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A NOTE ON THE NOTION
OF CHARACTERISTIC SUBOBJECT
IN THE MAL’TSEV AND PROTOMODULAR SETTINGS
DOMINIQUE BOURN
D´edi´e, pour son anniversaire, a` une grande dame des math´ematiques
Abstract. We introduce a categorical definition of characteristic sub- objects which is alternative to the one given in [14] for semi-abelian cat- egories and applies to the wider context of Mal’tsev categories. It allows, among other things, to set clearly this notion and to produce examples in the category Rg of non unitary rings and in the category TopGp of topological groups.
Introduction
In a recent work [14] A. Cigoli and A. Montoli investigated the notion of char- acteristic subobject in the context of semi-abelian categories. Mimicking strictly what happens in the categories Gp of groups and R-Lie of Lie R-algebras, their approach was based upon the notion of internal action as described in [4]. In any semi-abelian category the internal actions are in bijection with the split epimorphisms and, actually, it is possible to get rid of the concept of inter- nal action and to get straight to the notion of characteristic subobject just using split epimorphisms and hypercartesian morphisms related to the change of base functors associated with the fibration of points (Definition 2.1). This alternative approach produces alternative proofs and has, among other things, the following benefits:
1) it allows us to extend the notion to the conceptually lighter and wider context of Mal’tsev categories
Received: 29 April 2014 / Accepted: 24 December 2014.
2010 Mathematics Subject Classification. Primary 20D30, 17B05, 13A15, 18A20; Sec-
ondary 18D35,18C99.
Key words and phrases. Mal’tsev, protomodular and action representative category, hy-
percartesian monomorphism, characteristic subobject, commutator and centralizer.
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