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The model category structure on co-simplicial commutative $k$-algebras, $CAlg_k^\Delta$, with fibrations degreewise surjections: is it left proper?

Most of the literature considers the standard model category structure on (graded) commutative differential algebras. But this generalizes to all (not-necessarily commutative) dg-algebras. Details an …

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) …

Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition. What's a reference for that?

There is a standard (simplicial) model category structure on the category $Ab^\Delta \simeq Ch^\bullet_+(Ab)$ of co-simplicial abelian groups, whose fibrations are the degreewise surjections (and weak …

It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model stru …

For $k$ a field (the case I am interested in, but the question makes sense over any dga), $\mathrm{Ch}_\bullet(k)$ its projective model category of unbounded chain complexes (here), $\mathrm{sCh}_\ …

For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)preshe …