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Think of the category of structures in some signature $\Sigma$. The functor represented by a finitely presentable object $F$, say, corresponds to some term in the language generated by $\Sigma$. For e …

Since this result is derived by Lieberman, Rosicky, and Vasey as a corollary of some more sophisticated constructions with more sophisticated goals, I think it might be worth "compiling out" the proof …

Here's a general way to get an accessible category that likely will not have many colimits: let $C$ be a locally presentable category, and then let $C^{mono}$ be the category with the same objects as …

Some ideas, building off of Simon Henry and Ivan di Liberti's remarks: $Pr^L$ is in fact essentially a large (not huge) category (in either the ordinary or $\infty$ context). That is, let $\lambda$ …

Marc Hoyois answered in the comments: the answer is affirmative, by HTT 5.3.5.15.

Here's a further generalization: Claim: Let $\mathcal T$ be a triangulated category and with a $t$-structure such that $\mathcal T^\heartsuit$ is has coproducts, which are exact. Suppose there exi …

Here's something we can say which addresses a large class of examples. Let us say that a (possibly noncommutative) ring $R$ is not zero-dimensional if there exists $r \in R$ which is neither a right u …

A few years later, this is proven, by Ragimov and Schlank.