Weighted centrality, and a further approach to categorical commutativity

Shaumbwa, Vaino Tuhafeni (2019-12)

Thesis (PhD)--Stellenbosch University, 2019.

Thesis

ENGLISH ABSTRACT: We investigate weighted commutators, that is, weighted subobject commutator and weighted normal commutator, as well as commutators in the sense of Huq, Higgins, Ursini and Smith, which are all special cases of weighted commutators. One of the main aims is to establish further properties of weighted commutators, and explore new relationships among commutators. In a normal Mal'cev category C with finite colimits, we show that the Huq commutator of a pair of local representations (i.e. equivalence relations considered as subobjects in a category of points over fixed object) is the local representation of the Smith commutator of the equivalence relations corresponding to the original local representations. We also show that the weighted normal commutator can be obtained as the image of the kernel functor applied to the Huq commutator of another type of morphisms in a category of points over a fixed object. In addition, the weighted normal commutator is characterized as the largest monotone ternary operation C defined on subobjects in a finitely cocomplete normal Barr-exact Mal'cev category, such that: (a) C(X; Y; W ) X ^ Y ; (b) C(f(X); f(Y ); f(W )) = f(C(X; Y; W )); for subobjects (X; x); (Y; y); and (W; w) of an object A; and every morphism f whose domain is A. The weighted subobject commutator is characterized in a similar way, and furthermore, known characterizations of Higgins, Huq, and Ursini commutators are recovered as special cases. Another aim is to extend the notion of commuting morphisms to a more general context, and in particular, to a subtractive category with finite joins of subobjects, where we show that commuting morphisms are related to the notion of internal partial subtraction structures. Furthermore, we show that several results about central morphisms, commutative objects, and abelian objects, which usually require a category to be at least (strongly) unital, also hold in the context of (regular) subtractive category with finite joins of subobjects.

AFRIKAANSE OPSOMMING: Ons ondersoek geweegde kommutator, dit wil sê geweegde subobjekkommutator en geweegde normale kommutator, sowel as kommutators in die sin van Huq, Higgins, Ursini en Smith, wat almal spesiale gevalle van geweegde kommutator is. Een van die hoofdoelwitte is om verdere eienskappe van geweegde kommutator te vestig, en om nuwe verhoudings tussen kommutator te ondersoek. In 'n normale Mal'cev-kategorie C met eindige kolimiete, wys ons dat die Huqkommutator van 'n paar plaaslike voorstellings (dit wil sê ekwivalensieverhoudinge wat as subobjekte in 'n kategorie punte oor 'n vaste objek beskou word) die plaaslike voorstelling van die Smith is kommutator van die ekwivalensieverhoudinge wat ooreenstem met die oorspronklike plaaslike voorstellings. Ons wys ook dat die geweegde normale kommutator verkry kan word as die beeld van die "kernel" funktor wat op die Huq-kommutator van 'n ander soort morfismes toegepas word in 'n kategorie punte oor 'n vaste objek. Daarbenewens word die geweegde normale kommutator gekenmerk as die grootste monotone ternêre werking C gedefinieër op sub-objekte in 'n eindelik klaargemaakte normale Barr-exact Mal'cev-kategorie, sodat: (a) C(X; Y; W ) X ^ Y ; (b) C(f(X); f(Y ); f(W )) = f(C(X; Y; W )); vir subobjekte (X; x); (Y; y); en (W; w) van 'n objek A; en elke morfisme f waarvan die domein A is. Die geweegde subobjekkommutator word op 'n soortgelyke manier gekenmerk, en voorts word bekende karakterisering van Higgins, Huq en Ursini-kommutators as spesiale gevalle herwin. 'n Ander doel is om die idee van die pendel van morfismes uit te brei na 'n meer algemene konteks, en veral tot 'n subtraktiewe kategorie met 'n eindige samevoeging van sub-onderwerpe, waar ons wys dat die pendel-morfisme verband hou met die idee van interne gedeeltelike aftrekstrukture. Verder toon ons dat verskeie resultate oor sentrale morfismes, kommutatiewe objekte en abeliese objekte, wat gewoonlik vereis dat 'n kategorie ten minste (sterk) uniaal is, ook in die konteks van 'n (gereelde) aftrekkategorie met 'n beperkte samevoeging van subobjekte geld.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/107175
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