Notes on Restricted Inverse Limits of Categories

We describe the framework for the notion of a restricted inverse limit of categories, with the main motivating example being the category of polynomial representations of the group $GL_{\infty}$. This category is also known as the category of strict polynomial functors of finite degree, and it is the restricted inverse limit of the categories of polynomial representations of $GL_n$. This note is meant to serve as a reference for future work.