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Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested in it (especially with regards to stable homotopy theory)?

Or perhaps I might say, while I am reading Goodwillie's Calculus I, is there another paper out there that is introductory, perhaps with a long preface or introduction with intuitive ideas?

Thanks!

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    $\begingroup$ Also, for completeness, I will add Nicholas Kuhn's paper relating calculus of functors to chromatic homotopy theory: math.rochester.edu/u/faculty/doug/otherpapers/KuhnKinosaki.pdf $\endgroup$
    – Jonathan Beardsley
    Commented Jan 23, 2012 at 19:40
  • $\begingroup$ Goodwillie and Weiss have some nice survey papers for embedding calculus. Calculus of Embedddings Bull. Amer. Math. Soc. 33 (1996), 177-187. Embeddings from the point of view of immersion theory, Part I Geom. and Topology 3 (1999), 67-101. Spaces of smooth embeddings, disjunction and surgery, in Surveys on Surgery Theory vol. 2, Princeton University Press (2001), eds. Cappell, Ranicki and Rosenberg. They're all embedding calculus, though. $\endgroup$
    – Ryan Budney
    Commented Jan 23, 2012 at 20:34
  • $\begingroup$ There is a preprint version of "Caclculus I" which existed before the published version that gives a lot of intuition behind the ideas in its long introduction. The published version of "Calculus I is quite different. (By the way, Ryan, I am a co-author of the last reference ;-).) $\endgroup$
    – John Klein
    Commented Jan 23, 2012 at 23:26
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    $\begingroup$ Jon, I second Nick Kuhn's notes regarding chromatic homotopy theory. I think they do a great job of laying out the framework for Goodwillie Calculus. $\endgroup$
    – Hiro Lee Tanaka
    Commented Jan 23, 2012 at 23:37
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    $\begingroup$ Hi John, yes, after I wrote that I noticed I missed mentioning you. :) I've enjoyed reading your paper, BTW, and I wish MO had an edit feature for comments. Also, JonB, if you can get a copy of Goodwillie's dissertation, I find pages 1 through about 29 to be quite informative. Although the dissertation does not have the full formalism of calculus, it's nice to see how it originated in its "native" environment. The dissertation is AMS Memoirs Number 431. July 1990, Vol 86 (first of two). $\endgroup$
    – Ryan Budney
    Commented Jan 23, 2012 at 23:43

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It looks like the nLab article is pretty nice. There you'll find a list of references, including this:

Brian Munson, Introduction to the manifold calculus of Goodwillie-Weiss, arXiv:1005.1698

Is that the kind of thing you're looking for?

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    $\begingroup$ Wow these are great resources, thanks! I like this bit from Munson's paper: "We will frequently omit arguments which we find distract us from our attempts to be lighthearted." Right up my alley. $\endgroup$
    – Jonathan Beardsley
    Commented Jan 23, 2012 at 17:42
  • $\begingroup$ :-) That's a good bit of honest writing. $\endgroup$
    – Tom Leinster
    Commented Jan 23, 2012 at 17:45
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This is 7 years too late, but this survey (to appear in the Handbook of Homotopy Theory) is a really readable survey of Goodwillie calculus: https://arxiv.org/abs/1902.00803.

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You can get a pretty good birds-eye view by reading this 2004 Oberwolfach report (no. 17):

http://www.mfo.de/document/0414/OWR_2004_17.pdf

Calculus of Functors is the theme of the 2012 Talbot workshop, and one can hope that there will be reasonable attempts made to collect introductory and expository material and make it available. There may be much better answers to your question in a month or two (written 1/23/2012).

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There is a very good overview in "Cubical Homotopy Theory" chapter 10 which is available online here

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