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    • CommentRowNumber1.
    • CommentAuthorJon Beardsley
    • CommentTimeOct 17th 2014

    There’s quite a bit on the nn-lab about effective epimorphisms in \infty-toposes, so maybe I’m just not putting it together correctly, but I’m wondering if anyone here knows if there’s a characterization (in print somewhere, preferably) of effective epimorphisms for A A_\infty-ring spectra. In particular, I suspect there should be some kind of statement like: For f:ABf:A\to B, a morphism of connected A A_\infty-ring spectra, we can recover AA from the Amitsur complex (dually, the Cech nerve) of ff as long as π 0(f)\pi_0(f) is an isomorphism and π 1(f)\pi_1(f) is a surjection.

    This sort of statement is proven by Gunnar Carlsson in the context of what he calls “derived completion” but he works with the SS-algebra framework of Elmendorf, Kriz, May and Mandell, and it seems like it should be a much more general topos theoretic statement for an \infty-topos. Incidentally, has anyone written down anything about the topos of A A_\infty-ring spectra at all? At first glance it seems like most of Lurie’s work on the subject is for E E_\infty-rings.

    Thanks! Jon

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 19th 2014

    Do A_∞-ring spectra even form a ∞-topos? It does not appear, for example, that colimits in A_∞-ring spectra are universal (i.e., commute with pullbacks), which would prevent A_∞-ring spectra from forming an ∞-topos.