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I added to vertical categorification the comments that I'd made at MathOverflow, as Urs has requested. I'm not sure that I'm happy with where I put them and how I labelled them, but maybe it's better if other people judge that.
I think of "categorification" as including "laxification". Adding only invertible higher arrows I would call "groupoidification" (if that weren't already being used for something else...).
Yeah, I might do that too if that weren't already being used.
Except that a lot of examples called ‘categorification’ in the wild don't add noninvertible higher arrows; an example is moving from locales to Grothendieck topoi (or from topological spaces to ionaid, for that matter).
Conceptually, I think that it helps me to see adding higher arrows and allowing arrows (at whatever level) to be noninvertible as separate steps, even if you might want to do both.
I think the distinction is a fine one, but I would still prefer to allow "categorification" to refer to the whole shebang. What I'm objecting to is I think confined to the third paragraph here.
I would allow it, certainly, but I also need a term for not the whole shebang. So I said ‘categorification proper’. The main point of categorification (whatevery you call it), as I see it, is that.
But if you think that what I wrote can be better phrased, go ahead. Then I'll argue if I don't like it. (^_^)
Ok, I edited it a bit. What I object to is that "categorification proper" makes it sound like using "categorification" for the whole shebang is somehow "improper," i.e. not quite right.
The opposite of ‘proper’ in this sense, I think, is ‘greater’ (with the latter coming before the noun, as usual in English, rather than after). So they're both emotionally positive terms. (^_^) A random example from an Internet discussion board: ‘Did Magna Grecia (“Greater Greece”) ever replace Greece proper as the center of Greek civilization?’.
Anyway, your edits are fine, although I have put in a call for a good name. What do you think about the perspective that ‘greater categorification’ may be understood as the composite of two processes: ‘categorification proper’ and ‘laxification’, whatever we may decide to call these? (And how about that name ‘laxification’?; I expect you of all people to have an opinion on that!)
It's not entirely clear to me that the two can be completely disentangled, but it's interesting to think about. Are you thinking of always composing them in a specific order? E.g. we can get from sets to categories via groupoids, which is maybe what you are thinking of, or via preorders -- is going from preorders to categories a process of "adding higher morphisms but not allowing noninvertible things"?
Another possible word for "adding higher invertible morphisms" is "homotopification" since when carried to the limit it replaces all sets by -groupoids, i.e. homotopy types. Or from the type theorists' point of view it could be "intensionalization" since -groupoids "are" intensional types.
I'm not all that taken by "laxification," partly I think because "lax" things are often viewed as weird and technical (which I think lax functors and "lax categories" often are, although lax transformations and lax morphisms of algebras are certainly ubiquitous), while I think in some circles (such as algebraic geometry) there is already too much of a tendency to regard noninvertible higher cells as weird or pathological or not important. It could also be called "directification" or "directionalization" or "directing" (argh, English) since it involves making things "directed" that weren't before.
I don't really like any of these words, but I'm just thinking out loud.
is going from preorders to categories a process of "adding higher morphisms but not allowing noninvertible things"
Yes, if you mean not allowing additional noninvertible things. (If you do, then you get 2-posets, of course.)
Although when speaking of specific categories, you get a very different result by decatorifying to a poset and then delaxifying to a set, than you get by delaxifying to a groupoid and then decategorifying to a set, and the latter route is usually (but not always) more interesting.
Or from the type theorists' point of view it could be "intensionalization"
Maybe. I'm still a bit cautious about approaching higher categories through intensional type theory; while a type in intensional type theory is an -groupoid, I see no reason why most -groupoids should arise in this way. Indeed, adding an extensionality axiom to an intensional type theory is often consistent, in which case none of these -groupoids can be proved to be not discrete. (But I need to catch up on the latest ideas here.)
I'll bet that Urs would like ‘homotopification’; we should ask him.
I think because "lax" things are often viewed as weird and technical
But that's wrong, right? For people who already know and love categorification, you point out that (half of the time) they've been laxifying all along! Although that won't help with geometers who dislike noninvertible higher cells already.
Something based on ‘direct’ could be good, although none of those particular words sound nice to me either.
Although that won't help with geometers who dislike noninvertible higher cells already.
I do not understand the motivation for this comment. My experience with monoidal functors is just the opposite. In works in geometry monoidal functors are almost always lax, while most of the papers in category theory mainly concentrate and on pseudo-version as a default.
"groupoidification" (if that weren't already being used for something else...)
But except for this cafe circle I would not say that word "groupoidification" is generally accepted: I mean doing the linear algebra of correspondences ("spans") is a great and old idea (very popular in algebraic geometry: Fourier-Mukai transforms being the most well-known archetypal exsample) and does not need to include groupoids in the game, so the term is in my opinion misleading. Spanification is not yet used term; but I do not think one needs to have a term for linear algebra in some setup. Linear algebra has been many times generalized, from vector spaces to modules, sheaves of modules, additive categories etc. It still stays linear lagebra and the basic idea of Fourier-Mukai transform is still at its basis the idea of Fourier transform, never mind the sheaves, inner homs and other bells and whistles.
I'm presuming that by "decategorifying to a poset" you mean taking the reflection into posets, which is a left adjoint, and by "delaxifying to a groupoid" you mean taking the core, which is a right adjoint? Is this some sort of indication of a general principle? (Of course there is also a left adjoint landing in groupoids that forcibly inverts everything, but that is probably less interesting?)
Certainly, if there's going to be a type theory that is sufficient as an internal logic for -toposes, then adding an extensionality axiom to it must not be consistent. I think one important axiom of such a theory will be exactness, which in the -case means that any internal groupoid has a "quotient." In low dimensions, exactness and extensivity are what prevent degenerate models. For instance, any Heyting algebra is an exact Heyting category, but it is not extensive. And any extensive Heyting category is an extensive Heyting 2-category, but it is not (2-)exact.
As I said, I think some lax things are weird and technical, like lax algebras, lax categories, and lax functors. But lax transformations and lax morphisms of algebras are ubiquitous. So maybe you're right, but I'm still worried about the geometers. (-:
Zoran, I had a discussion with an algebraic geometer about Lurie's use of "-category" to mean -category, and the similar tendency to use "2-category" to mean "(2,1)-category". His response was that most algebraic geometers think of noninvertible 2-cells and higher as "pathological," so that adding the ",1" is just a "niceness condition" which one can easily omit to mention, as in saying "space" for "compactly generated space." I can't understand this, for the reasons you gave and others, but that's what he said, and the fact is that many geometers do use the words "-category" and (I believe) even "2-category" in this way.
I also don't think it's true that category-theory papers concentrate on pseudo/strong monoidal functors. Lax functors of bicategories are fairly exotic, but lax monoidal functors are so important that in many circles they are simply called "monoidal functors," with "strong" added when the comparison maps are invertible. Lax morphisms of algebras for 2-monads, of which monoidal categories are a special case, play a very important role in the theory and are studied in a lot of places.
Harry, I don't like "local" because there is another category-theoretic meaning of "local" that that could easily be confused with (and which confused me at first when I read your post), namely "applying to hom-sets." To me "local categorification" "obviously" means "categorifying the hom-sets."
Is this some sort of indication of a general principle?
H'm, maybe it is. It just seems the obvious thing to do, and it seemed to be the obvious thing to you as well. That actually strengthens the case that decategorifying from a category to a set is just as much ‘decategorification’ simple as from a category to a poset, since each is a left adjoint.
I think one important axiom of such a theory will be exactness
Interesting, since one potential way to distinguish a ‘type theory’ from a ‘set theory’ is that the former is usually not exact (so that one must pass to setoids to get an exact set theory).
@ Harry
‘groupoidal categorification’ is a tad long, but it also strikes me as a very clear term.
But laxification is not categorification at all, as I see it. At least not vertical categorification.
Laxification could be done at any level, moving say from a $(3,1)$-category to a $(3,2)$-category, not only at the top. But Mike may feel as you do; see discussion at vertical categorification.
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