nLab Adams category

Idea

The Adams category, named after (Adams 74), was historically one of the first constructions of the stable homotopy category, following ideas in (Boardman 65), see (Lewis-May-Steinberger 86, pages 1-2) for recollection of the historical development and critical comments on the definition.

The Adams category is defined to be the category of CW-spectra together with left homotopy classes (via cylinder spectra) of “eventually defined” functions between them.

This was originally advertised as being a definition not involving tools from category theory. Arguably this is also its main deficiency when it comes to working with it (this is the “polemic” of Lewis-May-Steinberger 86, preamble). For modern alternatives see at stable homotopy category.

References

The definition is due to

• Frank Adams, Part III, section 2 of Stable homotopy and generalised homology, 1974

following

• Michael Boardman, Stable homotopy theory, mimeographed notes, University of Warwick, 1965 onward

The term “Adams category” for this starts to be used for instance in

• Harold Hastings, On function spectra, Proceedings of the AMS, volume 44, Number 1, May 1974 (pdf)

An account following (Adams 74) is also in

• Robert Switzer, section 8 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

Comments on the historical development are in

• L. Gaunce Lewis, Peter May, M. Steinberger, preamble of Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics, 1986 (pdf)

in the spirit of

There is much to love in Adams' book, but not in the foundational part on CW spectra. (Peter May, MO comment)

More recent textbook accounts include

• Stanley Kochman, section 3.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

Lecture notes include

• Cary Malkiewich, section 2.1 of The stable homotopy category, 2014 (pdf)