Please use this identifier to cite or link to this item: `http://hdl.handle.net/10773/4852`
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dc.contributor.authorOliveira, Sandra Margarida Barretopt
dc.coverage.spatialAveiropt
dc.date.accessioned2011-12-27T15:25:19Z-
dc.date.available2011-12-27T15:25:19Z-
dc.date.issued2006pt
dc.identifier.urihttp://hdl.handle.net/10773/4852-
dc.description.abstractThe concept of a category was introduced by Eilenberg and Mac-Lane in 1945 with the aim to simplify certain aspects of the algebraic topology. However, the language of category theory proved to be useful in many other branches of mathematics as well and helps to understand better what is common to them. In fact, many constructions in mathematics have a similar description and category theory provides a unifom description of these constructions. In any mathematical theory, isomorphic objects are indistinguishctble in terrns of the theory and the objective of the theory is to identify and study constructions and properties that are invariant under isomorphism (thus, for example, algebra studies properties that they are not modified, or destroyed, when a group is substituted by another one isornorphic to it}. In fact, in category theory, "is isornorphic to" can be seen as a synonym of "is equal to" arid the major part of definitions and constructions in category theoty, do no! specify "uniquely", but only, as it will be seen, up to isomorphism. tn general, objects X and Y are called isomorphic if there exist arrows F:X-rY and G:Y-rX such that F.G=I e G.F=I. This concept can be extended to categories: given two categories C and D, C and D are called isomorphic, if there exist functors F:C-tD and G:D+C such that G.F=I and F.G='i. However, this notion of "similarity" is stricter than necessary. Under the point of view of category theory, it is more natural to require instead "a" to be isomorphic to "G(F(a))" and " b to be isomorphic to "F(G(b))"; and this is the notion of "equality" (equivalence} considered in this work. Of special inlerest to us are equivalences between (known) categories and duals of (known) categories. The knowledge of such and equivalence provides us with new infomation about involved categories, since in many "everyday" categories for instance products are easier to describe than coproduds. In the first chapter of this thesis we introduce the definition of a category and present severa1 examples. In the second chapter we study irt detail specjsi! morphisms and objects in a category and their properties. Furkhermore we study the concept of functor and (co)limit. In the third chapter we analyse the notions of natural transfomiation, equivalence and adjoint situation. In the fourth chapter we focus on the study of dual equivalences. We analyse the structure of a dual adjunction and provide techniques for their constructian. Finally, we give conditions which guarantee that we constructed adjunction is in fact an equivalence. We illustrate this procedure by examples, among them the dual equivalence between CompHauss and C*-AIg (where CompHauss is the category of compact and separated spaces and C*-AIg the category of C*- algebras).pt
dc.language.isoporpt
dc.relation.urihttp://opac.ua.pt/F?func=find-b&find_code=SYS&request=000209180pt
dc.rightsopenAccesspor
dc.subjectMatemáticapt
dc.subjectCategorias (Matemática)pt
dc.typemasterThesispt