For a finite spectrum , evaluation and coevaluation identify with . Cellular induction constructs the dual from the duals of stable cells and verifies these identities across cofiber sequences of spectra. In particular .
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Spanier–Whitehead duality is a concept in algebraic topology, named after the mathematicians Edwin Spanier and Frank W. Whitehead. It provides a duality between certain types of topological spaces regarding their homotopy and homology theories. More specifically, it relates the category of pointed spaces to the category of pointed spectra, allowing one to translate problems in unstable homotopy theory into stable homotopy theory, and vice versa.