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Dold-Kan type theorem for \(\Gamma\)-groups. (English) Zbl 0963.18006
Let \(\Gamma\) be the category of finite based sets. A \(\Gamma\)-group is a functor \(T\) from \(\Gamma\) to groups such that \(T(\{0\})\) is trivial. Segal’s infinite loop space machine uses \(\Gamma\)-spaces (which are defined in a similar way), and it is shown here that any \(\Gamma\)-space is stably weak homotopy equivalent to a discrete \(\Gamma\)-group.
Given a \(\Gamma\)-group, the author uses cross-effects to construct a group-valued functor on \(\Omega\), where \(\Omega\) is the category of non-empty finite sets and surjections. The main result says that an abelian \(\Gamma\)-group is equivalent to a functor on \(\Omega\), and an arbitrary \(\Gamma\)-group is equivalent to a functor on \(\Omega\) with additional structure related to commutators. The paper also contains a spectral sequence for the stable homotopy of abelian \(\Gamma\)-groups and some results on Dold-Puppe stable derived functors.

18E25 Derived functors and satellites (MSC2010)
55P47 Infinite loop spaces
18G40 Spectral sequences, hypercohomology
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