Goodwillie calculus and algebras over a spectral operad

133 Pages·2013·7.06 MB·English

Goodwillie calculus and algebras over a spectral
operad
by
Luis Alexandre Meira Fernandes Alves Pereira
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of ARCHNES
Ph.D. in Mathematics OF TECHNOLOGY
JUL 2 5 2013
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY LIBRARIES
June 2013
@ Luis Alexandre Meira Fernandes Alves Pereira, MMXIII. All rights
reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
Author ............................. ................
Department of Mathematics
-1 April 29, 2013
Certified by.........
Mark J. Behrens
Associate Professor
Thesis Supervisor
Accepted by ........ .........
Paul Seidel
Chairman, Department Committee on Graduate Theses
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Goodwillie calculus and algebras over a spectral operad
by
Lufs Alexandre Meira Fernandes Alves Pereira
Submitted to the Department of Mathematics
on April 29, 2013, in partial fulfillment of the
requirements for the degree of
Ph.D. in Mathematics
Abstract
The overall goal of this thesis is to apply the theory of Goodwillie calculus to the
category Algo of algebras over a spectral operad.
Its first part generalizes many of the original results of Goodwillie in [14] so that
they apply to a larger class of model categories and hence be applicable to Algo.
The second part then applies that generalized theory to the Algo categories. The
main results here are: an understanding of finitary homogeneous between such cate-
gories; identifying the Taylor tower of the identity in those categories; showing that
finitary n-excisive functors can not distinguish between Algo and Algo,, the cate-
gory of algebras over the truncated operad O<; and a weak form of the chain rule
between the algebra categories, analogous to the one found in [1].
Thesis Supervisor: Mark J. Behrens
Title: Associate Professor
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Acknowledgments
This work was in part supported by funding from the Fundagso para a Ciencia e a
Teenologia.
Other than that, I thank my advisor Mark Behrens for putting up with me for
these past few years, and my annoying friends and family for being so damn annoying
in a sometimes mildly endearing way. As they know who they are, and as I'd rather
not risk incurring anyone's wrath by forgetting a name, I'll leave it at that.
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Contents
1 Overview 13
1.1 Part . .................. .......... 13
1.2 Part 1......................... ........ 15
I General Goodwillie calculus 17
2 Setup for homotopy calculus 19
2.1 Cofibrant generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Functors .... ... .. ....................... 25
2.3 Simplicial categories . ......................... 25
2.4 Spectra on model categories ....................... 26
2.5 Composing functors with Quillen equivalences . . . . . . . . . . . . . 29
3 Goodwillie calculus: polynomial functors 31
3.1 Cartesian cubes and total homotopy fibers. . . . . . . . . . . . . . . . 31
3.2 Excisive functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 The Taylor tower 41
5 Goodwillie calculus and stable categories 47
5.1 Hofibers/hocofibers detect weak equivalences . . . . . . . . . . . . . . 47
5.2 Cross-effects . . . . . . . . . . . . . . . . ................ 50
6 Delooping homogeneous functors 57
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7 Factoring multilinear symmetric functors through spectra 61
II Goodwillie calculus in Algo 71
8 Basic definitions 73
8.1 Symmetric Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.2 Stable model structures on SpE . . . . . . . . . . . . . . . . . . . . . 74
8.3 Operads and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.4 Model structures on Algo . . . . . . . . . . . . . . . . . . . . . . . . 79
8.5 Spectra in spectra ... . ........................ 80
9 Spectra on A1g 83
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10 Homogeneous functors between ModA and ModB 95
10.1 Linear simplicial functors are spectral functors . . . . . . . . . . . . . 96
10.2 Characterization of n-homogeneous finitary functors .......... 107
11 Goodwillie Calculus in the Algo categories 113
11.1 The Goodwillie tower of IdAg . . . . . . . . . . . . . . . . . . . . . 114
.
11.2 n-excisive finitary functors can be "truncated" . . . . . . . . . . . . . 117
11.3 Proto chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A Cofibrancy of Q'(i) 125
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List of Figures
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