Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 comma 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 finite 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2010
    • (edited Oct 21st 2010)

    I added to Hochschild homology in the section Function algebra on derived loop space a statement and proof of the theorem that “the function complex C(X)C(\mathcal{L}X) on the derived loop space C(C)C(\mathcal{L}C) is the Hochschild homology complex of C(X)C(X)”.

    There is a curious aspect to this: we are to compute the corresponding pushout in \infty-algebras. But in the literature on Hochschild homology, the pushout is of course taken not in algebras, but in modules

    HH k(C(X)):=Tor k(C(X),C(X)) C(X)C(X). HH_k(C(X)) := Tor_k(C(X), C(X))_{C(X) \otimes C(X)} \,.

    So how is that HHHH-complex actually an derived algebra?

    The solution of this little conundrum is remarkably trivial using Badzioch-Berger-Lurie’s result on homotopy T-algebras .This tells us that we may model the \infty-algebras as simplicial copresheaves on our syntactic category TT, using the left Bousfield localizatoin of the injective model structure at maps that enforce the algebra property.

    But since we are computing a pushout and since the traditional bar complex provides a cofibrant resolution of our pushout diagram already in the unlocalized structure, and since left Bousfield localization does not affect the cofibrations, due to all these reasons we may (or actually: have to) compute the pushout of \infty-algebras as just a pushout in simplicial copresheaves.

    In particular it follows that the pushout of our product-preserving coproseheaves is not actually product-presrving itself. Instread, it is (the simplicial set underlying) the standard Hochschild complex. So everything comes together. We know that if we wanted to find the actual \infty-algebra structure on this, we’d have to form the fibrant replacement in the localized model structure. That would make a bit of machinery kick in and actually produce the \infty-algebra structure on the Hochschild complex for us.

    But if we don’t feel like doing that, we don’t have to. The homotopy groups of our simplicial copresheaf won’t change by that replacement.