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This question is a reference request for the following result or two results, which I believe are rather easy to prove.

Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\subset\mathcal K$ be a coreflective full subcategory. Assume that the coreflector $C\colon\mathcal K\to \mathcal A$ is an accessible functor (e.g., when viewed as a functor $\mathcal K\to\mathcal K$; this means that there exists a cardinal $\lambda$ such that $C$ preserves $\lambda$-directed colimits). Then

  1. The category $\mathcal A$ is locally presentable.

  2. If $\mathcal K$ is a Grothendieck abelian category and $\mathcal A$ is closed under kernels in $\mathcal K$, then $\mathcal A$ is a Grothendieck abelian category, too.

Is there any relevant reference? I was only able to find Corollary 6.29 in the book of Adámek and Rosický "Locally presentable and accessible categories". This corollary claims, among other things, that any coreflective full subcategory $\mathcal A$ in a locally presentable category $\mathcal K$ is locally presentable, if one assumes Vopěnka's principle.

My lemma above does not depend on Vopěnka's principle or any other set-theoretical assumptions. Part 1. of it is an elementary version of this corollary from the book of Adámek and Rosický. Is there any other/better reference?

Some context: part 2. of the lemma is a generalization of Lemma 3.4 from my preprint S.Bazzoni, L.Positselski "Matlis category equivalences for a ring epimorphism", https://arxiv.org/abs/1907.04973 .

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Since a coreflective full subcategory is the category of coalgebras for the induced idempotent comonad, 1. is answered in presentability rank of categories of coalgebras (the corresponding comonad is accessible).

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  • $\begingroup$ I think the author asks for a reference. $\endgroup$
    – Martin Brandenburg
    Commented Mar 1, 2020 at 9:53
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    $\begingroup$ I haven't looked, but I'm presuming that the linked paper shows that the coalgebras for an accessible comonad on an accessible category are accessible, which would be a perfectly adequate reference for (1). But it's equally the case that the coalgebras for an idempotent comonad are the inverter of the underlying copointed endofunctor. Inverters are PIE limits, so it follows from well known results which go back at least to Makkai and Pare that if the endofunctor is accessible, then the coreflective subcategory is accessible. Of course, it's also cocomplete if the original category is. $\endgroup$
    – Tim Campion
    Commented Mar 1, 2020 at 19:55
  • $\begingroup$ Actually, I think the canonical reference for the fact that accessible categories and accessible functors are closed under PIE limits might be Greg Bird's thesis, which should be available on Ross Street's website. $\endgroup$
    – Tim Campion
    Commented Mar 1, 2020 at 19:57
  • $\begingroup$ @TimCampion Thank you. It is becoming clearer now. $\endgroup$
    – Leonid Positselski
    Commented Mar 1, 2020 at 21:09
  • $\begingroup$ @JiříRosický Thank you. My understanding of it is much better now. $\endgroup$
    – Leonid Positselski
    Commented Mar 1, 2020 at 21:14

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