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I'm trying to find a reference which defines and discusses some properties of connections and flat connections on the cotangent complex in a homotopical setting. That is to say, a connection or flat connection on a morphism of highly structured ring spectra. This is the sort of thing I feel like might be written down somewhere by Jacob Lurie or John Francis, and I'm actively looking for it, but of course that's a lot of ground to cover. Also, it might be written down somewhere else (perhaps Illusie or someone defined a notion of a connection on the cotangent complex at some point that I can generalize to the setting of ring spectra).

Any help would be greatly appreciated.

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    $\begingroup$ I recommend you look up "Atiyah extension" in Illusie's first volume. There is a definition using the cotangent complex in place of the sheaf of relative differentials (as in the "usual" definition). $\endgroup$
    – Jason Starr
    Commented Sep 6, 2014 at 10:29
  • $\begingroup$ Thanks @JasonStarr, at the moment I'm only finding something about the Atiyah class. This seems to be an obstruction to supporting a connection, or something along these lines. Is that what you're referring to? $\endgroup$
    – Jonathan Beardsley
    Commented Sep 6, 2014 at 12:51

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