Sufficient sets of colimits in small categories

Question

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.

Answer 1

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 $X$.

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Reference:

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