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Normal functors and strong protomodularity. (English) Zbl 0947.18004

The author continues his study of protomodular categories, which are categories equipped with structure that makes them resemble the category of groups in some way; in particular, they permit one to define a notion of ‘normal subobject’ which plays the rôle that normal subgroups do in group theory (a monomorphism \(X\to Y\) is normal if there exists an equivalence relation \(R\) on \(Y\) such that \(X\) is ‘identified with a single \(R\)-equivalence class’).
In the present paper, he observes that an important rôle in the theory is played by functors between such categories which preserve finite limits and reflect isomorphisms and normality of subobjects; he calls these normal functors. He defines a category to be strongly protomodular if the change of base functors for its fibration of pointed objects are (not only conservative – that is the definition of protomodularity – but) normal. Not every protomodular category is strongly protomodular: an exception is the category whose objects are sets equipped with two group structures sharing a common identity element. But many familiar examples of protomodular categories are strongly so, and the author gives some examples of the effect that this condition has on their structure.

MSC:

18B99 Special categories
08B05 Equational logic, Mal’tsev conditions
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
18D35 Structured objects in a category (MSC2010)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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