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started monoidal Quillen adjunction
(still need to fill in the properties and theorems)
added the statement of two theorems:
recognition of lax monoidal Quillen adjunctions;
lifts to Quillen adjunctions on model categories of monoids
at monoidal Quillen adjunction I started writing out the detailed proof of the theorem that asserts that monoidal Quillen adjunctions lift to Quillen adjunctions for monoids under mild conditions.
The proof depends on a very technical lemma. The proof of that lemma I have not yet finished typing up. But I need to call it quits for today.
2am! you sure do.
I am in the process of filling the remaining gaps in the $n$Lab-writeup of the full Schwede-Shipley proof of the monoidal Dold-Kan correspondence. The main work happens at monoidal Quillen adjunction, so I am working on completing the proof given there.
Now I have completed the description of the adjuncton $(L^{mon} \dashv R)$ on categories of monoids induced from a right adjoint lax monoidal functor $R : C \to D$ on a monoidal category $C$ with colimits: lift to an adjunction of monoids.
Next comes the Quillen-property. And I may still have to add some general statements on colimits in categories of monoids in categories with colimits.
made slow progress with making explicit some of the intermediate steps that Schwede-Shipley gloss over. Now I am about one-third through the long proof of lemma 5.1 in their Equivalence of monoidal model categories .
I’ll also need to go through lemma 6.2 of their previous Algebras and modules in monoidal model categories which is used throughout.
All very technical and not very insightful.
I’ll have to interrupt now for a bit, will continue later.
I think I am pretty much through with typing up all the details of the proof of the monoidal Dold-Kan correspondence following Schwede-Shipley. Tried to make everything explicit – except the proverbial straightforward but tedious string manipulations arising whenever the computation of pushouts along free monoid maps in terms iterated pushouts as described at category of monoids gets invoked. I hope we can trust Schwede and Shipley that they checked this. There ought to be a shortcut around this messy issue…
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