Over the last century category theory has emerged as a major branch of pure mathematics, both as a topic in its own right and in application to a wide array of fields. Underlying that wide applicability is the notion that category theory provides a kind of universal language of mathematics. Concrete ideas and concepts in diverse branches of mathematics turn out to be expressions of the same categorical notion and expressing them in form reveals connections between different fields. An arguable cost to this treatment is the abstract nature of the resulting expressions. This has led to work in category theory, especially work that treats it as a topic in its own right, to be given the moniker of 'abstract nonsense'.
The relationship category theory has to other areas of mathematics provide a tangible way of understanding the meta-mathematical relationship between application, theory and abstraction. This relationship is an important driver of research, specific investigations are both motivated and enabled by the movement between these three levels of engagement with mathematical ideas. In this talk, I aim to illustrate and discuss that metamathematical process principally through the lens of my recent work adapting a categorical framing of classical model theory concepts to the non-classical setting.