Twisted Chain Complexes and Twisted Homology
Let T be a theory of coactions and let chain be the category of chain complexes as defined in (I.6.3). We introduce in this chapter the functor $$ K\;:\;Twist \to chain$$ which carries a presentation ∂ X to a chain complex d X concentrated in degree 1 and
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Springer-Verlag Berlin Heidelberg GmbH
Hans-Joachim Baues
Combinatorial Foundation of Homology and Homotopy
Applications to Spaces, Diagrams, Transformation Groups, Compactifications, Differential Algebras, Algebraic Theories, Simplicial Objects, and Resolutions
,
Springer
Hans-Joachim Baues Max-Planck-Institut fiir Mathematik Gottfried-Claren-Strasse 26 D-53225 Bonn, Germany e-mail: [email protected]
Cataloging in Publication Data applied for Die Deutsche Bibliothek - CIP- Einheitsaufnahme Baues, Hans J.:
Combinatorial foundation of homology and homotopy: applications to spaces, diagrams, transformation groups, compactifications, differential algebras, algebraic theories, simplical objects,and resolutions I Hans Joachim Baues.
(Springer monographs in mathematics) ISBN 978-3-642-08447-8 ISBN 978-3-662-11338-7 (eBook) DOI 10.1007/978-3-662-11338-7
Mathematics Subject Classification (1991): 55-02
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Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999
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Typesetting: The Author's TE---> sn. Sec Baues [AH] 11.5.7. In a similar way the functor 71"n+l(A,B) in (1.11) is defined.
Here t :
71"
bo
A functor M : GOp
---+
Ab
(1.12)
is called a (right) G-module. Hence l'l/I is a contravariant functor from G to Ab. If G is small (i.e. if the class of objects in G is a set) then such G-modules are the objects of the abelian category Mod( G). Morphisms are natural transformations. Hence by (1.11) we see that homotopy groups 71"n(A) and 71"n(A, B) are (IIA)modules and (II B)-modules respectively. Next we consider the functorial property of the fundamental groupoid. For this let Grd be the category of small groupoids. Morphisms are functors. For a groupoid G let Grd( G) be the following category. Objects are functors G ---+ H between groupoids which are the identity on objects (hence Ob G = Ob H). Morphisms are functors H ---+ K under G that is, commutative triangles in Grd:
G
/~
H----t>K
For each cofibration D
---+
X in Top? we obtain the object
c(X)
=
(II(D)
---+
II(X, D))
in Grd(II D) where II(X, D) is the restricted fundamental groupoid of X. Thi
