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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.

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