Page 78 - Textos de Matemática Vol. 46

68 D. BOURN
2) it gives rise to a conceptual proof of the characterization of the characteristic subobjects as those normal subobjects which are stable under composition by normal subobjects on the left hand side (Theorem 2.10)
3) it shows that the notion is stable under the passage to the fibres P tY C of the fibration of points, see Theorem 2.7 for more details
4) it extends in a clear way the notion of characteristic subobjects to the cate- gory Rg of non unitary rings or T opGp of topological groups where the question was up to now far from being cleared (Sections 2.3 and 2.4)
5) it produces some unexpected situations dealing with objects whose any sub- object is a characteristic one (Corollary 3.2)
6) in the exact action representative contexts, such as Gp and R-Lie, it allows to characterize the characteristic subobjects as those normal subobjects whose quotient maps have a (unique) extension to the action groupoids (Corollary 3.5)
7) it allows to characterize in simple terms those pairs of characteristic subob- jects whose commutator is characteristic as well (Theorem 5.3)
8) it brings some new enlightments about the notion of peri-abelian category (Section 5.3).
The article is organized along the following lines. Section 1) gives some recalls about hypercartesian monomorphisms. Section 2) introduces the alternative definition, the first stability properties and the first investigations in Rg and TopGp. Section 3) gives conceptual insights on the classical definition of the characteristic subobjects in groups and in Lie R-algebras dealing with the group AutX and the algebra DerX through the investigation of the characteristic subobjects in the action representative context. Section 4) is dealing with the centralizer of characteristic subobjects while Section 5) is dealing with the commutator of such pairs.
1. Hypercartesian morphisms
Let U : E ! F be any functor. Recall that a map f : X ! Y in E is hypercartesian with respect to U when, given any other map g : X0 ! Y in E with a factorization h : U(X0) ! U(X) in F such that U(g) = U(f)h, there isauniquefactorizationh¯:X0 !XinEsuchthatg=h¯fandU(h¯)=h. Accordingly any hypercartesian map whose image by U is an isomorphism is an isomorphism.
Proposition 1.1. 1) The hypercartesian maps with respect to U are stable under composition; a hypercartesian map above a monomorphism is itself a monomorphism.