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Stream: learning: questions

Topic: Free phi-cocompletions of categories

John Onstead (Jul 15 2025 at 11:16):

A large class of subcategories of the presheaf category are given by cocompleting the category freely under some class of colimits. Examples of this include Cauchy completion (absolute colimits), coproduct completion, Ind completion (filtered colimits), and sifted colimit completions. There is an explicit representation of these cocompletion: given some class $\phi$, the free cocompletion under those colimits is $\mathrm{Flat}_{\phi^c}[C^{op}, \mathrm{Set}]$, where $\phi^c$ is a certain "conjugate/complement" class to $\phi$ and "flat" denotes a $\phi^c$-limit preserving functor that also preserves these limits even when they don't exist. If $C$ already has all $\phi^c$ colimits, then the free $\phi$-cocompletion is just $\mathrm{Lim}_{\phi^c}[C^{op}, \mathrm{Set}]$.

John Onstead (Jul 15 2025 at 11:17):

For instance, for ind completion, $\phi$ is filtered colimits and $\phi^c$ is finite limits, so the notion of $\phi^c$ flatness is just the usual notion of flatness. For sifted colimits, $\phi^c$ is finite products. The free cocompletion itself is also a special case- if $\phi$ is all colimits, then $\phi^c$ must be absolute limits, and any presheaf preserves those (even the absolute limits that don't exist). But in all examples I have with explicit characterizations of $\phi^c$, $\phi$-colimits both distribute AND commute with $\phi^c$-limits in $\mathrm{Set}$, making it difficult to determine which one of these two is the actual correct characterization of the class $\phi^c$, since commutativity and distributivity diverge in general.

John Onstead (Jul 15 2025 at 11:18):

So my question is: what is the correct characterization of $\phi^c$: is it the class of limits that commute with $\phi$-colimits in $\mathrm{Set}$, or those that distribute with them? Thanks!

Fernando Yamauti (Jul 15 2025 at 15:33):

Perhaps https://rezk.web.illinois.edu/accessible-cat-thoughts.pdf and the references therein might be useful.

John Onstead (Jul 15 2025 at 19:27):

Fernando Yamauti said:

Perhaps https://rezk.web.illinois.edu/accessible-cat-thoughts.pdf and the references therein might be useful.

Thanks! This work seems highly informative too for some of the other things I'm curious about.
Based on the section for "regular classes", it seems the criteria is that the colimit functor on diagrams in class $\phi$ preserve $\phi^c$ limits, which is the notion of commutativity of limits and colimits. So the answer to my question is "commutativity", not "distributivity".
Edit: Actually what I was thinking of was not the regular classes section, but the paper's definition of $\mathrm{Filt}_U$ for some class of limits $U$. But the result is the same!

Kevin Carlson (Jul 15 2025 at 20:03):

“Distributivity” is still a not-too-well-studied concept I think; commutativity is usually going to be the answer to a question like this.

John Onstead (Jul 15 2025 at 23:33):

I have a follow up question to this. The article states that if $D$ has filtered colimits, $C \to D$ is a fully faithful inclusion into the presentable objects of $D$, and $C$ generates $D$ under filtered colimits, then $D \cong \mathrm{Ind}(C)$. It also states the subcategory of $\mathrm{Ind}(C)$ on presentable objects are precisely the retracts of $C$ (IE, the idempotent completion of $C$). But the article doesn't address the converse- for instance, perhaps it's also possible for a non-equivalent category to the ind completion to have its presentable objects equivalent to the idempotent completion of $C$.

John Onstead (Jul 15 2025 at 23:34):

Let $D$ be a category with filtered colimits and a dense subcategory $C \to D$, such that $D$ also contains the idempotent completion of $C$. Is it true that if the full subcategory of $D$ on presentable objects is precisely the idempotent completion of $C$, then automatically $D \cong \mathrm{Ind}(C)$?

Kevin Carlson (Jul 15 2025 at 23:55):

Are you perhaps missing the fact that idempotent splittings are filtered colimits?

John Onstead (Jul 16 2025 at 01:03):

Kevin Carlson said:

Are you perhaps missing the fact that idempotent splittings are filtered colimits?

Hmm, in that case, the Cauchy completion of $C$ would be expressed as filtered colimits of objects of $C$ in $D$. But I'm not sure how that extends to showing all of $D$ is generated under filtered colimit of objects of $C$.

Kevin Carlson (Jul 16 2025 at 01:06):

No, that doesn't resolve your question, it just makes one of your assumptions automatic which was otherwise making me feel awkward.

Kevin Carlson (Jul 16 2025 at 01:15):

I think you can probably cut out the nontrivial finite colimits of representables from a category of presheaves on an idempotent complete category without breaking existence of filtered colimits, because [intuitively, I don't know how to make this precise] the only interesting finite filtered colimits are idempotent splittings.

Kevin Carlson (Jul 16 2025 at 01:22):

Oh, maybe I do know; if $C$ is a finite filtered category, then by definition its identity functor has a cocone $\alpha$ with tip some object $t.$ The component $\alpha_t:t\to t$ is idempotent by the cocone condition, and one can check that the corresponding functor from the walking idempotent is final.

John Onstead (Jul 16 2025 at 02:24):

Kevin Carlson said:

No, that doesn't resolve your question, it just makes one of your assumptions automatic which was otherwise making me feel awkward.

Ah, I see, I didn't have to add the extra assumption that the idempotent completion of the category also embeds.

John Onstead (Jul 16 2025 at 05:53):

After doing some more work with help from your hints above, I've derived a relatively simple counter-example to my claim and therefore showed it to be false. Here's the basics. By a partial result in the paper, if $C$ embeds fully faithfully into the compact objects of $D$ a category with filtered colimits, then the canonical functor $\mathrm{Ind}(C) \to D$ via the universal property of Ind is fully faithful. By density of $C$ we derive a chain of inclusions: $\mathrm{Ind}(C) \to D \to \mathrm{Psh}(C)$. So $D$ is just an enlargement of the ind completion in presheaves, therefore, if we find such an enlargement that doesn't change the compact objects of the ind completion (which are already precisely the retracts of representables), we have an example of a non-equivalent category to ind which satisfies my criteria.

John Onstead (Jul 16 2025 at 05:53):

Your suggestion above made me remember that the compact objects of presheaves are the finite colimits of representables. Thus, if $D$ includes any of those, we've added in a new compact object and so we indeed change the compact objects of the category. But the solution is simple: take the ind completion and add in just a single infinite colimit of representables, and let that be $D$. This isn't a compact object in presheaves, and therefore adding it does not in fact change the compact objects of $C$. Hence, it is a counterexample and the above "conjecture" is false.

Fernando Yamauti (Jul 16 2025 at 06:37):

The correspondence I think you are trying to get only works for loc pres cat: it's Gabriel-Ulmer duality (also in the context of sound doctrines).

Fernando Yamauti (Jul 16 2025 at 06:40):

Also you must be careful about splitting idempotents since, for 1-cats, finite (co)limits are enough (and, therefore, you won't see the requirement of being idemp complete in the classical case of the duality), but, in the higher case, you need, at least, sequential colimits.

John Onstead (Jul 16 2025 at 10:24):

Fernando Yamauti said:

The correspondence I think you are trying to get only works for loc pres cat: it's Gabriel-Ulmer duality (also in the context of sound doctrines).

That's really interesting that Gabriel-Ulmer duality comes up in relation to the connection between filtered colimits and finite limits. But I guess it makes some sense- when $C$ has finite limits, then the action of taking the subcategory of presheaves that preserve finite limits is exactly the action of cocompleting under filtered colimits. From the article, it looks like more generally, when $\phi$ is a sound doctrine, then if $C$ has all $\phi$ limits then taking the subcategory of presheaves that preserve them is the act of cocompleting under the colimits that commute with $\phi$ limits in $\mathrm{Set}$.

John Onstead (Jul 16 2025 at 10:24):

The real surprise is then that there's a functor going the opposite way constituting an equivalence- that you can start with a "presentable category" for some class of colimits (IE, filtered in the usual case) and uniquely recover the cauchy complete category that it is the cocompletion of under that class of colimits. So this indeed reveals that the precise extra criteria to make my above statement work is for $D$ to be locally presentable.