We give here some examples of non pointed protomodular categories $\mathbb C$ satisfying a property similar to the property of representation of actions which holds for the pointed protomodular category $Gp$ of groups: any slice category of $Gp$, any category of groupoids with a fixed set of objects, any essentially affine category. This property gives rise to an internal construction of the center of any object $X$, and consequently to a specific characterization of the abelian objects in $\mathbb C$.
Keywords: Protomodular categories; representation of actions; internal groupoids; abelian objects; central relations and center
2000 MSC: 25A05,18E05
Theory and Applications of Categories,
Vol. 16, 2006,
No. 2, pp 46-58.
http://www.tac.mta.ca/tac/volumes/16/2/16-02.dvi
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http://www.tac.mta.ca/tac/volumes/16/2/16-02.pdf
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