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In a survey article Algebraic geometry in mixed characteristic, B. Bhatt writes

For instance, given a commutative ring $R$ with a finitely generated ideal $I$,

  • the assignment carrying $R$ to the $\infty$-category $D_\text{$I$−comp}(R)$ of derived $I$-complete $R$-complexes forms a stack for the flat topology (or even a suitably defined $I$-completely flat topology), unlike the corresponding assignment at the triangulated category level.

What would be a reference for the above sort of statement? By this I mean I don't need a reference for that exact fact, although that would be fine too. I know what the words individually mean, but I have never seen a place where I can get an idea for how to go about proving this.

This question is very similar to the following question, which at the time of writing has no answer:
2-categories for the working algebraic geometer

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    $\begingroup$ I asked the other question and I still don't know a good answer to it but... for infinity-categories, the book Elements of infinity-Category Theory by E. Riehl and D. Verity is very probably the closest to what you want. It explains infinity-categories in a language which is closer to usual category theory (which I imagine you're more familiar with) than the usual treatments (which are heavily homotopy-theoretic). $\endgroup$
    – Gabriel
    Commented Nov 16, 2022 at 9:40
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    $\begingroup$ If you want to learn nontrivial facts about derived complete (co)modules, read relevant papers by L. Positselski, J. Rosický, J. Adámek, H. Krause, S. B. Iyengar. (In general this area of homological algebra is called theory of cotorsion pairs.) If you want to add quasicategories to that, probably you're out of luck besides Bhatt's papers. $\endgroup$
    – Denis T
    Commented Nov 16, 2022 at 12:49
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    $\begingroup$ In the next sentence, Bhatt gives several references, to Lurie and to Toen and Vezzosi. These references seem pretty reasonable to me. I don't know where exactly to find this precise statement, but Toen and Vezzosi's work, and Lurie's Spectral Algebraic Geometry, are comprehensive books/monographs which are largely concerned (among other things) with formalizing and proving this sort of statement. $\endgroup$
    – Tim Campion
    Commented Nov 16, 2022 at 13:49
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    $\begingroup$ fwiw Appendix D of SAG has various results of this type. In this case you could try deducing it from D.6.3.1 and D.6.2.3. Another option is to mimic the methods: the most general type of descent argument is in HA.4.7.5.3. To check the hypotheses use the same idea as in SAG.D.6.4: if A is I-completely faithfully flat over R then the completed tensor product with A gives a faithfully flat monad on connective, I-complete R-modules. $\endgroup$
    – Dylan Wilson
    Commented Nov 16, 2022 at 17:35
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    $\begingroup$ as a general rule- first check SAG, then HA, then HTT. They are easiest to read in that order since each book builds up machinery in order to make "intuitive" arguments rigorous, therefore the books become more intuitive as you go along, and further removed from the specific combinatorics of simplicial sets. As far as I know, despite lots of new resources for the "category theory" of higher categories, there are not yet any other textbook-like references for higher/derived/spectral algebra and geometry. $\endgroup$
    – Dylan Wilson
    Commented Nov 16, 2022 at 17:38

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I don't want this question to hang around forever on the "unanswered queue," so let me add an answer, even though I think the comments largely answer it. My motivation here is to advertise a few other sources beyond those mentioned in the comments.

First, as the comments point out, Lurie has written a ton about this kind of statement, and especially Spectral Algebraic Geometry seems the most user-friendly to commutative algebraists and algebraic geometers (as of today, the most recent update is from Feb of 2018, SAG is 2319 pages, and Lurie writes that it's 67% complete). I'm sure it was a big part of what got Bhargav Bhatt excited enough to write the survey the OP cites.

Bhatt also cites Toen and Vezzosi's papers on homotopical algebraic geometry (well, Homotopical Algebraic Geometry II is more of a book than a paper, at 228 pages), and I've always found these to have a bit more of an algebraic geometry focus relative to Lurie's writings (which often strike me as more topological). In addition to HAG I and HAG II, I recommend Toen's "global overview" (from 2009) and Toen's "derived algebraic geometry" survey from 2014. Still, I could not find in these sources the statement Bhatt cites, so I think Dylan Wilson is probably right that the best way to get this statement is to deduce it from Appendix D of SAG.

Also, the comment pointing out work of Positselski is worth following. His paper "Remarks on derived complete modules and complexes" discusses different definitions of $D_{I-comp}(R)$, a comparison between them, and their formal properties. See also his slides, which include a discussion of the badly-behaved non-derived case that Bhatt mentioned.

Turning to sources not mentioned in the comments or in Bhatt's article, I recommend:

  1. The stacks project, e.g., this entry on derived completion following work of Greenlees and May. Even though this does not explicitly mention $\infty$-categories, much of the story for $\infty$-categories is inspired by work that came earlier using things like model categories.

  2. Kiran Kedlaya's notes on prismatic cohomology, especially this entry about derived completion. Reading this, it seems Kedlaya has the same philosophical approach to the subject that Bhatt has. This makes sense, because Kedlaya writes that these lecture notes were inspired by 2018 notes from a course Bhatt taught, and it seems plausible to me that the statement the OP asks about would be contained in those notes. Similarly, this paper by Bhatt and Scholze seems like required reading.

  3. As a homotopy theorist, I cannot help but advertise recent work by Barthel, Heard, and Valenzuela, which revisits the work of Greenlees and May in (1), works out the theory of derived completion for comodules, and puts both pieces together in the setting of stable $\infty$-categories, in a way I really like, conceptually.

For a commutative algebraist or algebraic geometer, I suppose (2) would be the easiest way to get deeper into the types of results Bhatt mentioned.

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