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similar to the horizontal categorification article, this article also uses query boxes, which is against the recommendation of the “Anything I shouldn’t do?” section of the writing in the nLab article. Should the query boxes be removed?
Yes! Thanks for the heads-up.
So I am hereby removing the query box that was sitting in the entry, copying it to here, just for the record:
+– {: .query} This all looks good, with one complaint: I don't think anyone would consider laxification (passage from sets to posets, groupoids to categories, categories to $2$-posets, etc) to be ’categorification’. That is, the term ’directed categorification’ works best for the combination of groupoidal categorification followed by laxification. Do you agree, Mike? —Toby
Mike Shulman: I actually think those are definitely a type of categorification. I can think of several places where people have explicitly categorified from categories to (1,2)-categories, either with some particular application in mind, or as a waypoint on the way to 2-categories. I can see that the term “directed categorification” might most easily be interpreted to refer to the combination, but I do think “pure directification” is also a form of categorification.
Toby: I think that examples of ’categorifying’ form categories to $2$-posets would help me understand. (Clear examples of ’categorifying’ from $(\infty,n)$-categories to $(\infty,n+1)$-categories would be maximally motivating, if you have them too.)
Of course, $2$-posets can be thought of as special $2$-categories, and that may come even more easily than thinking of posets as special categories. But moving from categories to $2$-posets or from sets to posets seems like such a different thing. It probably doesn't help that I don't like the term horizontal categorification either; categories may be a common generalisation of groupoids, posets, and monoids, but out of all of those three originals, only moving from posets (or something even simpler such as sets) to categories seems like ’categorification’ to me; only that moves us higher in the hierarchy of $n$-categories.
Mike Shulman: I think moving from $(\infty,n)$-categories to $(\infty,n+1)$-categories is definitely vertical categorification. It’s exactly like moving from $n$-categories to $(n+1)$-categories in so many ways. In particular, the collection of all $(\infty,n)$-categories is an $(\infty,n+1)$-category. Or consider the notion of symmetric monoidal (∞,1)-category; I think this is clearly a categorification of the notion of symmetric monoidal ∞-groupoid, i.e. infinite loop space. Not only can a definition of symmetric monoidal (∞,1)-category obtained from a definition of symmetric monoidal ∞-groupoid in a straightforward way, but it also satisfies a version of the microcosm principle: you can define symmetric monoidal objects in any symmetric monoidal (∞,1)-category. The beauty of working in the ∞-context from the start is that you don’t have to invent new coherence isomorphisms when you categorify, but I don’t think that makes it any less categorification.
For the case of (2,1)-categories, here are a few papers that come to mind: * Power and Kinoshita, “Lax naturality through enrichment” * Carboni and Walters, “Cartesian bicategories I” * or even the discussion of the Burali-Forti paradox in a (2,3)-topos here.
The point about “horizontal categorification” is, of course, that the intuition of “moving us higher in the hierarchy of $n$-categories” is only applicable, basically by definition, to vertical categorification. Horizontal categorification is supposed to be something different. I’d rather not call that “categorification” either, but it is certainly a process of “generalizing something to use categories.” However I do feel, as I’ve said, that “directification” does “move us higher in the hierarchy of $n$-categories”—it’s just that that “hierarchy” is not linear! Remember the periodic table of (n,r)-categories.
Toby: All right, yes, that helps. I'll sleep on it, but there's a good chance that you've convinced me. =–
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