Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2011
   • (edited Feb 24th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI think I can see the following, but am wondering if this is discussed in the literature: for $A$ a cochain dg-algebra in non-negative degree let $\Xi A \in Alg^\Delta$ be the cosimplicial algebra obtained as its image under dual Dold-Kan. Then regarded as an object in $[\Delta^{op}, Alg^{op}] \hookrightarrow [\Delta^{op}, (Alg^\Delta)^{op}]$ it should always be Reedy cofibrant, for the projective model structure on $Alg^\Delta$ where the fibrations are degreewise surjections: because the morphism out of the latching object in degree $n$ is dually a projection out of the algebra $(\Xi A)_n$ discarding some top degree generators, hence is an epi, hence a fibration in $Alg^\Delta_{proj}$ hence a cofibration $(Alg^\Delta_{proj})^{op}$. This statement should be of general relevance and is probably discussed somewhere. Does anyone have anything?

   I think I can see the following, but am wondering if this is discussed in the literature:

   for $A$ a cochain dg-algebra in non-negative degree let $\Xi A \in Alg^\Delta$ be the cosimplicial algebra obtained as its image under dual Dold-Kan. Then regarded as an object in $[\Delta^{op}, Alg^{op}] \hookrightarrow [\Delta^{op}, (Alg^\Delta)^{op}]$ it should always be Reedy cofibrant, for the projective model structure on $Alg^\Delta$ where the fibrations are degreewise surjections:

   because the morphism out of the latching object in degree $n$ is dually a projection out of the algebra $(\Xi A)_n$ discarding some top degree generators, hence is an epi, hence a fibration in $Alg^\Delta_{proj}$ hence a cofibration $(Alg^\Delta_{proj})^{op}$.

   This statement should be of general relevance and is probably discussed somewhere. Does anyone have anything?