Is every conservative, left exact left adjoint comonadic, $\infty$-categorically?

Question

Consider a conservative left adjoint $G : C \to D$ between complete 1-categories. By Beck's theorem, the following are equivalent:

1. $G$ is comonadic.
2. $G$ preserves $G$-split equalizers.

(2) is generally a bit finicky to check, but there are stronger conditions which can be easier to check and hold sometimes in practice. Following Barr and Wells, they go by names like "crude monadicity theorem". For example, because equalizers are finite limits, (2) follows from

3. $G$ preserves all finite limits.

Now let $F : A \to B$ be a conservative left adjoint between complete $\infty$-categories. Beck's theorem generalizes in the "obvious" way (I think this is due to Lurie? it's in Higher Algebra, and there's another proof due to Riehl and Verity), so that the following are equivalent:

4. $F$ is comonadic.
5. $F$ preserves $F$-split totalizations.

Unfortunately totalizations are not finite limits, so it's not clear that (5) follows from

6. $F$ preserves all finite limits.

This leads to a few

Questions:

1. If $F$ is a functor between complete $\infty$-categories which preserves finite limits, then does $F$ preserve $F$-split totalizations?

   A. What if $F$ is additionally assumed to be conservative and / or a left adjoint?

   B. What if $F$ is a conservative left adjoint between presheaf categories, or maybe between $\infty$-toposes, or even between arbitrary presentable $\infty$-categories? Or between presentable stable $\infty$-categories?

2. Alternatively, is there an even stronger condition than left exactness which is still weaker than preservation of all totalizations, which implies the Beck condition while being easier to check, and which might be frequently satisfied in practice? For instance, when $B$ is $Spaces$, for example, one might imagine asking for preservation of certain limits of towers -- say those satisfying some kind of connectivity hypothesis.

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My sense is it seems very unlikely that the answer to the first question should be "yes", but I am not at all sure how to build a counterexample. As more hypotheses on $F$ are added, I grow increasingly hopeful that something "magical" might happen and save us.

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Motivation: The fact that any left exact, conservative left adjoint is comonadic is very convenient in 1-topos theory, because that's the definition of a surjective geometric morphism. It would be nice if there were still just one reasonable choice for the meaning of "surjective $\infty$-geometric morphism".

Another place where left exactness is pretty cheap is when mapping between stable $\infty$-categories (where it already follows from being a left adjoint). A positive answer in this case would mean that the (co)monadicity theorem for stable $\infty$-categories is extremely nice! One would just need to check adjointness + conservativity.

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A bit of evidence:

It's a fact (if finite limits exist) that if a cosimplicial object $X_\bullet$ is split then the associated pro-object $(Tot_{\leq \bullet} X)$ is isomorphic (in the pro-category) to a constant pro-object. (I learned this from Akhil Mathew.) If $A_\bullet$ is a cosimplicial object and $FA_\bullet$ is "strongly split" in the sense that the associated pro-object $(Tot_{\leq \bullet} (FA))$ is literally constant, then by conservativity and preservation of finite limits, it follows that the pro-object $(Tot_{\leq \bullet} A)$ is also literally constant. It follows that the totalization is preserved in this case.

Of course, for Beck's theorem to kick in, this restricted case will not suffice.

Answer 1

The answer is no. That is, a conservative, exact left adjoint need not be comonadic, $\infty$-categorically.

For a counterexample, recall Thm 6.7 of Bousfield's Localization of spectra with respect to homology, which gives examples of ring spectra $E$ such that $L_E (H\mathbb Z) \to Tot(H\mathbb Z \wedge E^{\wedge \bullet+1})$ is not an equivalence. In the adjunction $E \wedge (-) : Spectra_E {}^\to_\leftarrow Mod_E : Forget$, the left adjoint is conservative and exact. If it were comonadic, then the totalization of that cosimplicial object (which is $E \wedge (-)$-split) would be canonically equivalent to $L_E(H\mathbb Z)$. Since it isn't, the adjunction is not comonadic.

(Here, $L_E : Spectra \to Spectra_E$ is the (homological) Bousfield localization with respect to $E$, and $Mod_E$ is $E$-modules).

So the moral of the story is that the failure of the $\infty$-categorical crude monadicity theorem lies in the difference between Bousfield localization and nilpotent completion.

It's conceivable that crude comonadicity might be rescued by restricting attention to $\infty$-topoi, but given the pervasiveness of these counterexamples of Bousfield, that seems unlikely to me.