Cross effects and calculus in an unbased setting

Abstract:

We study functors $F:\mathcal {C}_f\rightarrow \mathcal {D}$ where $\mathcal {C}$ and $\mathcal {D}$ are simplicial model categories and $\mathcal {C}_f$ is the category consisting of objects that factor a fixed morphism $f:A\rightarrow B$ in $\mathcal {C}$. 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 $\mathcal {C}_f$ to $\mathcal {S}$, a suitable category of spectra, to produce a tower of functors $\dots \rightarrow \Gamma _{n+1}F\rightarrow \Gamma _nF\rightarrow \Gamma _{n-1}F\rightarrow \dots \rightarrow F(B)$ whose $n$th term is a degree $n$ functor. We compare this tower to Goodwillie’s tower, $\dots \rightarrow P_{n+1}F\rightarrow P_nF\rightarrow P_{n-1}F\rightarrow \dots \rightarrow F(B)$, 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 $\mathcal {C}_f$.