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Let $C$ be a small category, and consider the class of diagrams $G:D\to C$, with $D$ a small category, that have colimits in $C$. This is a proper class even when $C$ is very small, e.g. whenever $D$ has a terminal object $t$, any functor $G:D\to C$ has a colimit $G(t)$, and there is a proper class of small categories with a terminal object.

However, those colimits feel kind of "trivial"; in some cases at least we can find a small set of diagrams that "carry all the nontrivial information" about colimit diagrams in $C$. For instance, if $C$ is a poset, then it suffices to consider injective functors $G$ (and we may as well take $D$ to be discrete as well), and these form an essentially small set. For a non-posetal $C$ we can't restrict to injective functors, since coproducts are not idempotent, but maybe there is some other restriction that works. Note that by Freyd's theorem, a non-posetal small category does have a bound on the cardinality of coproducts that it can admit; but this doesn't quite answer the question itself, since a particular colimit can exist even if the coproducts that would be necessary to construct it from coproducts and coequalizers do not.

Here are two ways to make the question precise:

  1. Given a small category $C$, is there a small set $L$ of diagrams $G:D\to C$ with colimits such that for any diagram $G':D'\to C$ with a colimit, there is a $(D,G)\in L$ and a final functor $F:D\to D'$ such that $G = G' \circ F$? (Edit: As pointed out by Dylan in the comments, this version is impossible. Take $C$ terminal and let $D$ vary over all ordinals; no small set of categories can be cofinal in all ordinals.)

  2. Given a small category $C$, is there a small set $L$ of diagrams $G:D\to C$ with colimits such that if a functor $H:C\to E$ preserves colimits of all diagrams in $L$, then it preserves all colimits that exist in $C$?

Any solution to question 1 is also a solution to question 2, but I'm not sure whether the converse holds. The mention of Freyd's theorem above suggests that a solution might require classical logic — I would find it more surprising if such a set existed for a non-posetal small complete category, although I don't immediately see an argument that it cannot.

One can of course also ask similar questions for enriched categories, internal categories, $\infty$-categories, and so on. Bonus points go to an answer that applies more generally in such contexts.

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    $\begingroup$ I'm confused about (1). Take $C$ to be a point, so every diagram has a colimit. Then doesn't (1) ask for a small collection of categories $\{D\}$ such that every category admits a final map from one of the $D$? That seems unlikely, right? $\endgroup$ – Dylan Wilson Jun 17 '19 at 12:19
  • $\begingroup$ I don't know what you mean about 'coproducts are not idempotent', but at the cost of possibly replacing C by something equivalent, you can replace the diagrams G by those that are injective on objects. I thought I'd convinced myself that diagrams that are faithful functors were enough, but now I'm not so sure. $\endgroup$ – David Roberts Jun 17 '19 at 12:56
  • $\begingroup$ @DylanWilson good point! So (1) is impossible as stated. Maybe there is some modification of it that is possible, but I don't see it immediately. So I guess (2) is the better question. $\endgroup$ – Mike Shulman Jun 17 '19 at 15:54
  • $\begingroup$ @DavidRoberts By "coproducts are not idempotent" I mean that for a general object $X$ in a general category, $X$ is not isomorphic to $X+X$ (whereas this is true in a poset). This is one reason why injective-on-objects functors are not enough, because (if $C$ is skeletal, say) $X+X$ might not be the colimit of any injective-on-objects functor (in a useful way). It's true that you can make any given functor isomorphic to an injective-on-objects one by changing $C$ to an equivalent category, but there's no equivalent category to $C$ that will do that for all functors simultaneously. $\endgroup$ – Mike Shulman Jun 17 '19 at 15:56
  • $\begingroup$ A small note: for (2), if we switch from colimits to limits, then since limits are detected by representables, it suffices to take $E = Set$. So the question can be rephrased as: given a small category $C$ and the (possibly large) sketch $L_C$ of all limit diagrams in $C$, does there exist a small sub-sketch with an equivalent category of models? It seems to me the first thing to ask is whether the category of models of $L_C$ is accessible. $\endgroup$ – Tim Campion Jun 17 '19 at 17:21
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I think the answer to (2) is affirmative under Vopenka's Principle. That is,

Claim:

Let $C$ be a small category, and assume Vopenka's Principle. Then there exists a small set of limit cones $L$ in $C$ such that for any category $D$ and any functor $F: C \to D$, $F$ preserves limits if and only if $F$ preserves the limit cones in $L$.

Notes:

  • I have no idea if VP is necessary, or if the statement has large cardinal strength at all.

  • I'm a little suspicious of the claim because of its reliance on the Lemma below. The reason I'm suspicious of the lemma is that it follows very easily from the work of Adamek and Rosicky and I'm surprised I haven't seen it stated before.

  • The proof shows the more general claim, as discussed in the comments, that under VP any limit-sketch on a small category has a small sub-sketch (on the same category) with equivalent models.

Proof: As noted in the comments, it suffices to consider the case $D = Set$, because $F: C \to D$ preserves limits (resp. preserves limits in $L$) if and only if $D(d,F-): C \to Set$ preserves limits (resp. preserves limits in $L$) for every $d \in D$.

Note that the category $Lim(C)$ of limit-preserving functors $C \to Set$ is the intersection in $Fun(C,Set)$ of the cateogries $Lim_\mu(C)$ of functors preserving $\mu$-small limits in $C$ for each $\mu$. So by the following lemma, we may take $L= L_\mu$ for some $\mu$.

Lemma:

Let $K$ be a locally presentable category, and assume Vopenka's Principle. Then every decreasing Ord-indexed sequence of reflective categories of $K$ stabilizes after a small number of steps.

Proof: Let $(L_\mu)_{\mu \in Ord}$ be such a sequence of localization functors. The proof of Thm 6.22 in [1] shows that even under weak Vopenka's Principle, the sequence $L_\mu(X)$ stabilizes after a small number $\mu_X$ of steps for any fixed $X$ (briefly -- if it doesn't stabilize, we get an embeding of $Ord^{op}$ into $X \downarrow K$).

So we just need to show that the number of steps is bounded independent of $X$. Here we use Thm 6.24 of [1], which tells us that by virtue of Vopenka's Principle, $L_\infty = \varinjlim_\mu L_\mu$ reflects onto an accessible, accessibly-embedded subcategory, and so $L_\infty$ is a $\lambda$-accessible functor for some $\lambda$ (viewed as an endofunctor of $K$). Thus, we can take a small subcategory $C \subseteq K$ which generates $K$ under $\lambda$-filtered colimits. Then if $\mu = \max(\lambda, \sup_{c \in C} \mu_c)$, then the colimit defining $L_\infty(X)$ stabilizes after $\mu$-many steps for every $X$.


Reference:

[1] Adámek, J., and J. Rosickỳ. Locally Presentable and Accessible Categories. London Math. Soc. Lect. Notes Ser. 189. Cambridge Univ Pr, 1994.

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  • $\begingroup$ I think the lemma also follows from Theorem 2.3 in "left-determined model categories and universal homotopy theories" (jstor.org/stable/1194855) applied to the trivial model structure. I had also gotten that far, but somehow didn't make the last connection. Thanks! $\endgroup$ – Mike Shulman Jun 24 '19 at 13:46
  • $\begingroup$ I have a suspicion that the stronger statement -- "Every limit sketch on a small category admits an equivalent small subsketch" -- should follow from having a proper class of strongly compact cardinals (maybe even weakly compact?). It just feels like a compactness statement in infinitary logic. This would be a considerably weaker hypothesis than VP (especially if it can be reduced to weakly compact) $\endgroup$ – Tim Campion Jun 24 '19 at 13:59
  • $\begingroup$ Hmm... it doesn't feel so much like a compactness statement to me, but I'm not an expert in that sort of stuff. I do find the lemma somewhat (and its homotopy-theoretic version) surprising; to be honest it makes me wonder a bit more than before about VP. (-: $\endgroup$ – Mike Shulman Jun 24 '19 at 14:49
  • $\begingroup$ I think I'd be worried if $VP$ implied that any increasing $ORD$-indexed sequence of reflective subcategories stabilized,but decreasing is fine with me.After all, if we think of the localization functor $C \to LC$ as a sort of quotient map, then increasing is like a "class-Artinian" condition while decreasing is like a (weaker) "class-Noetherian" condition (it's flipped from "descending chain of ideals" and "ascending chain of ideals" because we're looking at the "quotients";the "ideals" have no clear counterpart except maybe the class of morphisms left orthogonal to the localization). $\endgroup$ – Tim Campion Jun 27 '19 at 4:19

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