The homotopy theory of spectra is the prototypical (and universal) example of a

stable homotopy theory. In fact, spectra arise as the stabilization of the homotopy

theory of spaces. In this talk we discuss classical and some more recent character-

izations of abstract stable homotopy theories, thereby offering various conceptual

explanations for the passage from (pointed) spaces to spectra.

One of these characterizations emphasizes the idea that stabilization amounts to

dualizing certain bimodules with values in spaces, and it turns out that different

classes of bimodules give rise to the same stabilization theory. Moreover, there

are variants of these results for the passages from spaces to pointed spaces or

E-infinity monoids in spaces. These observations are the starting point for an on-going

project on a relative stabilization theory which is joint with Mike Shulman.

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