Categories for the Working Philosopher
Elaine Landry is working on putting together a book, Categories for the Working Philosopher. The list of topics that various contributors are working on is broad, including model theory, special relativity, quantum mechanics and ontology, biology, computer science, foundations, and more.
John Baez has posted an excellent abstract of his planned contribution. It talks about what happens when we attempt to formalize intuitive notions of equality. Some exciting relationships fall out, such as understanding equivalence as a path between two points in a topological space.
In short, by formalizing the concept of sameness using isomorphisms instead of equality, we bring group theory, geometry and topology closer to the foundations of mathematics, giving them a kind of inevitability that they might not otherwise seem to possess. We also see that these three subjects are tightly connected. A lot of important mathematics, and also theoretical physics, flows from this realization.
I recommend checking out the entire thing. John mentions in the comments that there is a (less exciting) paper in the same vein here, and all of this ties into the recent work on homotopy type theory.