Cross effects and calculus in an unbased setting
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- by Kristine Bauer, Brenda Johnson and Randy McCarthy; with an appendix by Rosona Eldred PDF
- Trans. Amer. Math. Soc. 367 (2015), 6671-6718 Request permission
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 $\mathcal {C}_f$.References
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Additional Information
- Kristine Bauer
- Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
- Email: bauerk@ucalgary.ca
- Brenda Johnson
- Affiliation: Department of Mathematics, Union College, 807 Union Street, Schenectady, New York 12308
- Email: johnsonb@union.edu
- Randy McCarthy
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801-2907
- Email: rmccrthy@illinois.edu
- Rosona Eldred
- Affiliation: Mathematisches Institut, University of Muenster, Einsteinstrasse 62, 48149 Münster, Germany
- MR Author ID: 1015629
- Email: eldred@uni-muenster.de
- Received by editor(s): January 5, 2011
- Received by editor(s) in revised form: June 29, 2012, February 6, 2013, April 11, 2013, and February 28, 2014
- Published electronically: November 4, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 6671-6718
- MSC (2010): Primary 55P65; Secondary 18G55, 18G30
- DOI: https://doi.org/10.1090/S0002-9947-2014-06447-7
- MathSciNet review: 3356951