Finite spectrum

A finite spectrum is a topological spectrum equivalent to a cell spectrum with finitely many cells. Such an object is compact in the stable homotopy category and has a Spanier-Whitehead dual. Compactness means maps out of it commute with filtered homotopy colimits of spectra.

The Spanier-Whitehead dual of a finite spectrum $A$ is $A^{\vee }=F(A,\mathbb{S})$. It is finite, and evaluation and coevaluation give the natural Spanier-Whitehead duality adjunction.

For a finite spectrum $A$, evaluation and coevaluation identify $[B,D\wedge A{]}_{\mathrm{st}}$ with $[B\wedge A^{\vee },D{]}_{\mathrm{st}}$. Cellular induction constructs the dual from the duals of stable cells and verifies these identities across cofiber sequences of spectra. In particular ${\pi }_k(D\wedge A)=[{\Sigma }^k{A}^{\vee },D{]}_{\mathrm{st}}$.