On matrix Toda brackets in Baues–Wirsching cohomology

Hardie, Kamps and Marcum have considered the matrix Toda brackets introduced by Barratt in the category of topological spaces from a $2$-categorical point of view. Baues and Dreckmann have shown that a class in the third Baues–Wirsching cohomology of a small category $\mathcal{C}$ governs every classical Toda bracket if the bracket is defined with a Toda category in $\mathcal{C}$. Our aim is to generalize such a relationship to that between the class in the cohomology and matrix Toda brackets in a $2$-category. Moreover, the non-triviality of the third cohomology is discussed via computation of a matrix Toda bracket in the category of cochain complexes on an additive category.

Keywords

track category, linear track extension, universal Toda bracket, triangulated category

2010 Mathematics Subject Classification

18D05, 18E30, 55U35

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Received 29 March 2017

Received revised 25 August 2017

Published 27 June 2018