nLab Baues-Wirsching cohomology

The development of the cohomology of categories.

The cohomology of (small) categories? went through several preliminary stages before the publication by Baues and Wirshing. In fact, the actual method and ideas that they use can be found earlier in an unpublished document by Charles Wells from 1979. That paper traces the theory back to Leech earlier in the 1970s for the related situation of cohomology of monoids. It handles non-Abelian coefficients as well as the Abelian ones that the Baues and Wirsching theory uses.



  • Jonathan Leech, \mathcal{H}-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975.

  • Charles Wells, Extension theories for categories (preliminary report), (available from here, 1979.)

Baues-Wirsching cohomology of small categories: Explicit references,

  • Hans Joachim Baues, Günther Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187–211, doi, MR87g:18013

  • Hans Joachim Baues, Winfried Dreckmann, The cohomology of homotopy categories and the general linear group, KK-Theory 3 (1989), no. 4, 307–338, doi, MR91d:18008

  • H. J. Baues, F. Muro, The homotopy category of pseudofunctors and translation cohomology, J. Pure Appl. Algebra 211 (2007), no. 3, 821–850, doi MR2008g:18009

  • Hans Joachim Baues, On the cohomology of categories, universal Toda brackets and homotopy pairs, KK-Theory 11 (1997), no. 3, 259–285, doi, MR98h:55020

  • Petar Pavešić, Diagram cohomologies using categorical fibrations, J. Pure Appl. Algebra 112 (1996), no. 1, 73–90, [doi]

  • Teimuraz Pirashvili, On the center and Baues-Wirsching cohomology of small categories, Georgian Math. J. 16 (2009), no. 1, 131–144, journal

  • Mamuka Jibladze, Teimuraz Pirashvili, Quillen cohomology and Baues-Wirsching cohomology of algebraic theories Cah. Topol. Géom. Différ. Catég. 47 (2006), no. 3, 163–205, numdam

  • Teimuraz Pirashvili, María Julia Redondo, Cohomology of the Grothendieck construction Manuscripta Math. 120 (2006), no. 2, 151–162, doi

Relation with Andre-Quillen cohomology and (S,O)(S,O)-cohomology of Dwyer-Kan is elucidated in

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