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This is almost exactly the same question as The (co)monadicity theorem relative to a presheaf topos, but the accepted answer there does not cover the full extension of Linton's monadicity that is a logical equivalence instead of an implication. Inspired by the note on nlab's monadicity page, that there is an extension of Linton's monadicity over Set to monadicity over presheaves.

The form I have in mind is a follows:

$U\colon D\to Set$ is monadic if and only if

  1. $U$ has a left adjoint,
  2. $D$ admits kernel pairs and coequalizers,
  3. A parallel pair $R \rightrightarrows S$ in $D$ is a kernel pair if and only if its image under $U$ is so, and
  4. A morphism $A\to B$ in $D$ is a regular epimorphism if and only if its image under $U$ is so.

The question is then, given a category $\mathbb{C}$, is there an extension of this If-an-only-If form of Linton's monadicity theorem, replacing $Set$ with presheaves on $\mathbb{C}$? For some context I am really interested in the cases when $\mathbb{C}$ is some direct category.

References are welcome.

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    $\begingroup$ I suggest checking whether this statement relies on Choice in $\mathbf{Set}$, ie that epis split. $\endgroup$
    – Paul Taylor
    Commented Jun 20 at 18:22
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    $\begingroup$ As Paul Taylor suggests, the proof in the case of $\mathrm{Set}$ seems to rely heavily on epimorphisms splitting. For instance, Linton proves a generalisation of his monadicity theorem for $\mathrm{Set}$-like categories in Theorem 3 of Applied functorial semantics, II, but requires epimorphisms in the base category to split. So I suspect there will not be a similar-looking theorem for presheaf categories. $\endgroup$
    – varkor
    Commented Jun 21 at 9:35
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    $\begingroup$ Perhaps even simpler than thinking about the proof and the axiom of choice, it looks to me like Linton's theorem implies very directly that $D$ is Barr exact, since all the properties needed for Barr exactness are created by $U$. But since monadic categories over presheaves are not always Barr exact or even regular, you'd have to dramatically modify the statement to have any chance here. In particular, you can't ask for anything like the creation of regular epis. Seems like you're going to get pushed back to $U$-split coequalizers and away from Linton's formulation. $\endgroup$
    – Kevin Carlson
    Commented Jun 21 at 21:07
  • $\begingroup$ I've been digging around but I'm having trouble understanding what that sentence could be referring to other than Duskin's theorem (which is both necessary and sufficient for presheaf categories, I've just edited that in), which doesn't really feel much like a form of Linton's theorem to me since it does involve $U$-split congruences. $\endgroup$
    – Kevin Carlson
    Commented Jun 21 at 22:21
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    $\begingroup$ I checked with Mike Shulman, who wrote that line in 2010, and he can’t immediately remember what he meant. $\endgroup$
    – Kevin Carlson
    Commented Jun 21 at 23:40

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