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Bousfield-Kan map

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                                      Contents

                                      Idea

                                      The Bousfield–Kan map(s) are comparison morphisms in a simplicial model category between two different “puffed up” versions of (co)limits over (co-)simplicial objects: one close to a homotopy (co)limit and the other a version of nerve/geometric realization.

                                      Definition

                                      Let CC be an (SSet,=×\otimes = \times)-enriched category and write

                                      Δ:ΔSet Δ op \Delta : \Delta \to Set^{\Delta^{op}}

                                      for the canonical cosimplicial simplicial set (the adjunct of the hom-functor Δ op×ΔSet\Delta^{op} \times \Delta \to Set).

                                      Write furthermore N(Δ/):ΔSet Δ opN(\Delta/-) : \Delta \to Set^{\Delta^{op}} for the fat simplex, the cosimplicial simplicial set which assigns to [n][n] the nerve of the overcategory Δ/[n]\Delta / [n].

                                      The Bousfield–Kan map of cosimplicial simplicial maps is a canonical morphism

                                      φ:N(Δ/())Δ \varphi : N(\Delta/(-)) \to \Delta

                                      of cosimplicial simplicial sets.

                                      This can also be regarded as a morphism

                                      φ:N(()/Δ op) opΔ. \varphi : N((-)/\Delta^{op})^{op} \to \Delta \,.

                                      This morphism induces the following morphisms between (co)simplicial objects in CC.

                                      Bousfield–Kan for simplicial objects

                                      For X:Δ opCX : \Delta^\op \to C any simplicial object in CC, the realization of XX is the coend

                                      |X|:=X Δ opΔ:= [n]ΔX nΔ n, |X| := X \otimes_{\Delta^{op}} \Delta := \int^{[n] \in \Delta} X_n \otimes \Delta^n \,,

                                      where in the integrand we have the copower or tensor of CC by SSet.

                                      Here the Bousfield–Kan map is the morphism

                                      X Δ opN(()/Δ op) opId X Δ opϕX Δ opΔ. X \otimes_{\Delta^{op}} N((-)/\Delta^{op})^{op} \stackrel{Id_X \otimes_{\Delta^{op}} \phi }{\to} X \otimes_{\Delta^{op}} \Delta \,.

                                      Bousfield–Kan for cosimplicial objects

                                      For X:ΔCX : \Delta \to C any cosimplicial object, its totalization is the Δ\Delta-weighted limit

                                      TotX:=lim ΔX [nΔ]X n Δ n, Tot X := lim^\Delta X \simeq \int_{[n \in \Delta]} X_n^{\Delta^n} \,,

                                      where in the integrand we have the power or cotensor X n Δ n=(Δ,X n)X_n^{\Delta^n} = \pitchfork(\Delta, X_n) of CC by SSet.

                                      Here the Bousfield–Kan morphism is the morphism

                                      TotXhom(Δ,X)hom(ϕ,Id X)hom(N(Δ/()),X). Tot X \simeq hom(\Delta,X) \stackrel{hom(\phi,Id_X)}{\to} hom(N(\Delta/(-)), X) \,.

                                      Properties

                                      Theorem

                                      If the simplicial object XX is Reedy cofibrant then its Bousfield–Kan map is a natural weak equivalence.

                                      If the cosimplicial object XX is Reedy fibrant then its Bousfield–Kan map is a natural weak equivalence.

                                      Proof

                                      This can be proven for instance using homotopy colimits in the Reedy model structure. Details are at Reedy model structure – over the simplex category .

                                      Relation to homotopy limits

                                      When the cosimplicial object XX is degreewise fibrant, then

                                      lim N(Δ/())XholimX lim^{N(\Delta/(-))} X \simeq holim X

                                      computes the homotopy limit of XX as a weighted limit (as explained there). Then the above theorem says that the homotopy limit is already computed by the totalization

                                      holimXlim ΔX. holim X \simeq lim^\Delta X \,.

                                      References

                                      The original reference is

                                      • Aldridge Bousfield and Dan Kan, Homotopy limits, completions and localizations Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.

                                      Reviews include

                                      • Hirschhorn, Simplicial model categories and their localization.

                                      The Bousfield–Kan map(s) are on p. 397, def. 18.7.1 and def. 18.7.3.

                                      Realization and totalization are defs 18.6.2 and 18.6.3 on p. 395.

                                      Notice that this book writes BB for the nerve!

                                      Last revised on September 19, 2011 at 13:20:06. See the history of this page for a list of all contributions to it.