On Mitchell's embedding theorem and some of its corollaries

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Title: On Mitchell's embedding theorem and some of its corollaries
Author: Leino, Joni
Contributor: University of Helsinki, Faculty of Science, Department of Mathematics and Statistics
Publisher: Helsingin yliopisto
Date: 2018
Language: eng
URI: http://urn.fi/URN:NBN:fi-fe201804208624
Thesis level: master's thesis
Discipline: Mathematics
Abstract: Abelian categories provide an abstract generalization of the category of modules over a unitary ring. An embedding theorem by Mitchell shows that one can, whenever an abelian category is sufficiently small, find a unitary ring such that the given category may be embedded in the category of left modules over this ring. An interesting consequence of this theorem is that one can use it to generalize all diagrammatic lemmas (where the conditions and claims can be formulated by exactness and commutativity) true for all module categories to all abelian categories.\\ The goal of this paper is to prove the embedding theorem, and then derive some of its corollaries. We start from the very basics by defining categories and their properties, and then we start constructing the theory of abelian categories. After that, we prove several results concerning functors, "homomorphisms" of categories, such as the Yoneda lemma. Finally, we introduce the concept of a Grothendieck category, the properties of which will be used to prove the main theorem. The final chapter contains the tools in generalizing diagrammatic results, a weaker but more general version of the embedding theorem, and a way to assign topological spaces to abelian categories. The reader is assumed to know nothing more than what abelian groups and unitary rings are, except for the final theorem in the proof of which basic homotopy theory is applied.

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