Consider a conservative left adjoint $G : C \to D$ between complete 1-categories. By [Beck's theorem](https://ncatlab.org/nlab/show/monadicity+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](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html), 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:**
- If $F$ is a functor between complete $\infty$-categories which preserves finite limits, then does $F$ preserve $F$-split totalizations?

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

    - 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?

- 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](https://arxiv.org/abs/1404.2156).) 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.