Resumen

Suppose you are given an algebra or a ring, and you are asked to investigate it. You are, of course, aware of representation theory, and you could attempt to understand it by studying its modules, having all the machinery of category theory and homological algebra at your disposal. Now, imagine you are given an additive category, and you are asked the same task: to investigate it. How could you proceed? This question certainly has many possible answers, but "representation theory" is still one of them. In this talk, I will explain what modules over an additive category are, and I will give some examples of how properties of an additive category can be understood in terms of its modules. We will also see a few instances of how this approach has been used, and also how it could be further explored, for example, in functional analysis, or in other areas.