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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 2

### On matrix Toda brackets in Baues–Wirsching cohomology

Pages: 315 – 339

DOI: http://dx.doi.org/10.4310/HHA.2018.v20.n2.a16

#### Authors

#### Abstract

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

Received 29 March 2017

Received revised 25 August 2017

Published 27 June 2018