TY - JOUR

T1 - Cross effects and calculus in an unbased setting

AU - Bauer, Kristine

AU - Johnson, Brenda

AU - McCarthy, Randy

AU - Eldred, Rosona

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2015

Y1 - 2015

N2 - We study functors F : Cf → D where C and D are simplicial model categories and Cf is the category consisting of objects that factor a fixed morphism f : A → B in C. We define the analogs of Eilenberg and Mac Lane’s cross effect functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of Deriving calculus with cotriples (by the second and third authors) from the setting of functors from pointed categories to abelian categories to that of functors from Cf to S, a suitable category of spectra, to produce a tower of functors · · · → Γn+1F → ΓnF → Γn−1F → ··· → F(B) whose nth term is a degree n functor. We compare this tower to Goodwillie’s tower, · · · → Pn+1F → PnF → Pn−1F → · · · → F(B), of n-excisive approximations to F found in his work Calculus II. When F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of Cf .

AB - We study functors F : Cf → D where C and D are simplicial model categories and Cf is the category consisting of objects that factor a fixed morphism f : A → B in C. We define the analogs of Eilenberg and Mac Lane’s cross effect functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of Deriving calculus with cotriples (by the second and third authors) from the setting of functors from pointed categories to abelian categories to that of functors from Cf to S, a suitable category of spectra, to produce a tower of functors · · · → Γn+1F → ΓnF → Γn−1F → ··· → F(B) whose nth term is a degree n functor. We compare this tower to Goodwillie’s tower, · · · → Pn+1F → PnF → Pn−1F → · · · → F(B), of n-excisive approximations to F found in his work Calculus II. When F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of Cf .

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U2 - 10.1090/s0002-9947-2014-06447-7

DO - 10.1090/s0002-9947-2014-06447-7

M3 - Article

AN - SCOPUS:84931079405

SN - 0002-9947

VL - 367

SP - 6671

EP - 6718

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 9

ER -