Algebraic & Geometric Topology

Homotopy theory of $G$–diagrams and equivariant excision

Emanuele Dotto and Kristian Moi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let G be a finite group. We define a suitable model-categorical framework for G–equivariant homotopy theory, which we call G–model categories. We show that the diagrams in a G–model category which are equipped with a certain equivariant structure admit a model structure. This model category of equivariant diagrams supports a well-behaved theory of equivariant homotopy limits and colimits. We then apply this theory to study equivariant excision of homotopy functors.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 1 (2016), 325-395.

Dates
Received: 2 September 2014
Revised: 11 April 2015
Accepted: 7 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841110

Digital Object Identifier
doi:10.2140/agt.2016.16.325

Mathematical Reviews number (MathSciNet)
MR3470703

Zentralblatt MATH identifier
1341.55001

Subjects
Primary: 55N91: Equivariant homology and cohomology [See also 19L47] 55P91: Equivariant homotopy theory [See also 19L47]
Secondary: 55P65: Homotopy functors 55P42: Stable homotopy theory, spectra

Keywords
equivariant homotopy excision

Citation

Dotto, Emanuele; Moi, Kristian. Homotopy theory of $G$–diagrams and equivariant excision. Algebr. Geom. Topol. 16 (2016), no. 1, 325--395. doi:10.2140/agt.2016.16.325. https://projecteuclid.org/euclid.agt/1510841110


Export citation

References

  • J Adámek, J Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189, Cambridge Univ. Press (1994)
  • G Biedermann, B Chorny, O R öndigs, Calculus of functors and model categories, Adv. Math. 214 (2007) 92–115
  • G Biedermann, O R öndigs, Calculus of functors and model categories, II, Algebr. Geom. Topol. 14 (2014) 2853–2913
  • A,J Blumberg, Continuous functors as a model for the equivariant stable homotopy category, Algebr. Geom. Topol. 6 (2006) 2257–2295
  • A,K Bousfield, D,M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer, Berlin (1972)
  • W Chachólski, J Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. 736, Amer. Math. Soc. (2002)
  • E Dotto, Equivariant calculus of functors and $\mathbb{Z}/2$–analyticity of real algebraic $K$–theory, J. Inst. Math. Jussieu (2015) Online publication
  • D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177–201
  • D Dugger, D,C Isaksen, Topological hypercovers and $\mathbb{A}\sp 1$–realizations, Math. Z. 246 (2004) 667–689
  • D Dugger, B Shipley, Enriched model categories and an application to additive endomorphism spectra, Theory Appl. Categ. 18 (2007) 400–439
  • A,D Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983) 275–284
  • P,G Goerss, J,F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
  • T,G Goodwillie, Calculus, II: Analytic functors, $K\!$–Theory 5 (1991/92) 295–332
  • T,G Goodwillie, Calculus, III: Taylor series, Geom. Topol. 7 (2003) 645–711
  • P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
  • S Jackowski, J Słomińska, $G$–functors, $G$–posets and homotopy decompositions of $G$–spaces, Fund. Math. 169 (2001) 249–287
  • G,M Kelly, Structures defined by finite limits in the enriched context, I, Cahiers Topologie Géom. Différentielle 23 (1982) 3–42
  • M,A Mandell, J,P May, Equivariant orthogonal spectra and $S$–modules, Mem. Amer. Math. Soc. 755, Amer. Math. Soc. (2002)
  • J,P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 155, Amer. Math. Soc. (1975)
  • J,P May, Enriched model categories and presheaf categories, preprint (2013)
  • C Rezk, A streamlined proof of Goodwillie's $n$–excisive approximation, Algebr. Geom. Topol. 13 (2013) 1049–1051
  • E Riehl, Categorical homotopy theory, New Mathematical Monographs 24, Cambridge Univ. Press (2014)
  • S Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120 (1997) 77–104
  • S Schwede, Lecture notes on equivariant stable homotopy theory, unpublished (2013) Available at \setbox0\makeatletter\@url http://www.math.uni-bonn.de/people/schwede/ {\unhbox0
  • B Shipley, A convenient model category for commutative ring spectra, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory”, (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc. (2004) 473–483
  • M Stephan, Elmendorf's theorem for cofibrantly generated model categories, Master's thesis, ETH Zurich (2010) Available at \setbox0\makeatletter\@url http://www.math.ku.dk/~jg/students/stephan.msthesis.2010.pdf {\unhbox0
  • J Thévenaz, P,J Webb, Homotopy equivalence of posets with a group action, J. Combin. Theory Ser. A 56 (1991) 173–181
  • R,W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109
  • R Villarroel-Flores, Equivariant homotopy type of categories and preordered sets, PhD thesis, University of Minnesota (1999) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304524888 {\unhbox0
  • R Villarroel-Flores, The action by natural transformations of a group on a diagram of spaces, preprint (2004)
  • F Waldhausen, Algebraic $K$–theory of spaces, from: “Algebraic and geometric topology”, (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419