In this YouTube video by Richard Southwell at the 2:30 timestamp, it is said that:
Fractal geometry can be studied quite profitably using category theory.
The fields seem very different to me but I am not very familiar with either field. I don't see how the study of composition in category theory would apply to the study of fractals, other than for specific types such as IFS fractals...
Googling this statement and searching stack exchange, I can't find any results that link the two fields.
So, how are fractals studied using category theory?
It might initially look implausible that there's a connection, I agree. After all, as Tom Leinster so nicely puts it in his lovely book, category theory “takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.” And you might suppose that the study of fractal geometry is mostly about just those fine details which become invisible at the level of category theory. Just look, for example, at the fine detail of the theorems about Julia sets in Barnsley's Fractals Everywhere.
But ...
But then fractals arise from self-similarities; and self-similarity seems on second thoughts to be the kind of high-level idea which might after all lend itself to some sort of category theoretic treatment. OK we might not get at some of the nitty-gritty theorems that way but we might get some higher-level results. And so we find, ahah, that there is indeed e.g. an interesting paper by none other than Tom Leinster about Self-Similarity, mentioning Julia sets and more, which brings categorial ideas to bear. See here. ....