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Jun, 2019
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Calculus of Functors

There appeared two important developements in stable homotopy theory in 90s. One is the analysis of chromatic phenomena by Hopkins and his collaborators. The other is the developement of a new technique, calculus of functors, by Goodwillie, so-called Goodwillie calculus.

Goodwillie’s idea is also useful in unstable homotopy theory, e.g. Weiss’ orthogonal calculus [ Wei95 ] . Goodwillie and Weiss developed so-called manifold calculus [ Wei99 GW99 ] in order to study spaces of embeddings.

Basic references are Goodwillie’s three papers [ Goo90 Goo92 Goo03 ] . Rezk [ Rez ] found a simplification of a proof in the third paper. There is an exposition by Kuhn [ Kuh ] . Munson’s [ Mun ] is written with emphasis on manifold calculus but it is based on the analogy to calculus and can serve as a good exposition.

Lurie proposes to use the language of higher category theory .

References