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Cross effects and calculus in an unbased setting


Authors: Kristine Bauer, Brenda Johnson and Randy McCarthy; with an appendix by Rosona Eldred
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
Published electronically: November 4, 2014
MathSciNet review: 3356951
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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$.


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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
Email: eldred@uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06447-7
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
Article copyright: © Copyright 2014 American Mathematical Society