Basically every branch of pure mathematics uses modelling of some kind, and the idea that pure mathematicians deal with 'exactness' and do not need to model or approximate the phenomena they are studying is surely a misconception.
There are so many examples that I am not sure where to start, but as an example, since the 1970s, diagram categories have been used to model loop spaces and recently new approaches have been developed to model iterated loop spaces $\Omega^n X$ for all $n$.