Cross effects and calculus in an unbased setting

Kristine Bauer, Brenda Johnson, Randy McCarthy, Rosona Eldred

Research output: Contribution to journalArticlepeer-review


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 .

Original languageEnglish (US)
Pages (from-to)6671-6718
Number of pages48
JournalTransactions of the American Mathematical Society
Issue number9
StatePublished - 2015

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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