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Abstract:

We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular, this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen's Theorem A for posets. © 2016, Filipp Levikov. Permission to copy for private use granted.

Registro:

Documento: Artículo
Título:Homotopy colimits of diagrams over posets and variations on a theorem of Thomason
Autor:Fernández, X.; Minian, E.G.
Filiación:Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina
Palabras clave:Finite topological space; Grothendieck construction; Homotopy colimit; Poset; Quillen's Theorem A
Año:2016
Volumen:18
Número:2
Página de inicio:233
Página de fin:245
DOI: http://dx.doi.org/10.4310/hha.2016.v18.n2.a13
Título revista:Homology, Homotopy and Applications
Título revista abreviado:Homology Homotopy Applic.
ISSN:15320073
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15320073_v18_n2_p233_Fernandez

Referencias:

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  • Barmak, J.A., On Quillen's Theorem A for posets (2011) J. Combin. Theory Ser. A, 118, pp. 2445-2453
  • Barmak, J.A., Minian, E.G., Simple homotopy types and finite spaces (2008) Adv. Math, 218 (1), pp. 87-104
  • Barmak, J.A., Minian, E.G., One-point reductions of finite spaces, h-regular CW-complexes and collapsibility (2008) Algebr. Geom. Topol, 8 (3), pp. 1763-1780
  • Bousfield, A.K., Kan, D.M., Homotopy Limits, Completions and Localizations (1972) Lecture Notes in Math, 304
  • Chachólski, W., Scherer, J., Homotopy Theory of Diagrams (2002) Mem. Amer. Math. Soc, 155 (736), pp. x and 90
  • Cohen, M.M., Simplicial structures and transverse cellularity (1967) Ann. of Math, 85, pp. 218-245
  • Maltsiniotis, G., La théorie de l'homotopie de Grothendieck (2005) Astérisque, 301, pp. vi and 140
  • May, J.P., Finite topological spaces (2003) Notes for REU, Chicago, , http://www.math.uchicago.edu/~may/MISCMaster.html
  • May, J.P., (2003) Finite spaces and simplicial complexes, , http://www.math.uchicago.edu/~may/MISCMaster.html, Notes for REU, Chicago
  • May, J.P., (2012) Finite Spaces and Larger Contexts, , http://math.uchicago.edu/~may/REU2012/, Book in progress, Chicago
  • McCord, M.C., Singular homology groups and homotopy groups of finite topological spaces (1966) Duke Math. J, 33, pp. 465-474
  • Quillen, D., Higher algebraic K-theory. I: Higher K-theories (1972) Lecture Notes in Math, 341, pp. 85-147
  • Stong, R.E., Finite topological spaces (1966) Trans. Amer. Math. Soc, 123, pp. 325-340
  • Thomason, R.W., Homotopy colimits in the category of small categories (1979) Math. Proc. Cambridge Philos. Soc, 85 (1), pp. 91-109
  • Vogt, R.M., Homotopy limits and colimits (1973) Math. Z, 134, pp. 11-52
  • Welker, V., Ziegler, G., Zivaljević, R., Homotopy colimits-comparison lemmas for combinatorial applications (1999) J. Reine Angew. Math, 509, pp. 117-149

Citas:

---------- APA ----------
Fernández, X. & Minian, E.G. (2016) . Homotopy colimits of diagrams over posets and variations on a theorem of Thomason. Homology, Homotopy and Applications, 18(2), 233-245.
http://dx.doi.org/10.4310/hha.2016.v18.n2.a13
---------- CHICAGO ----------
Fernández, X., Minian, E.G. "Homotopy colimits of diagrams over posets and variations on a theorem of Thomason" . Homology, Homotopy and Applications 18, no. 2 (2016) : 233-245.
http://dx.doi.org/10.4310/hha.2016.v18.n2.a13
---------- MLA ----------
Fernández, X., Minian, E.G. "Homotopy colimits of diagrams over posets and variations on a theorem of Thomason" . Homology, Homotopy and Applications, vol. 18, no. 2, 2016, pp. 233-245.
http://dx.doi.org/10.4310/hha.2016.v18.n2.a13
---------- VANCOUVER ----------
Fernández, X., Minian, E.G. Homotopy colimits of diagrams over posets and variations on a theorem of Thomason. Homology Homotopy Applic. 2016;18(2):233-245.
http://dx.doi.org/10.4310/hha.2016.v18.n2.a13