When attempting to understand a category of modules, it is natural to ask whether it is possible to classify indecomposable modules with certain properties.  

In general, it may be extremely difficult to answer a classification problem. In this talk I will discuss three approaches to classification arising in my work.  Firstly I will talk about locally noetherian localisations of the functor categories and how this can reduce the problem of classifying of all indecomposable modules to the classification of Sigma-pure-injective modules.  Next I will discuss how the classification of indecomposable pure-injective modules amounts to the classification of uniform objects in a functor category.  Finally I will connect the previous approaches to the functorial filtration classification technique.

I will illustrate the above by mentioning examples from joint work with K. Arnesen, D. Pauksztello and M. Prest as well as work in progress with R. Bennet-Tennenhaus.