Teorie di pretorsione e Operatori di chiusura

The study of torsion theories resulted as a fundamental tool in module theory, homology theory and in abelian category theory. The main problem related to the possibility of giving a more general notion of torsion theory is that it requires the category C to be pointed (i.e. C has a zero object), which isn’t always the case. This has led to several attempts to generalize the notion of torsion theories to arbitrary categories, one of which is pretorsion theories. A classical result in the theory of abelian categories establishes a bijection between the hereditary torsion theories and the idempotent closure operators. This correspondence has been extended to the non-additive context of homological categories. The aim of this work is to give a further extension of both these approaches, and for this case we introduce the notion of Z -normal category C , where Z is a full replete subcategory of C . This definition allows us to put ourselves in a much more general context that incorporates trivially the other approaches. We then study the relation between closure operators and pretorsion theories. We will then find the correspondence between pretorsion theories and hereditary closure operators. sending a topological group X to the quotient of X by the topological closure of the trivial sub- group 0 of X , D.Bourn and M.Gran proved that the homological closure of a kernel exactly co- incides with its topological closure. The aim of this work is to give a further extension of both these approaches, and for this case we introduce the notion of Z -normal category C , where Z is a full replete subcategory of C . This definition allows us to put ourselves in a much more general context. Indeed, when the ideal of Z -trivial morphisms is the ideal of zero morphisms, we get the homological case, while when it consist of all the morphisms in the category we get the regular one in [4]. When the category C is homological, it is possible to define the closure operator on nor- mal subobjects, i.e. on kernels. Of course in a non-pointed category this is no longer possi- ble. The natural thing to do in our context, is to replace the zero object with a class of "trivial objects", that is a full replete subcategory Z and to define the closure operator on Z -kernels (non-pointed version of kernels). We determine in the conditions on such a closure operator that make it correspond to a (Z -normal epi)-reflective subcategory (i.e. every component of the adjunction is a Z -cokernel) F of C . Furhermore, throughout the whole work, differently from the homological case [3], we will never need the category to be protomodular; hence classical results such as Short Five Lemma will not hold in the base category C . So, in order to replace some useful classical diagram lem- mas, sometimes we will need to require some additional hypothesis on the ideal of morphisms. We then restrict ourselves to Birkhoff subcategories. We first introduce the notion of Z -ideal determined category , inspired by the notion of ideal determined category of Janelidze. We then determine in the conditions that a closure operator has to satisfy to make it correspond to a Birkhoff subcategory. To conclude we give a detailed example in the category PreOrd.