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Macquarie University  Department of Mathematics

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Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

July 18-21 2005, Australian National University, Canberra

Tue, 19 July:  16:15 - 16:55

Moore normalization and Dold--Kan theorem for semi-abelian categories
Bourn, Dominique (Université du Littoral)

The usual Dold--Kan theorem asserts that, if $\mathbb A$ is an abelian category, the Moore normalization functor $N : \mathrm{Simpl}\,\mathbb A \rightarrow \mathrm{Ch}\,\mathbb A$ from simplicial objects in $\mathbb A$ to chain complexes in $\mathbb A$ is an equivalence of categories. This normalization construction $N : \mathrm{Simpl}\,\mathbb C \rightarrow \mathrm{Ch}\,\mathbb C$ is clearly possible as soon as the category $\mathbb C$ is pointed and finitely complete. Certainly, it is far from being an equivalence

It is proved here that when the category $\mathbb C$ is semi-abelian (such as, for instance, the category $\mathrm{Gp}$ of groups or the category $\mathrm{Rg}$ of rings) this functor $N$ is necessarily monadic. This must be thought of as the semi-abelian formulation of the Dold--Kan theorem. The fact that the category $\mathbb C$ is semi-abelian is a quasi-necessary condition for this monadicity theorem. Indeed, assuming that the category $\mathbb C$ is pointed (which is necessary to define the functor $N$), the existence of the left adjoint is equivalent to the existence of coproducts in $\mathbb C$, and the fact that $N$ is conservative is equivalent to the fact that the category $\mathbb C$ is protomodular.

Typeset PDF of this abstract.