When homological stability fails, representation theory comes along and tries to pick up the pieces. The resulting theory of representation stability was observed by Church and Farb, and has applications to topology, combinatorics and algebraic geometry. Together with Ellenberg and Nagpal this theory was recast in a more algebraic light, resulting in the introduction of FI-modules. One major advantage of this point of view is that it allows the application of homological techniques which can often greatly simplify problems. Another is that this theory fits into a more general framework, introduced by Sam and Snowden, known as representations of combinatorial categories.

We will start by briefly reviewing the relevant representation theory of symmetric groups (a subject beautiful in its own right!) before motiving representation stability with concrete examples. This will pave the way to the definition of FI-modules, and a brief exposition of that theory. In particular, FI-modules satisfy a Noetherian property that powers a lot of this subject. Finally, and with time permitting, I will introduce the combinatorial categories of Sam and Snowden, an application of which will be a simple combinatorial proof of the aforementioned Noetherian property.