The Geometry of Iterated Loop Spaces
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271
J. P. May University of Chicago. Chicago/USA
The Geometry of Iterated Loop Spaces
Springer-Verlag Berlin · Heidelberg· New York 1972
AMS Subject Classifications (1970): 55D35. 55D40. 55F35. 55B20
ISBN 3-540-05904-0 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-05904-0 Springer-Verlag New York - Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlllg Berlin . Heidelberg 1972. Library of Congress Catalog Card Number'
Offsetdruck::JuliulBeltz, Hemsbath
Printed in Germany.
Preface
This is the first of a series of papers devoted to the study of iterated loop spaces.
Our goal is to develop a simple and
coherent theory which encompasses most of the known results about such spaces.
We begin with some history and a description of the
desiderata of such a theory. First of all, we require a recognition principle for n-fold loop spaces.
That is, we wish to specify
structure such that a space only if space.
internal
X possesses such structure if and
X is of the (weak) homotopy type of an n-fold loop For the case n= 1, Stasheff's notion [28] of an
is such a recognition principle.
space
Beck [5] has given an elegant
proof of a recognition principle, but, in practice, his recognition principle appears to be unverifiable for a space that is not given a priori as an n-fold loop space. very convenient recognition
In the case n =00, a
is given by Boardman and
Vogt's notion [8] of a homotopy everything space, and, in [7], Boardman has stated a similar recognition principle for n Z.
9
g(d) = 'l(c; d. d)v.
where v E E
. .• Z· (X X X)J - X J X X J on X J.
if d E ' (j) and
Z
Zj
gives the evident shuffle map
An examination of the definitions shows that
Z·
E X J. then and
If
is embedded in
by
0' -
f and g are
J
z) = eZ.(g(d). z}, J
O'(Z••••• Z). in the notation of Definition
l.1(c). then f and g are E.-equivariant. J
J
Our hypotheses guarantee that
homotopic. and the result follows.
Z.
Operads and monads In this section, we show that an operad
C determines a
matical structure, namely a monad, and that algebras over the derived monad.
Cspaces can be replaced by
We shall also give a preliminary state
ment of the approximation theorem. of
simpler mathe-
The present reformulation of the notion
Cspace will lead to a simple categorical construction of classifying
spaces for
enspaces in section 9.
We first recall the requisite cate
gorical definitions. Definition 2.1.
A monad (C,!J., TJ) in a category
(covariant) functor C: functors !J.: C
Z
J .... T
,.,. J
consists of a
together with natural transfor
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