Relative commutator theory in semi-abelian categories. (English) Zbl 1264.18014
Let \(\mathcal A\) be a semi-abelian category, i.e., \(\mathcal A\) is pointed, exact and protomodular. If \(f:N\rightarrow A\) is a normal monomorphism then \(N\) is a normal subobject of \(A\). It is proved if \(N\) and \(M\) are normal subobjects of \(A\) then the join of \(N\) and \(M\) as a normal subobject of \(N\vee M\) coincides with their join as a subobject of \(N\cup M\). The other properties of normal subobjects of \(A\) are given. A full reflective subcategory \(\mathcal B\) of \(\mathcal A\) closed under subobjects and regular quotients is called a Birkhoff subcategory of \(\mathcal A\). A commutator defined to a relative Birkhoff subcategory \(\mathcal B\) of \(\mathcal A\) is studied. This commutator characterizes Janelidze and Kelly’s \(\mathcal B\)- central extensions. When \(\mathcal B\) is determined by the abelian objects of \(\mathcal A\) then it coincides with Huq’s commutator. If \(\mathcal A\) is a variety of \(\Omega\)-groups then it coincides with the relative commutator.
Reviewer: Václav Koubek (Praha)
MSC:
| 18E25 | Derived functors and satellites (MSC2010) |
| 20F12 | Commutator calculus |
| 20J99 | Connections of group theory with homological algebra and category theory |
| 20K35 | Extensions of abelian groups |
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