nLab Hans-Joachim Baues

Hans-Joachim Baues (d. May 8, 2020) was a German topologist and the author of

• Algebraic Homotopy, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989);
• Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991);
• Homotopy Types, in I.M.James, ed., Handbook of Algebraic Topology, 1–72, Elsevier, (1995);
• Homotopy type and homology, Oxford University Press, 1996
• (with Antonio Quintero) Infinite Homotopy Theory, K-monographs in mathematics, Volume 6, Kluwer, 2001;

among other works.

A part of his work developed the ideas of Whitehead's basic programme for algebraic homotopy, using a mix of cofibration categories and categories with a particular type of cylinder functor, that he called I-categories. These are treated in separate entries.

Baues also systematically developed aspects of secondary homological algebra (secondary cohomological operations, secondary Steenrod algebra, secondary chain complexes, secondary triangulated category) which is a form of categorification of the usual homological algebra. He developed them together to give sound applications in the computations with Adams-Novikov spectral sequence and calculations of homotopy groups of spheres.

He is also well known for his development of the “cohomology of small categories” with Wirsching, sometimes called Baues-Wirsching cohomology, although versions of this theory existed prior to his contribution.

MPIM Obituary

Selected writings

On cohomology of monoids in monoidal categories (specifically in distributive monoidal categories):

• Hans-Joachim Baues, Mamuka Jibladze, Andy Tonks, Cohomology of monoids in monoidal categories, in: Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) 137-166 [doi:10.1090/conm/202, preprint:pdf]