

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, socalled Goodwillie calculus.
Goodwillie’s idea is also useful in unstable
homotopy theory, e.g. Weiss’ orthogonal
calculus
[
Wei95
]
. Goodwillie and Weiss developed socalled 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
.
[Goo90]
Thomas G. Goodwillie. Calculus. I. The
ﬁrst derivative of pseudoisotopy theory.
K
Theory
, 4(1):1–27, 1990.
[Goo03]
Thomas G. Goodwillie. Calculus. III. Taylor
series.
Geom. Topol.
, 7:645–711 (electronic), 2003.
[Goo92]
Thomas G. Goodwillie. Calculus. II. Analytic
functors.
K
Theory
, 5(4):295–332, 1991/92.
[GW99]
Thomas G. Goodwillie and Michael Weiss.
Embeddings from the point of view of immersion
theory. II.
Geom. Topol.
, 3:103–118 (electronic), 1999.
[Kuh]
Nicholas J Kuhn. Goodwillie towers and
chromatic homotopy: an overview,
arXiv:math/0410342
.
[Mun]
Brian A. Munson. Introduction to the manifold
calculus of GoodwillieWeiss,
arXiv:1005.1698
.
[Rez]
Charles Rezk. A streamlined proof of
Goodwillie’s
n
excisive approximation,
arXiv:0812.1324
.
[Wei95]
Michael Weiss. Orthogonal calculus.
Trans. Amer. Math. Soc.
, 347(10):3743–3796, 1995.
[Wei99]
Michael Weiss. Embeddings from the point of view of
immersion theory. I.
Geom. Topol.
, 3:67–101 (electronic), 1999.


