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Stream: learning: questions

Topic: Closures of categories under limits

John Onstead (Jul 01 2025 at 11:25):

Recently, I entered into a correspondence with a category theorist who has been giving me problems to work on to help improve my knowledge. Here's the most recent one I was given:
Problem: For a small category $C$, let $\mathrm{Lim}(C)$ be the full subcategory of $\mathrm{Psh}(C)$ on all small limits of representables- therefore, the limit closure of the representables. Show that $\mathrm{Lim}(C)$ is not just complete (as it obviously is by definition) but also cocomplete.
He gave the following example: "Let $C$ already have all limits, therefore $\mathrm{Lim}(C) \cong C$. But any small complete category is a preorder, and any complete preorder is also cocomplete. Hence the theorem holds in this trivial case."

John Onstead (Jul 01 2025 at 11:26):

The proof seems quite straightforward. If a category is locally presentable, it's cocomplete, hence we can prove this by showing $\mathrm{Lim}(C)$ is locally presentable. Using Theorem 2.48 of Adamek and Rosicky (The Reflection Theorem), any subcategory of a locally presentable category is itself locally presentable when closed under limits and $k$-filtered colimits for some $k$. Obviously, $\mathrm{Psh}(C)$ is locally presentable, and $\mathrm{Lim}(C)$ is by definition closed under limits, so already two conditions are met. We can then assume Vopenka's principle to make the final condition redundant and the proof is complete. In fact, the weak Vopenka's principle says that any limit-closed subcategory of a locally presentable category is reflective, so this tautologically proves the above result.

John Onstead (Jul 01 2025 at 11:27):

But my "tutor" wasn't satisfied and claimed that cocompleteness of $\mathrm{Lim}(C)$ can be proven without Vopenka's principle. I was eventually able to switch tactics and apply SAFT to the inclusion of $\mathrm{Lim}(C)$ into presheaves, and it seemed to satisfy him. But while this does prove cocompleteness by showing $\mathrm{Lim}(C)$ is a reflective subcategory of presheaves, it doesn't get us to showing $\mathrm{Lim}(C)$ is locally presentable, and I'm still curious about if that can be shown without Vopenka. So I wanted to know: is this true, and if so, how can this be shown? Thanks!

John Onstead (Jul 03 2025 at 11:46):

Here's a fun simple question. Both the Kaurobi completion and free coproduct completion have two definitions- the explicit one and the subcategory of presheaves one. In the explicit definition, it gives an actual way to construct the objects of these explicitly from objects in $C$- IE, the objects of the free coproduct completion are indexed sets of objects of $C$ ("formal colimits" of objects of $C$), while those of the karoubi completion are objects of $C$ equipped with idempotents. This explicit perspective allows for the easy proof of things that might be harder with just the "subcategory of presheaves" one.

John Onstead (Jul 03 2025 at 11:47):

My question is: is there a similar explicit characterization of the objects of $\mathrm{Lim}(C)$? My only thought was perhaps "formal limits of objects of $C$", but that'd probably get something more like the free completion of $C$ rather than $\mathrm{Lim}(C)$.

Nathanael Arkor (Jul 03 2025 at 12:49):

There's [[free strict cocompletion]], which I think does what you're intending?

Mike Shulman (Jul 03 2025 at 15:03):

John's Lim(C) is not the free completion, but the closure of the representables under limits in the ordinary Yoneda embedding. In particular, the functor $C\to \mathrm{Lim}(C)$ preserves all existing limits in $C$. I don't know offhand a universal property of this.

Nathanael Arkor (Jul 03 2025 at 15:04):

Ah thanks, somehow I missed that this was a continuation of a previous thread.

Mike Shulman (Jul 03 2025 at 15:09):

I'm not sure to what extent it is that, but he included his definition of Lim(C) in the first post here.

Kevin Carlson (Jul 03 2025 at 18:02):

I wonder, how does this closure relate to the free conservative completion of $C$, ie the universal complete guy into which $C$ maps continuously?

Mike Shulman (Jul 03 2025 at 18:03):

Yeah, that's the obvious thing to compare it to. But I don't know the answer; I'd be surprised if they were the same.

John Onstead (Jul 03 2025 at 21:05):

Kevin Carlson said:

I wonder, how does this closure relate to the free conservative completion of $C$, ie the universal complete guy into which $C$ maps continuously?

Ah I believe that’s the subcategory of presheaves that send colimits to limits. Interestingly I discussed this category on previous topics. It’s not the same as the closure under limits, but they are related!

John Onstead (Jul 03 2025 at 21:06):

Let's say we had some presheaf $F$ that is a small limit of representables $\mathrm{lim}_iR_i$, and we apply $F(\mathrm{col}_jX_j)$ to some colimit in $C$. First, we compute limits pointwise in $\mathrm{Set}$, so we get $\mathrm{lim}_iR_i(\mathrm{col}_jX_j)$. Representables send colimits to limits, so we get $\mathrm{lim}_i\mathrm{lim}_jR_i(X_j)$. Limits commute with limits, so we end up with $\mathrm{lim}_j\mathrm{lim}_iR_i(X_j)$. But recalling the definition of $F$ this is just $\mathrm{lim}_jF(X_j)$, and therefore we find $F$ sends colimits to limits! So that means not only does $\mathrm{Lim}(C)$ embed into the category of limit preserving presheaves, but because the embedding of $C$ into the latter preserves colimits and limits, so too does the embedding of $C$ into $\mathrm{Lim}(C)$!

John Onstead (Jul 04 2025 at 00:26):

I did figure out another characterization of $\mathrm{Lim}(C)$- it's equivalent to the category whose objects are categories of cones in $C$. Given an index category $J$, and a diagram $F: J \to C$, the category $\mathrm{Cone}(F)$ is equivalent to the comma category $(\Delta / F)$ in $[J, C]$. The category of all such cone categories and the fibered functors between them is equivalent to $\mathrm{Lim}(C)$ via the Grothendieck construction (that sends a presheaf to the category of elements of that presheaf). This just follows from the universal property of limits that morphisms into them correspond to cones over their diagrams.