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 -