Elmendorf constructions for 𝐺-categories and 𝐺-posets

Rubin, Jonathan

doi:10.1090/proc/15563

Elmendorf constructions for $G$-categories and $G$-posets
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Proc. Amer. Math. Soc. 149 (2021), 4041-4056 Request permission

Abstract:

We introduce new Elmendorf constructions for equivariant categories and posets, and we prove that they are compatible with the classical topological one. Our constructions are more concrete than their model-categorical counterparts, and they give rise to new proofs of the Elmendorf theorems for equivariant categories and posets.

References

A. J. Blumberg and M. A. Hill, Equivariant stable categories for incomplete systems of transfers, J. London Math. Soc. (2021), DOI 10.1112/jlms.12441.
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
Anna Marie Bohmann, Kristen Mazur, Angélica M. Osorno, Viktoriya Ozornova, Kate Ponto, and Carolyn Yarnall, A model structure on $G\mathcal Cat$, Women in topology: collaborations in homotopy theory, Contemp. Math., vol. 641, Amer. Math. Soc., Providence, RI, 2015, pp. 123–134. MR 3380072, DOI 10.1090/conm/641/12861
A. D. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), no. 1, 275–284. MR 690052, DOI 10.1090/S0002-9947-1983-0690052-0
P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125
Daniel M. Kan, On c. s. s. complexes, Amer. J. Math. 79 (1957), 449–476. MR 90047, DOI 10.2307/2372558
J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
Peter May, Inna Zakharevich, and Marc Stephan, The homotopy theory of equivariant posets, Cah. Topol. Géom. Différ. Catég. 58 (2017), no. 2, 82–114 (English, with English and French summaries). MR 3700671
Robert J. Piacenza, Homotopy theory of diagrams and CW-complexes over a category, Canad. J. Math. 43 (1991), no. 4, 814–824. MR 1127031, DOI 10.4153/CJM-1991-046-3
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
George Raptis, Homotopy theory of posets, Homology Homotopy Appl. 12 (2010), no. 2, 211–230. MR 2721035
Marc Stephan, On equivariant homotopy theory for model categories, Homology Homotopy Appl. 18 (2016), no. 2, 183–208. MR 3551501, DOI 10.4310/HHA.2016.v18.n2.a10
R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91–109. MR 510404, DOI 10.1017/S0305004100055535
R. W. Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle 21 (1980), no. 3, 305–324. MR 591388