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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?

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    $\begingroup$ It's been said that $1$ is the loneliest number. That's pretty special. $\endgroup$
    – Asaf Karagila ♦
    Commented Dec 20, 2014 at 18:03
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    $\begingroup$ Perhaps 1 is just the average n-category number. $\endgroup$
    – Myshkin
    Commented Dec 20, 2014 at 18:37
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    $\begingroup$ “I didn't invent categories to study functors; I invented them to study natural transformations.” – Saunders Mac Lane. So, apparently the inventor already thought that it wasn't about the objects, nor the morphisms, nor even the functors, but higher categorical. $\endgroup$
    – jmc
    Commented Dec 20, 2014 at 18:54
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    $\begingroup$ The general rule of thumb appears to be that the prevalence of $n$-categories is very rapidly decreasing as a function of $n$ (except for the special case $n=\infty$). Indeed, ordinary sets are 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
    Commented Dec 20, 2014 at 19:49
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    $\begingroup$ Is it a step at all? I would describe it more as a pocket of mathematics, rather than something general. In certain contexts a little category theory helps, but it's far from a general tool for mathematics. $\endgroup$
    – Ryan Budney
    Commented Dec 20, 2014 at 23:49

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You should increase your category level if you think it'll help you understand something you're thinking about. Otherwise, don't.

To the extent that there's something special about $1$-categories, it might be that on the one hand it's a big step up from $0$-categories and on the other hand it's a low enough category level that you don't need to worry about coherence conditions. Already at the $2$-categorical level an important distinction arises between weak and strict $2$-categories (when I say "$2$-category" I mean the weak thing by default because the alternative does not generalize to higher values of $2$) that takes some time to appreciate, although you can attempt to ignore it by strictifying, and at the $3$-categorical level you can't even always strictify..

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    $\begingroup$ "the alternative does not generalize to higher values of 2", what?! $\endgroup$
    – Fernando Muro
    Commented Dec 21, 2014 at 9:50
  • $\begingroup$ @Fernando: my understanding is that the standard convention is that "bicategory" means weak and "$2$-category" means strict, but this convention doesn't generalize past "tricategory" and so forth to an arbitrary category number, and it seems perverse to have "$n$-category" mean strict by default. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 21, 2014 at 10:20
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    $\begingroup$ for me it's strict, but regardless of conventions weak and strict 2-categories generalize to higher dimensions. $\endgroup$
    – Fernando Muro
    Commented Dec 21, 2014 at 15:36
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Taking the homotopical point of view, a 1-category is a space where an imbedding of a more than one dimensional cell is determined by the boundary. On the one hand this makes it easier to calculate and make concrete statements, on the other it is often rather limiting and unnatural. A good example of this is the derived category, which can be seen as the homotopy category of a higher category.

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