The Adams spectral sequence is available in any triangulated category
equipped with a projective or injective class. Higher Toda brackets can also
be defined in a triangulated category, as observed by B Shipley based on
J Cohen’s approach for spectra. We provide a family of definitions of
higher Toda brackets, show that they are equivalent to Shipley’s and show
that they are self-dual. Our main result is that the Adams differential
in any Adams spectral sequence can be expressed as an
–fold Toda
bracket and as an
order cohomology operation. We also show how the result simplifies
under a sparseness assumption, discuss several examples and give
an elementary proof of a result of Heller, which implies that the
–fold
Toda brackets in principle determine the higher Toda brackets.
Keywords
triangulated category, Adams spectral sequence, Toda
bracket, cohomology operation, differential, higher order
operation, projective class
