Enriching representable functors

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2002-01-01
Authors
Rothbauer, Stefan
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As George Bergman pointed out at the International Conference of Mathematicians in Vancouver (1974) category theory can be a very efficient way to determine all possible operations on sets. In fact, under certain conditions the operations on the values of a (representable) set valued functor are in one to one correspondence with the cooperations on the representing object and the relations that hold between those operations can be determined by analyzing the relations that hold between the cooperations on the representing object. In this thesis the ideas of Bergman described above together with the approaches described below are applied to the group of units of an arbitrary ring and the idempotent elements of commutative rings. Algebra, one of the oldest and still most popular branches of mathematics, is the study of algebraic structures. An algebraic structure, loosely speaking, is a object together with one or more operations on it. If there is more than one operation on the set these operations often satisfy certain relations. The class of objects which share certain properties is called a category. Some of the most common such categories consist of objects which posses an underlying set (e.g. groups, rings, fields, vector spaces). Often these sets can be regarded as members of different categories and accordingly are endowed with a number of operations between which relations hold. In the case that a particular object is part of different categories, one can try (more often than not successfully) to make use of certain facts that are true in one category to gain knowledge about other objects in the second category. One other approach to analyze algebraic structures is to look at their substructures (whether those lie within this category or in a different one) and gain knowledge about the original object by composing the pieces of the puzzle gained by analyzing the sub objects. Frequently the opposite approach works quite as well, i.e. to look at the objects which contain an object as subobjects.

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