Cross effects and calculus in an unbased setting

We study functors F from C_f to D where C and D are simplicial model categories and C_f is the full subcategory of C consisting of objects that factor a fixed morphism f from A to B. We define the analogs of Eilenberg and Mac Lane's cross effects 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 [10] from the setting of functors from pointed categories to abelian categories to that of functors from C_f to D to produce a tower of functors whose n-th term is a degree n functor. We compare this tower to Goodwillie's tower of n-excisive approximations to F found in [8]. When D is a good category of spectra, and 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 C_f.

[1]  A. Mauer-Oats.  Goodwillie Calculi , 2013, 1304.5662.

[2]  Michael Cole,et al.  Rings, Modules, and Algebras in Stable Homotopy Theory , 2007 .

[3]  P. Goerss,et al.  Model Categories and Simplicial Methods , 2006, math/0609537.

[4]  W. Dwyer,et al.  Homotopy theory of small diagrams over large categories , 2006, math/0607117.

[5]  N. Kuhn.  Goodwillie towers and chromatic homotopy: an overview , 2004, math/0410342.

[6]  T. Goodwillie.  Calculus III: Taylor Series , 2003, math/0310481.

[7]  R. McCarthy,et al.  Deriving calculus with cotriples , 2003 .

[8]  A. Mauer-Oats.  Algebraic Goodwillie calculus and a cotriple model for the remainder , 2002, math/0212095.

[9]  D. Sinha.  The topology of spaces of knots: cosimplicial models , 2002, math/0202287.

[10]  J. Scherer,et al.  Homotopy theory of diagrams , 2001, math/0110316.

[11]  Mark Hovey,et al.  Symmetric spectra , 1998, math/9801077.

[12]  T. Goodwillie.  Calculus II: Analytic functors , 1991 .

[13]  D. M. Kan,et al.  Homotopy Limits, Completions and Localizations , 1987 .

[14]  Saunders MacLane,et al.  On the Groups H(Π, n), II: Methods of Computation , 1954 .

[15]  J. Jardine,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[16]  Daniel Dugger.  A PRIMER ON HOMOTOPY COLIMITS , 2008 .

[17]  Michael A. Mandell,et al.  Interactions between Homotopy Theory and Algebra , 2007 .

[18]  Philip S. Hirschhorn.  Model categories and their localizations , 2003 .

[19]  C. L. REEDY,et al.  HOMOTOPY THEORY OF MODEL CATEGORIES , 1974 .

[20]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[21]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .