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The stub entry model structure on simplicial Lie algebras used to point to model structure on simplicial algebras. But is it really a special case of the discussion there?
Quillen 69 leaves the definition of the model structure to the reader. Is it with weak equivalences and fibrations those on the underlying simplicial sets? Is this a simplicially enriched model category?
Is this a simplicially enriched model category?
Yes, that’s a special case of theorem I in Reedy’s thesis, from 1974. I have added this to the entry, here.
I have made explicit (here) that taking normalized chains gives a Quillen adjunction
$(N^* \dashv N) \;\colon\; (LieAlg_k^{\Delta^{op}})_{proj} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^*}{\longleftarrow}} {\bot} (dgLieAlg_k)_{proj} \,.$I keep claiming that Quillen claimed that there is a model structure on reduced rational simplicial Lie algebras whose weak equivalences and fibrations are those on the underlying simplicial sets.
But Quillen’s classical text is very vague about this. In “Rational homotopy theory” (web) he just writes below the analogous theorem 4.7 for reduced simplicial commutative Hopf algebras:
We leave to the reader to formulate a similar theorem for $[$reduced simplicial Lie algebras $]$.
So I should check. But if anyone has a sanity check to offer, I’d be grateful.
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