Cosimplicial models for the limit of the Goodwillie tower

Eldred, Rosona¹

¹Department of Mathematics, University of Hamburg, Bundesstr. 55, D-20146 Hamburg, Germany

Algebraic and Geometric Topology, Tome 13 (2013) no. 2, pp. 1161-1182

Résumé

We call attention to the intermediate constructions TnF in Goodwillie’s Calculus of homotopy functors, giving a new model which naturally gives rise to a family of towers filtering the Taylor tower of a functor. We also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor F under general assumptions.

Détail

DOI : 10.2140/agt.2013.13.1161

Classification : 55P65, 55P60, 55P10
Keywords: cosimplicial, Goodwillie Calculus, homotopy functor, homotopy limit, cofinal

Bibliographie

[1] G Arone, M Kankaanrinta, A functorial model for iterated Snaith splitting with applications to calculus of functors, from: "Stable and unstable homotopy (Toronto, ON, 1996)", Fields Inst. Commun. 19, Amer. Math. Soc. (1998) 1

[2] K Bauer, R Eldred, B Johnson, R Mccarthy, Geometric models for Taylor polynomials of functors, in preparation

[3] K Bauer, B Johnson, R Mccarthy, R Eldred, A cotriple model for calculus in an unbased setting, submitted to Trans. Amer. Math. Soc.

[4] A K Bousfield, E M Friedlander, Homotopy theory of $\Gamma $–spaces, spectra, and bisimplicial sets, from: "Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II", Lecture Notes in Math. 658, Springer (1978) 80

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

[6] D Dugger, A primer on homotopy colimits, preprint (2008)

[7] W G Dwyer, D M Kan, An obstruction theory for diagrams of simplicial sets, Nederl. Akad. Wetensch. Indag. Math. 46 (1984) 139

[8] S Eilenberg, N Steenrod, Foundations of algebraic topology, Princeton Univ. Press (1952)

[9] R Eldred, Cosimplicial invariants and Calculus of homotopy functors, PhD thesis, University of Illinois at Urbana-Champaign (2011)

[10] P G Goerss, Barratt's desuspension spectral sequence and the Lie ring analyzer, Quart. J. Math. Oxford Ser. 44 (1993) 43

[11] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser (1999)

[12] T G Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, $K$–Theory 4 (1990) 1

[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, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)

[16] M J Hopkins, Formulations of cocategory and the iterated suspension, from: "Algebraic homotopy and local algebra (Luminy, 1982)", Astérisque 113, Soc. Math. France (1984) 212

[17] M J Hopkins, Some problems in topology, DPhil thesis, University of Oxford (1984)

[18] B Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347 (1995) 1295

[19] A J Mauer-Oats, Goodwillie calculi, PhD thesis, University of Illinois at Urbana-Champaign (2002)

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

[21] B E Shipley, Convergence of the homology spectral sequence of a cosimplicial space, Amer. J. Math. 118 (1996) 179

[22] D P Sinha, The topology of spaces of knots: cosimplicial models, Amer. J. Math. 131 (2009) 945