlier references, but that result could be seen as a consequence of Prop 4.23 (see also 4.22) of my paper Flat vs. filtered colimits in the enriched context, joint with Lack. There, among other things, we characterize Cauchy completions for $\mathcal V$-categories enriched over, what we call, a "locally dualizable" base; the main point being that such a base has a strong generator $\mathcal G$ made of dualizable objects.
$\mathbf{Vect}$ is an example of these, and the strong generator $\mathcal G$ is given by the unit; therefore the $\mathcal G$-copowers used in Prop. 4.23 are trivial (i.e. $P\cdot \mathcal C(-,C)\cong \mathcal C(-,C)$ in the statement). Using that the absolute=Cauchy weights are the objects of the Cauchy completion, this gives the result you wanted.

Thanks! Since I really need the result for Vect rather than Ab, this will do the job quite nicely!

I was also curious about it for RMod for an arbitrary commutative ring, and I guess this too has a strong generator given by the unit R, which is dualizable.

John Baez (Jun 21 2023 at 20:18):

In Proposition 4.22 you say a $\mathcal{V}$-category is Cauchy complete iff it has finite direct sums, $\mathcal{G}$-copowers, and splitting of idempotents. This sounds like a very practical criterion, but I don't see where you define "has $\mathcal{G}$-copowers" - the phrase first shows up in Remark 4.11. Is the idea that $\mathcal{G}$ is an arbitrary strong generator consisting of dualizable objects? Did you fix such a set somewhere?

John Baez (Jun 21 2023 at 20:32):

Also, I don't really know what it means for a $\mathcal{V}$-category to have "finite direct sums". Are these biproducts? Is there a way to define biproducts at this level of generality? I know how to do it if $\mathcal{V}$ has a zero object.

Giacomo (Jun 22 2023 at 07:24):

Regardig $\mathcal G$, it is part of Defn 4.3 where we give the hypotheses on the base of enrichment; and yes, you can take it to be any strong generator made of dualizable objects. By saying that a $\mathcal V$-category has $\mathcal G$-copowers we simply mean that it has copowers by objects in $\mathcal G$ (maybe the notation is not as standard as I thought). Then, the equivalence $(2)\Leftrightarrow (3)$ in 4.22 says that it doesn't really matter what strong generator you choose at the beginning: you are always getting all copowers by dualizable objects.
About finite direct sums. Yes, I mean biproducts (but I personally don't like the name, as it makes me think about the "weak" 2-limits). A quick way to define them would be to say that $\mathcal V$ is enriched over commutative monoids and has finite coproducts. Which is equivalent to saying that it has a zero object and binary biproducts (as they are usually defined in the additive context).

John Baez (Jun 22 2023 at 14:22):

Okay, thanks very much! I understood the term "$\mathcal{G}$-copowers" for some arbitrary set of objects $\mathcal{G}$, but I had trouble quickly finding where you assert your standing hypotheses on this set $\mathcal{G}$ (or on your base of enrichment). I see now that Definition 4.3 has what I want.

I put some of this information into the nLab article [[Cauchy complete category]].