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Gabriel–Zisman Cohomology and Spectral Sequences

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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.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Imma Gálvez-Carrillo

  2. School of Mathematics and Actuarial Science, Pure Mathematics Group, University of Leicester, University Road, Leicester, LE1 7RH, UK

    Frank Neumann & Andrew Tonks

Authors
  1. Imma Gálvez-Carrillo
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  2. Frank Neumann
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  3. Andrew Tonks
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Corresponding author

Correspondence to Andrew Tonks.

Additional information

Communicated by M. Batanin.

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Imma Gálvez-Carrillo was partially supported by Spanish Ministry of Science and Catalan government Grants PID2019-103849GB-I00, 2017 SGR 932, MTM2017-90897-REDT, MTM2016-76453-C2-2-P (AEI/FEDER, UE), MTM2015-69135-P, and Andrew Tonks by MTM2016-76453-C2-2-P (AEI/FEDER, UE) all of which are gratefully acknowledged. Frank Neumann thanks the Centre de Recerca Matemàtica (CRM) in Bellaterra, Spain for inviting him during the research programme Homotopy Theory and Higher Categories (HOCAT), where this work was initiated.

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Gálvez-Carrillo, I., Neumann, F. & Tonks, A. Gabriel–Zisman Cohomology and Spectral Sequences. Appl Categor Struct 29, 69–94 (2021). https://doi.org/10.1007/s10485-020-09609-7

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