I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms between objects instead of just the objects themselves. You might say that, as I have mathematically matured, my tendency has been to think more at the level of $1$-categories instead of at the level of $0$-categories.

Phrasing it like that made me wonder: is this just the first step? Should I 'really' be thinking not about morphisms themselves, but about morphisms between morphisms? Indeed, of the couple of examples I can think of off the top of my head where such $2$-morphisms arise quite naturally, it does seem to be the 'right' thing to do to think about morphisms up to isomorphism (the best example I have in mind is the difference between isomorphism and equivalence of categories, the former not being quite as useful).

Of course, once you've decided to think $2$-categorically, why not think $3$-categorically or $4$-categorically, or hell, why not $\infty$-categorically?

I apologize for this being a subjective question, but: To what extent should I begin to train myself to think in terms of higher category theory, or, is there indeed something special occurring at the 'normal' level of $1$-categories that makes this level in particular the 'right' way to think about things?

setsare vastly more common in day-to-day mathematics than categories, in the sense that an average mathematics paper will contain a significantly larger number of different sets than categories. Categories are in turn vastly more common than 2-categories, and so on. In the other direction, truth values are more common than sets. $\endgroup$ – Dan Petersen Dec 20 '14 at 19:492more comments