Geodesic
Homotopy theory of G–diagrams and equivariant excision
Dotto, Emanuele1 ; Moi, Kristian2
1Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
2Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 325-395
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2016.16.325
Classification : 55N91, 55P91, 55P65, 55P42
Keywords: equivariant homotopy, excision

Dotto, Emanuele  1   ; Moi, Kristian  2

1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
2 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark 
@article{10_2140_agt_2016_16_325,
     author = {Dotto, Emanuele and Moi, Kristian},
     title = {Homotopy theory of {G{\textendash}diagrams} and equivariant excision},
     journal = {Algebraic and Geometric Topology},
     pages = {325--395},
     year = {2016},
     volume = {16},
     number = {1},
     doi = {10.2140/agt.2016.16.325},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.325/}
}
TY  - JOUR
AU  - Dotto, Emanuele
AU  - Moi, Kristian
TI  - Homotopy theory of G–diagrams and equivariant excision
JO  - Algebraic and Geometric Topology
PY  - 2016
SP  - 325
EP  - 395
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.325/
DO  - 10.2140/agt.2016.16.325
ID  - 10_2140_agt_2016_16_325
ER  - 
%0 Journal Article
%A Dotto, Emanuele
%A Moi, Kristian
%T Homotopy theory of G–diagrams and equivariant excision
%J Algebraic and Geometric Topology
%D 2016
%P 325-395
%V 16
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.325/
%R 10.2140/agt.2016.16.325
%F 10_2140_agt_2016_16_325
Dotto, Emanuele; Moi, Kristian. Homotopy theory of G–diagrams and equivariant excision. Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 325-395. doi: 10.2140/agt.2016.16.325

[1] J Adámek, J Rosický, Locally presentable and accessible categories, 189, Cambridge Univ. Press (1994)

[2] G Biedermann, B Chorny, O Röndigs, Calculus of functors and model categories, Adv. Math. 214 (2007) 92

[3] G Biedermann, O Röndigs, Calculus of functors and model categories, II, Algebr. Geom. Topol. 14 (2014) 2853

[4] A J Blumberg, Continuous functors as a model for the equivariant stable homotopy category, Algebr. Geom. Topol. 6 (2006) 2257

[5] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972)

[6] W Chachólski, J Scherer, Homotopy theory of diagrams, 736, Amer. Math. Soc. (2002)

[7] E Dotto, Equivariant calculus of functors and Z∕2–analyticity of real algebraic K–theory, J. Inst. Math. Jussieu (2015)

[8] D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177

[9] D Dugger, D C Isaksen, Topological hypercovers and A1–realizations, Math. Z. 246 (2004) 667

[10] D Dugger, B Shipley, Enriched model categories and an application to additive endomorphism spectra, Theory Appl. Categ. 18 (2007) 400

[11] A D Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983) 275

[12] P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999)

[13] T G Goodwillie, Calculus, II : Analytic functors, K–Theory 5 (1991/92) 295

[14] T G Goodwillie, Calculus, III : Taylor series, Geom. Topol. 7 (2003) 645

[15] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003)

[16] S Jackowski, J Słomińska, G–functors, G–posets and homotopy decompositions of G–spaces, Fund. Math. 169 (2001) 249

[17] G M Kelly, Structures defined by finite limits in the enriched context, I, Cahiers Topologie Géom. Différentielle 23 (1982) 3

[18] M A Mandell, J P May, Equivariant orthogonal spectra and S–modules, 755, Amer. Math. Soc. (2002)

[19] J P May, Classifying spaces and fibrations, 155, Amer. Math. Soc. (1975)

[20] J P May, Enriched model categories and presheaf categories, preprint (2013)

[21] C Rezk, A streamlined proof of Goodwillie’s n–excisive approximation, Algebr. Geom. Topol. 13 (2013) 1049

[22] E Riehl, Categorical homotopy theory, 24, Cambridge Univ. Press (2014)

[23] S Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120 (1997) 77

[24] S Schwede, Lecture notes on equivariant stable homotopy theory, unpublished (2013)

[25] B Shipley, A convenient model category for commutative ring spectra, from: "Homotopy theory : relations with algebraic geometry, group cohomology, and algebraic K–theory" (editors P Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 473

[26] M Stephan, Elmendorf’s theorem for cofibrantly generated model categories, Master’s thesis, ETH Zurich (2010)

[27] J Thévenaz, P J Webb, Homotopy equivalence of posets with a group action, J. Combin. Theory Ser. A 56 (1991) 173

[28] R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91

[29] R Villarroel-Flores, Equivariant homotopy type of categories and preordered sets, PhD thesis, University of Minnesota (1999)

[30] R Villarroel-Flores, The action by natural transformations of a group on a diagram of spaces, preprint (2004)

[31] F Waldhausen, Algebraic K–theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318

Cité par Sources :