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started a minimum at Bousfield-Friedlander theorem
(the model category theoretic incarnation of idempotent $\infty$-monads)
cross-linked with brief paragraphs to Bousfield-Friedlander model structure
I have spelled out the proof of the existence of the model structure at Bousfield-Friedlander theorem – except for the proof of the lemma labeled A.8 (iii) in Bousfield-Friedlander; but for that I at least added a pointer to Goerss-Jardine chapter X, lemma 4.4, where full details are given.
I have typed out the remaining proof of the second lemma that enters the proof of the Bousfield-Friedlander theorem.
I have typed out the proof of the proposition (here) that in a Bousfield-Friedlander-type Bousfield localization, the fibrations are precisely the “homotopy $Q$-modal morphisms”, so to say.
I directly followed the proof of Goerss-Jardine’s theorem X.4.8, just trying to make it a little more transparent.
For completeness, we should somewhere have direct proof of all the properties of “naive homotopy pullbacks” that are being invoked. Notably preservation under retracts. Maybe I’ll write that out later.
I have expanded the second part of the proof of the proposition (here) which characterizes the fibrations in the $Q$-model structure.
The previous version had been a bit quick towards the end. (In Goerss-Jardine it is done even quicker, but here it is good to actually spell it out.)
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