Cosimplicial invariants and Calculus of homotopy functors

Goodwillie Calculus;Homotopy Theory;Algebraic Topology;homotopy functor;homotopy limit

Eldred, Rosona

关键词: Goodwillie Calculus;    Homotopy Theory;    Algebraic Topology;    homotopy functor;    homotopy limit;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/29665/Eldred_Rosona.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The origin of these investigations was the successful attempt by myself and coauthors to generalize rational equivalences of two constructions which suggest possible definitions of deRham cohomology of “brave new” rings, one of Rezk (using a cosimplicial resolution) and the other by Waldhausen (using a variant of Goodwillie’s Taylor tower). The proof of agreement of these constructions relies heavily on the fact that the functors involved take values in Spectra.Goodwillie conjectured the extension of these results to include the case of functors taking value in Spaces. The main result of my thesis is a proof of this conjecture (Theorem 7.1.2), using significantly different methods than in the stable setting of the joint work. This makes strong use of the intermediate constructions T_n F in Goodwillie’s Calculus of homotopy functors. I give a new model which naturally gives rise to a new family of towers filtering the Taylor Tower of a functor. I also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor F under general assumptions.