Skip to main content
Top
Published in: Applied Categorical Structures 1/2021

10-10-2020

Gabriel–Zisman Cohomology and Spectral Sequences

Authors: Imma Gálvez-Carrillo, Frank Neumann, Andrew Tonks

Published in: Applied Categorical Structures | Issue 1/2021

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference André, M.: Limites et fibrés. C. R. Acad. Sci. Paris, Sér. A. 260, 756–759 (1965)MATH André, M.: Limites et fibrés. C. R. Acad. Sci. Paris, Sér. A. 260, 756–759 (1965)MATH
3.
go back to reference Beilinson, A.A.: Residues and adeles. Funct. Anal. Pril. 14(1), 44–45 (1980); English transl. in: Funct. Anal. Appl. 14(1), 34–35 (1980) Beilinson, A.A.: Residues and adeles. Funct. Anal. Pril. 14(1), 44–45 (1980); English transl. in: Funct. Anal. Appl. 14(1), 34–35 (1980)
4.
go back to reference Bergner, J.E., Osorno, A.M., Ozornova, V., Rovelli, M., Scheimbauer, C.I.: The edgewise subdivision criterion for 2-Segal objects. Proc. Am. Math. Soc. 148, 71–82 (2020)MathSciNetCrossRef Bergner, J.E., Osorno, A.M., Ozornova, V., Rovelli, M., Scheimbauer, C.I.: The edgewise subdivision criterion for 2-Segal objects. Proc. Am. Math. Soc. 148, 71–82 (2020)MathSciNetCrossRef
5.
go back to reference Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)CrossRef Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)CrossRef
7.
go back to reference Ciblis, C., Redondo, M.J.: Cartan–Leray spectral sequence for Galois coverings of linear categories. J. Algebra 284, 310–325 (2005)MathSciNetCrossRef Ciblis, C., Redondo, M.J.: Cartan–Leray spectral sequence for Galois coverings of linear categories. J. Algebra 284, 310–325 (2005)MathSciNetCrossRef
8.
go back to reference Cisinski, D.-C.: Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10(2), 195–244 (2003)MathSciNetCrossRef Cisinski, D.-C.: Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10(2), 195–244 (2003)MathSciNetCrossRef
10.
go back to reference Dyckerhoff, T., Kapranov, M.: Higher Segal Spaces. Lecture Notes in Mathematics, vol. 2244. Springer, Berlin (2019)CrossRef Dyckerhoff, T., Kapranov, M.: Higher Segal Spaces. Lecture Notes in Mathematics, vol. 2244. Springer, Berlin (2019)CrossRef
11.
12.
go back to reference Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer, New York (1967)CrossRef Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer, New York (1967)CrossRef
13.
go back to reference Gálvez-Carrillo, I., Kock, J., Tonks, A.: Decomposition spaces, incidence algebras and Möbius inversion I: basic theory. Adv. Math. 331, 952–1015 (2018)MathSciNetCrossRef Gálvez-Carrillo, I., Kock, J., Tonks, A.: Decomposition spaces, incidence algebras and Möbius inversion I: basic theory. Adv. Math. 331, 952–1015 (2018)MathSciNetCrossRef
14.
go back to reference Gálvez-Carrillo, I., Neumann, F., Tonks, A.: André spectral sequences for Baues-Wirsching cohomology of categories. J. Pure Appl. Algebra 216, 2549–2561 (2012)MathSciNetCrossRef Gálvez-Carrillo, I., Neumann, F., Tonks, A.: André spectral sequences for Baues-Wirsching cohomology of categories. J. Pure Appl. Algebra 216, 2549–2561 (2012)MathSciNetCrossRef
15.
go back to reference Gálvez-Carrillo, I., Neumann, F., Tonks, A.: Thomason cohomology of categories. J. Pure Appl. Algebra 217, 2163–2179 (2013)MathSciNetCrossRef Gálvez-Carrillo, I., Neumann, F., Tonks, A.: Thomason cohomology of categories. J. Pure Appl. Algebra 217, 2163–2179 (2013)MathSciNetCrossRef
16.
go back to reference Gelfand, S.I., Manin, YuI: Methods of Homological Algebra. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2003)CrossRef Gelfand, S.I., Manin, YuI: Methods of Homological Algebra. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2003)CrossRef
17.
go back to reference Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory, Progress in Mathematics, vol. 174. Birkhäuser Verlag, Basel (1999)CrossRef Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory, Progress in Mathematics, vol. 174. Birkhäuser Verlag, Basel (1999)CrossRef
18.
go back to reference Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. (2) 9, 119–221 (1957)MathSciNetMATH Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. (2) 9, 119–221 (1957)MathSciNetMATH
19.
go back to reference Grothendieck, A.: Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à 11. Séminaire de Géométrie Algébrique (SGA1), 1960/61. Troisième édition, corrigée. Publications Institut des Hautes Études Scientifiques, Paris (1963) Grothendieck, A.: Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à 11. Séminaire de Géométrie Algébrique (SGA1), 1960/61. Troisième édition, corrigée. Publications Institut des Hautes Études Scientifiques, Paris (1963)
20.
21.
23.
go back to reference Illusie, L.: Complexe Cotangent et Déformations II. Lecture Notes in Mathematics, vol. 283. Springer, Berlin (1972)MATH Illusie, L.: Complexe Cotangent et Déformations II. Lecture Notes in Mathematics, vol. 283. Springer, Berlin (1972)MATH
24.
go back to reference Latch, D.M.: The uniqueness of homology for the category of small categories. J. Pure Appl. Algebra 9, 221–237 (1977)MathSciNetCrossRef Latch, D.M.: The uniqueness of homology for the category of small categories. J. Pure Appl. Algebra 9, 221–237 (1977)MathSciNetCrossRef
25.
go back to reference MacLane, S.: Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics, vol. 5. Springer, New York (1998) MacLane, S.: Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)
27.
go back to reference Parshin, A.N.: On the arithmetic of two-dimensional schemes I, repartitions and residues. Izv. Akad. Nauk SSSR Ser. Mat. 40(4), 736–773 (1976); English transl. in: Math. USSR Izv. 10(4), 695–729 (1976) Parshin, A.N.: On the arithmetic of two-dimensional schemes I, repartitions and residues. Izv. Akad. Nauk SSSR Ser. Mat. 40(4), 736–773 (1976); English transl. in: Math. USSR Izv. 10(4), 695–729 (1976)
28.
29.
go back to reference Quillen, D.: Higher algebraic \(K\)-theory. I. Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147. Lecture Notes in Math. 341, Springer, Berlin (1973) Quillen, D.: Higher algebraic \(K\)-theory. I. Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147. Lecture Notes in Math. 341, Springer, Berlin (1973)
30.
go back to reference Richter, B.: From Categories to Homotopy Theory, Cambridge Studies in Advanced Mathematics, vol. 188. Cambridge University Press, Cambridge (2020)CrossRef Richter, B.: From Categories to Homotopy Theory, Cambridge Studies in Advanced Mathematics, vol. 188. Cambridge University Press, Cambridge (2020)CrossRef
31.
go back to reference Roos, J.E.: Sur les foncteurs dérivés de \(\lim \). Applications. C. R. Acad. Sci. Paris, Sér. A. 252, 3702–3704 (1961)MATH Roos, J.E.: Sur les foncteurs dérivés de \(\lim \). Applications. C. R. Acad. Sci. Paris, Sér. A. 252, 3702–3704 (1961)MATH
32.
go back to reference Schneider, P., Stuhler, U.: Representation theory and sheaves on the Bruhat–Tits building. Publ. Math. IHÉS 85, 97–191 (1997)MathSciNetCrossRef Schneider, P., Stuhler, U.: Representation theory and sheaves on the Bruhat–Tits building. Publ. Math. IHÉS 85, 97–191 (1997)MathSciNetCrossRef
33.
Metadata
Title
Gabriel–Zisman Cohomology and Spectral Sequences
Authors
Imma Gálvez-Carrillo
Frank Neumann
Andrew Tonks
Publication date
10-10-2020
Publisher
Springer Netherlands
Published in
Applied Categorical Structures / Issue 1/2021
Print ISSN: 0927-2852
Electronic ISSN: 1572-9095
DOI
https://doi.org/10.1007/s10485-020-09609-7

Other articles of this Issue 1/2021

Applied Categorical Structures 1/2021 Go to the issue

Premium Partner