Diagrammatics in Categorification and Compositionality

In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends—categorification and compositionality—in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment—from a pedagogically novel perspective that introduces almost all notion via diagrammatic language—of the categorical machinery with which we may express the broader no- tions that found the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. Finally, the work culminates in a manifesto on the philosophical role of category theory.