Normalities and commutators

Abstract

We first compare several algebraic notions of normality, from a\ud categorical viewpoint. Then we introduce an intrinsic description of\ud Higgins' commutator for ideal-determined categories, and we define a new\ud notion of normality in terms of this commutator. Our main result is to\ud extend to any semi-abelian category the following well-known\ud characterization of normal subgroups: a subobject K is normal in A if.\ud and only if, {[A, K] <= K. (C) 2010 Elsevier Inc. All rights reserved.

What is an ideal a zero-class of?

“…We observe that this notion is a generalisation of the notion of kernel of a morphism: indeed, kernels are exactly zero-classes of effective equivalence relations. It is also easy to see that, in the pointed case, the definition above is equivalent to the one introduced by Bourn in [4]: see [15] and Example 3.2.4, Proposition 3.2.12 in [2].…”

“…Later these notions have been considered in a categorical context [11,12,15]. Clots were characterised as zero-classes of internal reflexive relations, and ideals were characterised as regular images of clots.…”

Central reflections and nilpotency in exact Mal’tsev categories

“…The co-smash product X ⋄ Z coincides in semiabelian categories with the second cross-effect cr 2 (X, Z) of the identity functor, cf. Definition 5.1 and [54,38,39]. Since the co-smash product is in general not associative (cf.…”

“…In a semi-abelian (or homological [4]) category, an object X is n-folded if and only if the image of the kernel K[θ X,...,X ] under the folding map δ X n+1 : X + · · · + X ։ X is trivial. Recall [38,39,41,54] that this image is by definition the so-called Higgins commutator [X, . .…”

“…In a σ-pointed exact Mal'cev category, abelianisation I 1 is the universal endofunctor J 1 of degree ≤ 1 (cf. Mantovani-Metere [54]). For n > 1 however, the universal endofunctor J n of degree ≤ n is in general a proper quotient of the n-th Birkhoff reflection I n (cf.…”