nLab Spanier-Whitehead category

Idea

Motivated by the Freudenthal suspension theorem, the suspension category of (Spanier-Whitehead 53) has as objects pointed CW-complexes, and as hom-sets it has colimits

$$Hom(X,Y) \coloneqq \underset{\longrightarrow}{\lim}_q [\Sigma^q X , \Sigma^q Y]$$

over homotopy classes of continuous functions between their arbitrary high suspensions.

More generally one considers the category whose objects are pairs $(X,n)$ of a pointed CW-complex $X$ and an integer $n$ and whose hom-sets are

$$Hom((X_1,n_1), (X_2,n_2)) \coloneqq \underset{\longrightarrow}{\lim}_{q \gt max({\vert n_1\vert},{\vert n_2\vert})} [\Sigma^{q+n_1} X_1 , \Sigma^{q+n_2} X_2] \,.$$

This is a triangulated category.

But the Spanier-Whitehead category lacks other desirable properties, for instance it does not have all coproducts and the canonical functor from the homotopy category of pointed topological spaces does not preserve the coproducts that already exist. As a consequence, in particular a Brown representability theorem does not hold in the SW-category.

Later it was realized (see e.g. Whitehead 62) that this all this is fixed by regarding the SW-category for finite CW complexes as a full subcategory on the (shifted) suspension spectra inside the larger category of spectra: the stable homotopy category (e.g. Schwede 12, chapter II theorem 7.2). As such it is the full subcategory on the finite spectra (e.g. Schwede 12, chapter II theorem 7.4).

Related concepts

• Spanier-Whitehead duality

• prestable (∞,1)-category

References

The definition is due to

• Edwin Spanier, George Whitehead, A first approximation to homotopy theory Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 655-660.

Survey:

• George Whitehead, Some aspects of stable homotopy theory, International Confress of Mathematics 1962 (pdf)

• H. R. Margolis, Spectra and the Steenrod algebra, volume 29 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1983

• Stefan Schwede, chapter II, section 7 of Symmetric spectra, 2012 (pdf)

• Michael Hopkins (notes by Akhil Mathew), Lecture 20 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

Discussion in the abstract generality of categories equipped with an abstract suspension-like functor is in

• Alex Heller, Stable homotopy categories, Bull. Amer. Math. Soc. Volume 74, Number 1 (1968), 28-63. (Euclid)

• Ivo Dell'Ambrogio, The Spanier-Whitehead category is always triangulated (pdf)