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
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.