Philosophical interest in n-categories may be characterised as belonging to one of two kinds.
Metaphysical: The formation of a new language which may prove to be as important for philosophy as predicate logic was for Bertrand Russell and the analytic philosophers he inspired.
Illustrative of mathematics as intellectual enquiry: Such a reconstitution of the fundamental language of mathematics reveals much about mathematics as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts.
Higher category theory provides a new foundation for mathematics - logical and philosophical.
Higher category theory refines the notion of sameness to allow more subtle variants. It advocates the avoidance of evil.
There ought to be a categorified logic, or 2-logic. There are some suggestions that existing work on modal logic is relevant. Blog discussion: I, II, III, IV. Mike Shulman’s project: 2-categorical logic.
Higher category theory may provide the right tools to take physics forward. A Prehistory of n-Categorical Physics See also physics.
More speculatively, category theory may prove useful in biology.
“Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self–renewing discourse.” (Yuri Manin)
To categorify mathematical constructions properly, one must have understood their essential features. This leads us to consider what it is to get concepts ‘right’. Which kind of ‘realism’ is suitable for mathematics? Which virtues should a mathematical community possess to further its ends: a knowledge of its history, close attention to instruction and the sharing of knowledge, a willingness to admit to what is currently lacking in its programmes?
Gavin: This is a sensitive topic. I do not know if this is the right place for it, but I feel that Lab would be lacking balance if it did not have some part that attempted to address the fact Category Theory has been responsible for some pretty wretched mathematics. To go into too much detail might be discourteous. I was delighted by the beauty of category theory when I first met it (in the library of the Courant Institute in 1962, when I was supposed to be writing a thesis in Theoretical Physics). However, I soon realized that in the mathematical community of the UK, at least, at that time, too great an indulgence in abstraction was viewed with suspicion. It is a suspicion that I share, to some extent. I remember coming across an article, some years later, in a Physics journal, written by a well-respected physicist, in which large chunks of a paper by Applegate on the Tierney Tower of adjoint functors were repeated verbatim; and evidently with the corrupt intention of combombulating readers. Category theory is easily misused, either for unnecessary flights into the Empyrean, or for more evil El-Naschiean purposes. But I guess that most nLabians are perfectly aware of all this. Since those early days category theory has become a lingua franca for almost every branch of mathematics. I too love abstraction; but I am just a little bit wary of it.
If you came across a journal called Category Theory and Prosopography would you smell a rat? The difficulty with cross-subject journals is that they make it too easy for authors to evade proper peer-review. Told that the mathematics in his paper is crap, the author can try to wriggle out by asserting that it is a masterpiece of prosopography. The shock and awe of the reputation of mathematics makes it less likely that prosopographers will condemn his paper from their side. It is just because Category Theory binds together widely disparate branches of mathematics that it lends itself to the snake-oil merchant.