Waldhausen K-theory of spaces via comodules

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• 作者：Kathryn Hess^a ; ^{kathryn.hess@epfl.ch" class="auth_mail" title="E-mail the corresponding author} ; Brooke Shipley^b ; ^{bshipley@math.uic.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 55U35 ; secondary ; 16T15 ; 18C15 ; 19D10 ; 16T15 ; 55P42 ; 55P43
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：26 February 2016
• 年：2016
• 卷：290
• 期：Complete
• 页码：1079-1137
• 全文大小：870 K

文摘

Let X   be a simplicial set. We construct a novel adjunction between the categories R_X of retractive spaces over X   and Comod_X+ of X₊$X_{+}$-comodules, then apply recent work on left-induced model category structures  and  to establish the existence of a left proper, simplicial model category structure on Comod_X+ with respect to which the adjunction is a Quillen equivalence after localization with respect to some generalized homology theory E_⁎${\mathcal{E}}_{⁎}$. We show moreover that this model category structure on Comod_X+ stabilizes, giving rise to a model category structure on Comod_Σ^∞X+, the category of Σ^∞X₊${\mathrm{\Sigma }}^{\infty }{X}_{+}$-comodule spectra.

It follows that the Waldhausen K-theory of X  , A(X)$A(X)$, is naturally weakly equivalent to the Waldhausen K  -theory of , the category of homotopically finite Σ^∞X₊${\mathrm{\Sigma }}^{\infty }{X}_{+}$-comodule spectra, where the weak equivalences are given by twisted homology. For X simply connected, we exhibit explicit, natural weak equivalences between the K  -theory of and that of the category of homotopically finite Σ^∞(ΩX)₊${\mathrm{\Sigma }}^{\infty }{(\mathrm{\Omega }X)}_{+}$-modules, a more familiar model for A(X)$A(X)$. For X   not necessarily simply connected, we have E_⁎${\mathcal{E}}_{⁎}$-local versions of these results for any generalized homology theory E_⁎${\mathcal{E}}_{⁎}$.

For H   a simplicial monoid, Comod_Σ^∞H+ admits a monoidal structure and induces a model structure on the category Alg_Σ^∞H+ of Σ^∞H₊${\mathrm{\Sigma }}^{\infty }{H}_{+}$-comodule algebras. This provides a setting for defining homotopy coinvariants   of the coaction of Σ^∞H₊${\mathrm{\Sigma }}^{\infty }{H}_{+}$ on a Σ^∞H₊${\mathrm{\Sigma }}^{\infty }{H}_{+}$-comodule algebra, which is essential for homotopic Hopf–Galois extensions of ring spectra as originally defined by Rognes [27] and generalized in [15]. An algebraic analogue of this was only recently developed, and then only over a field [5].