# 2-track algebras and the Adams spectral sequence

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## Abstract

In previous work of the first author and Jibladze, the \(E_3\)-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the \(E_3\)-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms \(E_r\). In this paper, we introduce 2-track algebras and tertiary chain complexes, and we show that the \(E_4\)-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.

## Keywords

Adams spectral sequence Tertiary Toda bracket 2-track groupoid Bigroupoid Double groupoid## Mathematics Subject Classification

55T15 18G50 55S20## Notes

### Acknowledgements

We thank the referee for their helpful comments. The second author thanks the Max-Planck-Institut für Mathematik Bonn for its generous hospitality, as well as David Blanc, Robert Bruner, Dan Christensen, and Dan Isaksen for useful conversations.

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