Pontryagin's famous duality theorem establishes a strong relationship between locally compact abelian hausdorff (LCA) groups and their dual groups, providing deep insights into the structure and behavior of these groups. In the talk we will give a proof of the duality theorem based on ideas from category theory, which is guided by the structure within the category LCA.The duality theorem will first be established for the subcategory of compactly generated abelian Lie-Groups.Through the study of "formal" limits and colimits the duality is expanded to a duality between the subcategories of discrete abelian and compact abelian LCA groups. From there, a study of exact sequences will result in the full duality theorem for the category LCA. https://www.uni-muenster.de/AGKramer/index.php?name=TSWS24&menu=teesemi&lang=de