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Jun 3, 2021 at 20:24 comment added Denis Nardin @Tintin For spectra this is just the standard axioms of a t-structure (the functor is the 0-truncation functor from connective spectra to the heart). For ring spectra, a reference is Higher Algebra, 7.1.3.15, but it's probably simpler to prove by identifying connective $E_\infty$-ring spectra with models for a certain Lawvere theory, where you can just apply the corresponding statement for spaces.
Jun 3, 2021 at 13:37 comment added Tintin Thank you very much, Denis. That indeed is a nice solution. One more thing, do you know a reference where I can read about the adjunction between groups/rings and spectra/ring spectra? The references I see only do the construction of $H$ and I am interested in reading in detail the adjoint map $1\to H\pi_0$, I don't see how to do it. Thanks in advance
Jun 3, 2021 at 13:24 vote accept Tintin
May 27, 2021 at 20:03 history edited Denis Nardin CC BY-SA 4.0
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May 27, 2021 at 16:13 history answered Denis Nardin CC BY-SA 4.0