What is Categorification?

October 22, 2008

Posted by John Baez

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Some folks are starting to talk more and more about “categorification”. Others are getting more and more puzzled by what this word means.

Let me tell you what it means.

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Actually, I’ll just tell you what it means to categorify a given set. Then you can guess for yourself what it means to categorify a function between sets — and I’ll tell you if your answer is right, if you want.

Since a huge amount of math is about sets and functions, this will give you some idea of what it means to categorify a given mathematical idea, or a given branch of mathematics. For example: to categorify the Jones polynomial, an invariant of knots.

Okay:

To categorify a set $S$ is to find a category $C$ and a function

$p: Decat(C) \to S$

where $Decat(C)$ is the set of isomorphism classes of objects of $C$ .

The classic example: $S$ is the natural numbers, $C$ is the category of finite sets, $p$ is ‘cardinality’.

The other classic example: $S$ is the natural numbers, $C$ is the category of finite-dimensional vector spaces, $p$ is ‘dimension’.

In these examples $p$ is one-to-one and onto, which is very nice.

Can you guess what I mean by saying “if we categorify the natural numbers to the category of finite sets, addition gets categorified to disjoint union?”

Emmy Noether categorified the concept of ‘Betti number’. Here $S$ is the set of natural numbers, $C$ is the category of finitely generated abelian groups, and $p$ is ‘rank’.

Extending this idea a bit further, we can categorify the set $S$ of Laurent polynomials with natural number coefficients using category $C$ of bounded chain complexes of finitely generated abelian groups. Here $p$ is ‘Poincaré polynomial’.

Khovanov categorified the set $S$ of Laurent polynomials with integer coefficients using the category $C$ of bounded chain complexes of $\mathbb{Z}/2$ -graded finitely generated abelian groups. This was the first easy step towards his real accomplishment: categorifying a bunch of polynomial invariants of knots.

In these examples $p$ is onto, but not one-to-one. If $p$ isn’t onto, categorification becomes too easy to be interesting, unless we use other tricks, like the ‘Grothendieck group’ trick.

There are many other important examples… but if you know some math, you can probably find your own!

Posted at October 22, 2008 3:33 PM UTC

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Re: What is Categorification?

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Maybe one can argue that there is a single use of the word which deserves to be addressed as the single correct one, but maybe also the word is commonly being used in different ways.

Apart from the

categorification = right inverse to taking equivalence classes / taking $\pi_0$

which you describe in the above entry there is also

categorification = relaxing structures up to coherent homotopy

E.g. 2-groups as “categorified groups”, $L_\infty$ -algebras as “categorified Lie algebras” etc.

I recalled this point discussed on Jeffrey Morton’s blog at Two Kinds of Categorification.

Maybe we need more descriptive terms! Categorification in the first sense is maybe more descriptively described as “categorical resolution” (as in: stacky quotient), whereas the second is maybe better “homotopification”.

(Some people say “vertical categorification” for this second meaning and then “horizontal categorification” for “oidification”, i.e. for passing from single to many object versions (group tp groupoid, algebra to algebroid).)

Posted by: Urs Schreiber on October 22, 2008 8:35 PM | Permalink | Reply to this

Re: What is Categorification?

Categorification in the first sense is maybe more descriptively described as “categorical resolution” (as in: stacky quotient), whereas the second is maybe better “homotopification”.

YES, especially for the first, perhaps homotopical resolution for the second

Posted by: jim stasheff on October 23, 2008 3:25 PM | Permalink | Reply to this

Re: What is Categorification?

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Urs wrote:

Maybe one can argue that there is a single use of the word which deserves to be addressed as the single correct one, but maybe also the word is commonly being used in different ways.

It’s used in lots of ways and I think that’s fine. I just wanted to provide an ultra-quick introduction to the word, for people who are puzzled by it — in particular, for people who are confused about how Khovanov’s usage relates to mine. I wanted to emphasize that there’s no discrepancy.

Apart from the

categorification = right inverse to taking equivalence classes / taking $\pi_0$

which you describe in the above entry there is also

categorification = relaxing structures up to coherent homotopy

E.g. 2-groups as “categorified groups”, $L_\infty$ -algebras as “categorified Lie algebras” etc.

Right. But for the bewildered nonexperts (the people I’m addressing here), I should point out that the second usage is intimately linked to the first one:

A 2-group is a category $C$ equipped with some extra structure. If we take the set of isomorphism classes of objects in a 2-group, we get a group! So, for any group we can seek 2-groups that categorify it in the sense described in my post. So, by a kind of extension of language, we say ‘the concept of 2-group categorifies the concept of group’.

A Lie 2-algebra is a category $C$ equipped with some extra structure. If we take the set of isomorphism classes of objects in a Lie 2-algebra, we get a Lie algebra! So, for any Lie algebra we can seek Lie 2-algebras that categorify it in the sense described in my post. So, by a kind of extension of language, we say ‘the concept of Lie 2-algebra categorifies the concept of Lie 2-algebra’.

Posted by: John Baez on October 23, 2008 4:04 PM | Permalink | Reply to this

Re: What is Categorification?

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Seems like Decat is a special case of $p$ : Decat takes isomorphism classes of objects, whereas $p$ takes a larger equivalence, making the map many-to-one. So I’d say this:

To categorify a set $S$ , we find a category $C$ and an equivalence $E$ such that $Ob(C) / E = S$ .

Then we can consider the idea of, say, a group being a set with structure, and then categorifying is looking for a category $C$ with structure and an equivalence $E$ that gets along with it, so that decategorifying a 2-group by taking isomorphism classes gives a group.

Posted by: Mike Stay on October 24, 2008 12:28 AM | Permalink | Reply to this

Re: What is Categorification?

If a scientist is aware of category theory and is introduced to the word ‘categorification’, they often make the natural assumption that it means ‘re-expressing conventional concepts using the language of category theory’. For people who might be entertaining the idea of learning category theory and making it a part of their research, understanding the benefits of this sort of ‘categorifiction’ is crucial, and people are often interested in finding out more about it.

Actually, I think this would be a better meaning for the word. Maybe we could replace the current word ‘categorification’ with something like ‘levelling up’!

Posted by: Jamie Vicary on October 22, 2008 9:23 PM | Permalink | Reply to this

Re: What is Categorification?

Jamie wrote:

Actually, I think this would be a better meaning for the word. Maybe we could replace the current word ‘categorification’ with something like ‘levelling up’!

Too late, I think.

Posted by: John Baez on October 23, 2008 3:45 PM | Permalink | Reply to this

Re: What is Categorification?

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Maybe we could replace the current word

Too late, I think.

But maybe one should openly admit and emphasize that the term category itself is really bad. It sounds like “drawer”, a place to put away stuff in, and that’s what it was originally indended to allude to: to those large categories which contain sets with certain structures and all structure-preserving maps between them.

But the real power of categories comes of course from realizing that they are objects of interest in their own right, not just jars to stow awy in your stuff. Just yesterday I read in Ronnie Brown’s upcoming book the curious observation that small groupoids and their theory generalizing group theory were not mentioned in Eilenberg and Mac Lane’s original article on categories. Maybe had they mentioned it, they would have thought of another term instead of “category”, whose everyday meaning is rather unrelated to many of the groupoids one runs into in nature.

A better descriptive term for categories would be combinatorial directed spaces, albeit too long, of course.

But it would help. Instead of telling the physicist on the street:

“hey, you should really think about categorifying your formalism”,

which will leave him or her puzzled and confused, one could say:

“hey, did you know that the objects which you are interested in can naturally be regarded as living in the following combinatorial directed space…”

and the basic idea would immediately be clear intuitively.

Posted by: Urs Schreiber on October 24, 2008 9:14 AM | Permalink | Reply to this

Re: What is Categorification?

I wonder how much thought Eilenberg and Mac Lane put into the choice of the term ‘category’. The story goes that they took it from Kant, but it’s not so obvious what a mathematical category has to do with

…the condition of the possibility of objects in general, that is, objects as such, any and all objects, not specific objects in particular.

Your ‘drawer’ image is in line with the common use of the term, but as the Wikipedia article says

Our contemporary definition of “category” is “a division into groups.” For Kant and Aristotle, though, the meaning of the word was in accordance with its original definition: “that which can be said about objects in general, or all possible objects.”

Kant has 12 categories arranged in 4 classes:

Quantity: Unity, Plurality, Totality
Quality: Reality, Negation, Limitation
Relation: Inherence and Subsistence (substance and accident), Causality and Dependence (cause and effect), Community (reciprocity)
Modality: Possibility, Existence, Necessity

Yet, Mac Lane certainly read philosophy.

Posted by: David Corfield on October 24, 2008 10:18 AM | Permalink | Reply to this

Stuff, structure, Poperties, and Kant; Re: What is Categorification?

Okay, now let’s try to tie Kant to the approach discussed several times in the n-Category Cafe.

* STUFF
measured by Quantity:
* Unity,
* Plurality,
* Totality

* STRUCTURE of Stuff
described by Relation:
* Inherence and Subsistence (substance and accident),
* Causality and Dependence (cause and effect),
* Community (reciprocity)

* PROPERTIES of Structures of Stuff
deals with Quality:
* Reality,
* Negation,
* Limitation
and interpreted via Modality:
* Possibility,
* Existence,
* Necessity

Any thoughts on this attempt, or notions of how to improve it?

Posted by: Jonathan Vos Post on November 2, 2008 1:55 AM | Permalink | Reply to this

Brian D. Josephson comments; Re: Stuff, structure, Poperties, and Kant; Re: What is Categorification?

Prof. Brian D. Josephson emailed me from Cavendish Lab to comment:

I’d interpret it differently. Kant was interested in our built-in reasoning and categories of thinking (“pure reason” as in the title of his book) generally. Category theorists on the other hand say that category theory reflects how we think. So the connection would be more all-encompassing than the listing quoted above suggests.

Health warning: I am no expert in any of these matters!

Posted by: Jonathan Vos Post on November 2, 2008 5:08 PM | Permalink | Reply to this

Re: Stuff, structure, Poperties, and Kant; Re: What is Categorification?

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Jonathan wrote:

* PROPERTIES of Structures of Stuff

deals with Quality:

* Reality,

* Negation,

* Limitation

and interpreted via Modality:

* Possibility,

* Existence,

* Necessity

Any thoughts on this attempt, or notions of how to improve it?

Since Kant is famous for having said ‘existence is not a property’, I’m not this game works well.

I don’t know Kant’s philosophy well enough to know if any game like this is worth playing.

Posted by: John Baez on November 3, 2008 8:39 PM | Permalink | Reply to this

Re: Stuff, structure, Poperties, and Kant; Re: What is Categorification?

Kant derived his list of categories from his Table of judgments. Most people looking at that table now will find it of interest solely as an historical artefact, telling us much about the state of logic at the time and about Kant’s thought in general.

If you want to do something useful for categorified logic, a better furrow to plough would be to extract the internal language of a 2-category, especially that of a 2-topos.

Posted by: David Corfield on November 4, 2008 9:09 AM | Permalink | Reply to this

Re: What is Categorification?

Surely you are joking, Dr. Feynman

and since when do most math terms carry their street meaning, even remotely ;-)

Posted by: jim stasheff on October 24, 2008 2:47 PM | Permalink | Reply to this

Re: What is Categorification?

I once read a remark by a logician, I forget who, that refers to a distinction between the ‘sacred’ and ‘profane’ traditions in logic. He thought of the sacred tradition as associated to foundations, set theory, Goedel, and so forth. The writer apparently emerged out of it when he took courses on recursion theory from Hartley Rogers and started treating logic casually, like most other areas of mathematics.

My feeling is that many mathematicians’ view of category theory went through a similar transition in the course of our times, perhaps the last several decades. Regardless of what input there was from philosophy, it’s hard to avoid the impression that Maclane’s categories were somewhat on the sacred side, conjuring up images of whole universes of some profound sort, and a functor was really an awful transformation. (Consequently, one couldn’t allow too many of them.) These days, a typical category looks much more like a graph.

This perspective will probably have matured when people think nothing of defining single-use categories and functors for solving concrete problems, pretty much in the way that an analyst cooks up an auxiliary function to prove a bound.

Posted by: Minhyong Kim on October 24, 2008 11:48 PM | Permalink | Reply to this

Re: What is Categorification?

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Urs wrote:

… maybe one should openly admit and emphasize that the term category itself is really bad. It sounds like “drawer”, a place to put away stuff in, and that’s what it was originally indended to allude to…

A better descriptive term for categories would be combinatorial directed spaces, albeit too long, of course.

It’s amusing to note that ‘space’ can also mean place where you put stuff.

I guess the difference is that a ‘category’ is a kind of metaphorical drawer where things are logically organized, while a ‘space’ is a kind of actual drawer where things are geometrically organized.

But the difference between ‘metaphor’ and ‘actuality’ is always rather blurry in mathematics.

Topos theory already tried hard to destroy the distinction between logic and geometry — the title of Mac Lane and Moerdijk’s book Sheaves in Geometry and Logic makes that clear.

But now you’re talking about a somewhat different way of thinking of a category as a space: you’re imagining a category not as a collection of sheaves on a poset of open subsets, but simply as a collection of ‘paths’.

Posted by: John Baez on January 16, 2009 6:27 PM | Permalink | Reply to this

Re: What is Categorification?

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It’s amusing to note that ‘space’ can also mean place where you put stuff.

True. But not every space is used that way.

Sure, there is for instance the space of all $G$ -torsors, the space of all vector spaces, etc. All these are spaces. All these correspond to the kind of categories which motivated (as far as I undertand) the term “category”.

But there are many other spaces which we don’t want to think of naturally as just a “space of things of certain kind”, or worse: “space of things of the kind ‘sets with extra structure”. And these correspond to categories for which the term “category” is not really suggestive. I’d say.

Topos theory already tried hard to destroy the distinction between logic and geometry — the title of Mac Lane and Moerdijk’s book Sheaves in Geometry and Logic makes that clear.

But now you’re talking about a somewhat different way of thinking of a category as a space: you’re imagining a category not as a collection of sheaves on a poset of open subsets, but simply as a collection of ‘paths’.

Hm, I wouldn’t say so. The way I am thinking of a cateory as a directed space is the way we think of a groupoid as a space: a combinatorial model which remembers only certain points and only certain paths. Categories are combinatorial models for directed spaces.

Sheaves raise both concept to a generalized level. A sheaf, a priori, is a generalized space (a “parameterized space”), modeled locally on the spaces which form the objects of the site that the sheaf is defined on.

So, a sheaf on Top is a generalized toplogical spaces. A sheaf on Grpd is a generalized groupoid.

These two sentences together incidentally make an otherwise esoteric point appear very naturally: “sheaf on Grpd?” you might ask. But isnt Grpd a 2-category? Yes, and if we are serious about it we should acknowledge this – and arrive will at “derived” generalized spaces.

And “derived” is just antother example for unsuggestive terminology for a fundamrntal concept which would deserve better.

Posted by: Urs Schreiber on January 17, 2009 1:26 PM | Permalink | Reply to this

Re: What is Categorification?

since when have we been concerned about using a word with a totally disparate street meaning
?

Posted by: jim stasheff on January 17, 2009 1:52 PM | Permalink | Reply to this

Re: What is Categorification?

Jamie Vicary wrote in part:

If a scientist is aware of category theory and is introduced to the word ‘categorification’, they often make the natural assumption that it means ‘re-expressing conventional concepts using the language of category theory’.

I would call this ‘expressing the concept in arrow-theoretic terms’ or ‘internalisation’.

The latter term is especially for when you take a concept about sets and functions (or possibly spaces and continuous maps, etc), express it in these terms (that is as about objects and morphisms in a specific category, such as the category of sets and functions) and then say ‘What if I interpret this in an arbitrary category C?’. (The last step is internalisation proper; you can leave it out if you’re feeling cautious.)

Posted by: Toby Bartels on October 24, 2008 5:14 AM | Permalink | Reply to this

Re: What is Categorification?

What about what I do: trying to rewrite a topic like knot theory from a category-theoretic perspective. Specifically, I try to view knots and links as special kinds of arrows in the category of tangles, and I try to find functors that give well-known knot invariants when restricted to these special arrows.

I don’t think this is “internalization” in your sense, and yet it fits Jamie’s narrative. In fact, at first I called the process “categorification” myself. Luckily I was at a school where I was able to learn (what we consider to be) the proper definition of categorification before I got too far with that. But I still have no good term for what I do.

Posted by: John Armstrong on October 24, 2008 12:40 PM | Permalink | Reply to this

Re: What is Categorification?

So this is the ‘embedding entities in a nice category’ approach. Better a good category of things than a category of good things.

I once posed the question, and had no response, of whether you do that the next level up. Do you ever select a good 2-category of things over a 2-category of good things?

Posted by: David Corfield on October 24, 2008 12:52 PM | Permalink | Reply to this

Re: What is Categorification?

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David wrote:

I once posed the question, and had no response, of whether you do that the next level up. Do you ever select a good 2-category of things over a 2-category of good things?

Sorry for not answering this earlier! My answer would have been “Of course we should: it’s the same principle at work! Namely, we prefer a good context over a lousy context that has been chosen to contain only good objects. The good context may contain some lousy objects — but so what? We are free to state theorems about just the good objects if we like. Working in a good context will make it easier to prove those theorems.”

But, just a couple days ago I bumped into a nice example, from Eugene Lerman’s paper on orbifolds. He’s trying to study orbifolds as objects of a 2-category. But what 2-category?

There’s the 2-category of orbifolds, but this sits in a larger and in some ways nicer 2-category of Lie groupoids and Hilsum-Skandalis maps — which in turn sits in the even larger and even nicer 2-category of differentiable stacks! If you read Eugene’s paper you can see him getting pulled towards these larger 2-categories because they are nicer contexts for reasoning, even though they contain more lousy objects.

I had some nice conversations with Eugene in Champaign last week, in which he explained this stuff.

Posted by: John Baez on October 28, 2008 6:09 AM | Permalink | Reply to this

Re: What is Categorification?

I see Eugene has this kind of rationale:

This leaves one to localize Lie groupoids at equivalences as a 2-category. There are several ways of carrying this out. The two ways that I find most appealing are (1) replacing functors by bibundles (and natural transformations by equivariant maps of bibundles) and (2) embedding groupoids into the 2-category of stacks.

So dealing with the problem that smooth functors which are equivalences may have no smooth inverse.

But what other nice properties of 2-categories are out there to attract us? I suppose the answer to this is ‘Work out a categorified form of doctrines to include 2-toposes, symmetric monoidal 2-categories, and so on’.

Posted by: David Corfield on October 28, 2008 10:34 AM | Permalink | Reply to this

Re: What is Categorification?

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David writes:

So dealing with the problem that smooth functors which are equivalences may have no smooth inverse.

That’s the reason for going from the 2-category of Lie groupoids with smooth functors as morphisms to either the 2-category of Lie groupoids with Hilsum–Skandalis maps as morphisms or the 2-category of differentiable stacks, or more general stacks.

But which solution is ‘better’ — and why?

I’d like to hear people’s opinions on this — especially people who work routinely on orbifolds or Lie groupoids.

From a 2-categorical perspective, the advantage of stacks is that they form a 2-topos. So, they behave almost as nicely as Cat in many ways.

I suppose the answer to this is ‘Work out a categorified form of doctrines to include 2-toposes, symmetric monoidal 2-categories, and so on’.

Luckily the Australians have already spent the last 20 years doing this. A lot of material sitting there, waiting for working mathematicians to use it.

(By the way, David: if you think the phrase ‘Hilsum–Skandalis map’ sounds mighty esoteric, I agree with you. But the actual concept is very nice, and doesn’t deserve to be saddled with such a name. It’s a span where the left leg is a very nice sort of equivalence.)

Posted by: John Baez on October 28, 2008 4:38 PM | Permalink | Reply to this

Re: What is Categorification?

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(By the way, David: if you think the phrase ‘Hilsum–Skandalis map’ sounds mighty esoteric, I agree with you. But the actual concept is very nice, and doesn’t deserve to be saddled with such a name. It’s a span where the left leg is a very nice sort of equivalence.)

By the way: Morita morphisms of groupoids internal to Spaces are precisely those out of “global acyclic fibrations” with respect to the folk model structure.

A (global) acyclic fibration of $\omega$ -categegories is a functor $f : C \to D$ which is $k$ -surjective for all $k$ , in the sense of Baez-Shulman.

This means that for all $k$ the universal morphism $C_k \to P_k$ into the pullback $P_k$ given by the pullback diagram $\array{ P_k &\to& D_k \\ \downarrow && \downarrow \\ C_{k-1} \times C_{k-1} &\stackrel{f_{k-1} \times f_{k-1}}{\to}& D_{k-1} \times D_{k-1} }$ is epi.

Restricting this to 1-groupoids internal to Spaces=Sheaves yields the notion of Morita morphism, which says that on objects $f_0$ has to admit local sections and on morphisms $C_1$ is the pullback itself, $C_1 \simeq P_1$ .

Posted by: Urs Schreiber on October 28, 2008 8:58 PM | Permalink | Reply to this

Re: What is Categorification?

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I can’t say that I work so heavily with Lie groupoids, but here’s my two cents. If we are good category theorists, then the way in which we invert our smooth equivalences shouldn’t matter, up to biequivalence. Sometimes it is handy to use different localisations. In my thesis there will be a result which says there are as many bicategories of anafunctors for groupoids internal to a site as there are Grothendieck pretopologies for that site, and they are all biequivalent to each other, as well as to Pronk’s bicategory of fractions construction from 1996. If Cech-style thinking is your thing, use open covers. If you like local diffeomorphisms, use them, and likewise with surjective submersions. They all invert smooth equivalences. The other option is to pass to stacks, and while they have fantastic properties, I personally don’t feel I have a geometric grip on what stacks look like. That being said, I suppose I am inadvertently quoting some poor algebraic geometer of the 60’s who was faced with the onset of schemes!

Posted by: David Roberts on October 29, 2008 5:27 AM | Permalink | Reply to this

Re: What is Categorification?

if you think the phrase ‘Hilsum–Skandalis map’ sounds mighty esoteric, I agree with you.

Some of us call them ‘saturated smooth anafunctors’, but that’s kind of esoteric too.

Posted by: Toby Bartels on October 30, 2008 4:52 AM | Permalink | Reply to this

Re: What is Categorification?

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if you think the phrase ‘Hilsum–Skandalis map’ sounds mighty esoteric, I agree with you.

Some of us call them ‘saturated smooth anafunctors’, but that’s kind of esoteric too.

I think an anafunctor $X \to C$ is saturated if not only the left leg is an acyclic fibration (namely a $k$ -surjective functor for all $k$ ) but also the right leg is an acyclic fibration onto its image $C' \subset C$ .

$X \lt\stackrel{\simeq}{\leftarrow} \mathbf{Y} \stackrel{\simeq}{\to}\gt C'$

I like this formulation in terms of the model structure on categories. It makes everything sound less esoteric to me, because it puts it into a more general picture. And it indicates how to generalize to higher categories.

Posted by: Urs Schreiber on October 30, 2008 2:01 PM | Permalink | Reply to this

Re: What is Categorification?

Hilsum-Skandalis maps are not anafunctors. They are equivalence classes of bibundles. You could take local sections of the principal leg of the bibundle and factor a bibundle into a span, but that’s a different story…

Posted by: Eugene Lerman on October 30, 2008 5:53 PM | Permalink | Reply to this

Re: What is Categorification?

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Hilsum-Skandalis maps are not anafunctors. They are equivalence classes of bibundles. You could take local sections of the principal leg of the bibundle and factor a bibundle into a span, but that’s a different story…

Last time we talked about that here. I haven’t really thought about it since then, but I suspect that the reasoning is this:

As we discussed in the context of groupoidification (the statement is somewhere in the TWFs) an action of a groupoid $C$ on something is equivalently described in a faithful functor from some groupoid $V//C$ to $C$ , in which case we say $V//C$ is the corresponding action groupoid. Its objects are what the groupoid acts on, and its morphisms connect the object related by a groupoid action.

Now, a saturated anafunctor is one, I think, where both legs of the span are in particular faithful functors. So this is the action groupoid for the simultaneous action of two groupoids.

I haven’t checked this in more detail right now, and hopefully Toby can help out, but I suspect in this sense a saturated anafunctor really “is” a Hilsum-Skandalis morphism: rather: it is the span of action groupoids for the action which defines the Hilsom-Skandalis bibundle.

Have to run now…

Posted by: Urs Schreiber on October 30, 2008 10:39 PM | Permalink | Reply to this

Re: What is Categorification?

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I’m hoping Toby will chime in here, but I think he meant that a Hilsum–Skandalis map ‘is’ a saturated anafunctor in the sense that we can turn one into the other and vice versa.

Toby wants to write a paper about this stuff, and this is the sort of thing that deserves to be straightened out.

Posted by: John Baez on November 1, 2008 4:52 AM | Permalink | Reply to this

Re: What is Categorification?

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I think [Toby] meant that a Hilsum–Skandalis map ‘is’ a saturated anafunctor in the sense that we can turn one into the other and vice versa.

Potentially, I might mean only that (between given Lie groupoids) the category of Hilsum–Skandalis maps is equivalent to the category of saturated anafunctors. This is enough to say that the concepts are equivalent and you may use one or the other. However, as Urs points out in the comment that he links above,

I think for the moment, where we are concerned with the peculiarities of the details of anafunctors [or any of these other concepts —Toby], it pays to point out that some equivalences in the game here are actually isomorphisms, while others are not.

And this one is an isomorphism.

In fact, given additionally a span

(1)

G _ 0 \leftarrow S \rightarrow H _ 0

of spaces, which is a feature of both anafunctors and bibundles, the additional structure to define a saturated anafunctor or a Hilsum–Skandalis morphism consists of simply a bunch of maps with some properties, nothing more. This is the context where we are happy to use ‘is’, just as we may say that a monoid with inverses ‘is’ the same thing as an associative loop, even though the definitions of these terms don’t refer to exactly the same maps. (To be precise, such ‘is’ refers to a bijection between sets of possible structures on the appropriate common stuff, a concept accessible to Bourbaki and requiring no category theory as such.)

Posted by: Toby Bartels on November 10, 2008 3:05 AM | Permalink | Reply to this

Re: What is Categorification?

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Anafunctors are not bibundles, and bibundles are not anafunctors —in general. However, saturated anafunctors are bibundles, and Hilsum–Skandalis maps are anafunctors, and in fact they are each other.

Both a bibundle and an anafunctor are (among other things) a span

(1)

G _ 0 \leftarrow S \rightarrow H _ 0

of spaces. Most of the rest of the definitions don’t correspond; a bibundle has action maps

(2)

G _ 1 \times _ { G _ 0 } S \to S

and

(3)

S \times _ { H _ 0 } H _ 1 \to S

(satisfying various properties), while an anafunctor has a functoriality map

(4)

S \times _ { G _ 0 } G _ 1 \times _ { G _ 0 } S \to H _ 1

(satisfying various properties).

However, if the bibundle is a Hilsum–Skandalis map, then the functoriality map can be defined as

(5)

( s , g , t ) \mapsto s / ( g \cdot t ) ,

where ‘⋅’ refers to the action of G and ‘/’ refers to the action of H (which is free and transitive so has division). Conversely, if the anafunctor is saturated, then the action maps can be defined to map (g, s) to the unique specification over the source of g of the identity of the value of s in H, while (s, h) maps to the unique specification of h over the value of s in G.

Of course, you have to check that these maps satisfy the relevant properties; that is, as they say, tedious but straightforward.

Posted by: Toby Bartels on November 10, 2008 2:33 AM | Permalink | Reply to this

Re: What is Categorification?

John, I agree that your example is a problem; since it involves functors between categories, it’s really more than internalisation. But since it’s not going up a level, I wouldn’t call it categorification either.

Full disclosure: I’m one of those people that says ‘categorial’ on the grounds that ‘categorical’ already has a meaning (especially in logic).

Hum… the word ‘categorialisation’ just popped into my head. I think that I will push it onto this page, in case anybody wants it.

Posted by: Toby Bartels on October 28, 2008 1:55 AM | Permalink | Reply to this

Re: What is Categorification?

David sez:

Do you ever select a good 2-category of things over a 2-category of good things?

Do you want to use cobordisms between tangles, or Reidemeister moves? I’d like the latter, because then you get knot covariants. But the former seems more natural in certain ways. That’s where you get 2-tangles.

Toby sez:

I wouldn’t call [your example] categorification either.

Well, sure. I accept that. My question isn’t “is this categorification”, but “what the heck should I call it?”

Posted by: John Armstrong on October 28, 2008 4:16 AM | Permalink | Reply to this

Re: What is Categorification?

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John A. wrote:

My question isn’t “is this categorification”, but “what the heck should I call it?”

You’re taking the monoid of knots and saying “Hey, this monoid is just the homset hom(1,1) sitting inside the category of tangles!”

This is a realization ‘riding’ the realization that monoids are 1-object categories.

The generalization from monoids to 1-object categories has been called ‘horizontal categorification’ — in contrast to the usual ‘vertical’ form — since we’re fattening up the set of objects instead of adding on higher morphisms.

It’s also been called ‘oidization’, since a group is a groupoid with one object, a ring is a ringoid with one object, an algebra is an algebra with one object, and so on.

(The fact that a monoid is a category with one object — well, that spoils the linguistic pattern, but we’ve all agreed not to rename categories ‘monoidoids’.)

A classic example of oidization is taking the algebra of $n \times n$ matrices and embedding it in the algebroid of all matrices. Another is taking the permutation group $S_n$ and embedding it in the groupoid of finite sets. Both these are incredibly powerful.

You’re doing something very similar.

Posted by: John Baez on October 28, 2008 4:53 PM | Permalink | Reply to this

algebroid is an algebra? Re: What is Categorification?

Should “an algebra is an algebra with one object” be “an algebroid is an algebra with one object”?

That’s by analogy with the usage that Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups.

As Yvette Kosmann-Schwarzbach puts it:

Lie algebroids were first introduced and studied by J. Pradines [“Theorie de Lie pour les groupoides differentiables. Calcul differentiel dans la categorie
des groupoides infinitesimaux”, Comptes rendus Acad. Sci. Paris 264 A (1967), 245-248.], following work by C. Ehresmann
and P. Libermann on differentiable groupoids (later called Lie groupoids). Just as Lie algebras are the infinitesimal objects of Lie groups, Lie algebroids are the infinitesimal objects of Lie groupoids. They are generalizations of both Lie algebras and tangent vector bundles.

Posted by: Jonathan Vos Post on October 29, 2008 3:16 AM | Permalink | Reply to this

Re: algebroid is an algebra? Re: What is Categorification?

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Should “an algebra is an algebra with one object” be “an algebroid is an algebra with one object”?

You probably meant to say it the other way round and then, yes:

an algebra is an algebroid with a single object.

Generally: an X is an Xoid with a single object.

Posted by: Urs Schreiber on October 29, 2008 9:40 PM | Permalink | Reply to this

Re: algebroid is an algebra? Re: What is Categorification?

Yes, that’s what I meant.

So I’m co-correct, or rect.

Posted by: Jonathan Vos Post on October 30, 2008 1:37 AM | Permalink | Reply to this

Re: What is Categorification?

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There is also categorification as in “finding a conceptual explanation for a set $S$ as living inside a higher-categorical structure”. Eg., the Hochschild cohomology of an algebra or a space can be conceptually thought of as the 2-endomorphisms of the identity in a certain 2-category.

I think it defeats the whole purpose of “categorification” to start nailing it down as some or other thing. That’s decategorifying! To me, “categorification” is simply “finding a conceptual explanation of something in a way which pays respect to the processes between those somethings”.

Posted by: Bruce Bartlett on October 22, 2008 9:27 PM | Permalink | Reply to this

Re: What is Categorification?

I completely agree with you, Bruce, that ‘categorification’ should be used in a loose and flexible way, since 1) that’s how it’s already used, and 2) we need a word with this broad scope.

However, we also need a very short story to tell people who are mystified by the word ‘categorification’.

The Wikipedia article is actually quite good, but it should be supplemented by a bunch of examples, and more detailed discussion.

Posted by: John Baez on October 23, 2008 3:50 PM | Permalink | Reply to this

Re: What is Categorification?

Bruce – I completely agree with your comment, and with the perspective on Hochschild cohomology (I think fairly generally dimensional reduction in TFTs, which is a form of decategorification, sends a category to its Hochschild (co)homology) but I think the assertion that Hochschild cohomology is derived endomorphisms of the identity for (an enhanced form of) the derived category goes back at least to Toen’s Inventiones paper arXiv:0408337. It’s a great paper,
proving in particular the functors from D(X) to D(Y) for schemes X,Y are given by D(X x Y) (where again D means an enhanced version of the derived category).

Posted by: David Ben-Zvi on October 23, 2008 10:11 PM | Permalink | Reply to this

Re: What is Categorification?

Yes, sorry about that reference. I knew it wasn’t the first paper to conceive of Hochschild (co)homology in that way, but I didn’t actually know what was, and I just wanted some or other link I could write down. If I had read the paper properly in the first place, I would have seen that they do indeed mention Toen’s paper and credit him appropriately.

Posted by: Bruce Bartlett on October 23, 2008 11:11 PM | Permalink | Reply to this

Re: What is Categorification?

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The Hochschild example (Bruce, David B.-Z.) is interesting (also) in that it involves a good mix of the various notions that were mentioned:

- one starts with an abelian category

- then one passes to its derived category = homotopy category of complexes, thereby “ $\infty$ -cateforifying” in the “homotopical resolution”-sense

- one then notice that certain natural operations in that $\infty$ -categorical context reproduce familiar operations that were originally conceived in the age before categorical enlightenment, thereby providing a “categorical interpretation” akin to what Jamie mentioned.

With the new systematic understanding of the unenlighted concept, one then goes ahead (as for instance Kapranov and Ganter did originally and now Bruce does in his upcoming thesis for the “endomorphism of the identity idea”) and internalizes the concept in lots of other context, as Toby mentioned, to find first that lots of different phenoema in the world are all examples of “categorical traces” and then discovering plenty of new such phenomena.

Posted by: Urs Schreiber on October 24, 2008 9:13 AM | Permalink | Reply to this

Re: What is Categorification?

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Urs, just to correct a point you made, when you said

(as for instance Kapranov and Ganter did originally and now Bruce does in his upcoming thesis for the “endomorphism of the identity idea”)

When you say “endomorphism of the identity idea” do you mean “categorical trace idea”? If that’s the case then I’d just like to pick you up on the word “originally”. I met Nora just before that paper came out and we discovered we were working on the same idea of categorical trace but for rather different reasons. This is mentioned in the introduction to their paper.

Of course Bruce was only just starting work on his PhD at the time, which is why he’s just finishing his thesis two and a half years later.

I’m not making any claims about being the first to consider the categorical trace, the idea might well have been ‘in the air’, but I just wanted to set the record straight.

Posted by: Simon Willerton on October 24, 2008 6:15 PM | Permalink | Reply to this

Re: What is Categorification?

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do you mean “categorical trace idea”?

Yes.

So in particular that’s more than just endos of the identity. Endos is really “categorical dimension”, as probably also you saw originally!

I met Nora just before that paper came out and we discovered we were working on the same idea of categorical trace but for rather different reasons. This is mentioned in the introduction to their paper.

Thanks for emphasizing this. I suppose I knew it but didn’t say it properly.

Of course Bruce was only just starting work on his PhD at the time

Sure. I suppose I have a skewed internal representation of this history due to my intensive interaction with Bruce and his nice explanation of your ideas to me! (This is not to say that it is Bruce’s fault, of course!!)

I know of this beautiful set of notes that you and Bruce have on this stuff for quite a while now. I gather this is now close to being revealed? That might also help to set the record straight.

Anyway, sorry again for misrepresenting the origin of the idea of the categorical trace. It wasn’t with bad intentions.

Posted by: Urs Schreiber on October 25, 2008 5:01 PM | Permalink | Reply to this

Re: What is Categorification?

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Urs said

It wasn’t with bad intentions.

Of course I didn’t for a nanosecond suspect that it was!

It’s very easy for one to get especially defensive where one’s graduate students are concerned.

Posted by: Simon Willerton on October 26, 2008 9:09 PM | Permalink | Reply to this

Re: What is Categorification?

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There are now $n$ Lab-entries for vertical categorification and horizontal categorification.

Posted by: Urs Schreiber on December 1, 2008 8:05 PM | Permalink | Reply to this

Re: What is Categorification?

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horizontal categorification

When you horizontally categorify a monoidal category, I guess you get a bicategory. What bicategory do you get by horizontally categorifying Hilb? Not 2Hilb, since that’s vertical categorification. The direct sum and tensor product play roles akin to $+$ and $*$ in a ring, so I assume it would have some kind of ringoid-like structure.

Posted by: Mike Stay on December 2, 2008 12:55 AM | Permalink | Reply to this

Re: What is Categorification?

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What bicategory do you get by horizontally categorifying Hilb?

One possible answer: you get $Bimod_{vN}$ , the bicategory whose objects are (von Neumann) algebras, whose morphisms are bimodules for these with composition being (some refined version of) tensor product of bimodules, and whose 2-morphisms are bimodule homomorphisms.

$Hilb$ sits inside this as the endomorphisms of the ground field $\mathbb{C}$ .

Forgetting about the subtlety with the extra structure on Hilbert spaces, the same kind of situation is obtained just for plain vector spaces: we have

$Vect_k = End_{Bimod}(k) \,.$

Posted by: Urs Schreiber on December 2, 2008 8:54 AM | Permalink | Reply to this

Re: What is Categorification?

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Thanks, that makes sense: the morphisms are rather like Hilbert spaces, but with a source and target!

Posted by: Mike Stay on December 3, 2008 12:57 AM | Permalink | Reply to this

Re: What is Categorification?

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I think this is actually a better formalism for talking about quantum computation than Hilb, since in computation it’s typically the rewriting of terms that takes time, not the composition of terms. In this framework, types would map to von Neumann algebras, terms hom(A,B) to bimodules, and rewrite rules to bimodule homomorphisms.

Posted by: Mike Stay on December 3, 2008 1:31 AM | Permalink | Reply to this

Re: What is Categorification?

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Mike: if you want to hear Urs’ answer explained at much greater length by your friendly neighborhood thesis advisor, try this. It’s a long paper, but the only parts relevant to your question are the introduction, Chapter 3 and the conclusions.

Many of the subtleties here arise from the fact that we’re dealing with infinite-dimensional Hilbert spaces and 2-Hilbert spaces. If you’re willing to work with a version of Hilb that consists only of finite-dimensional Hilbert spaces, things simplify drastically!

After all, this baby version of Hilb is equivalent to the monoidal category Vect where the objects are finite-dimensional complex vector spaces and the morphisms are linear operators. This monoidal category then sits nicely inside the 2-category of Kapranov–Voevodsky 2-vector spaces, usually called 2Vect. And then, 2Vect is equivalent, as a 2-category, to the finite-dimensional version of 2Hilb that I introduced here.

In short: in this finite-dimensional baby case, the monoidal category Hilb horizontally categorifies to the 2-category 2Hilb.

(And 2Hilb is a monoidal 2-category, and I expect this horizontally categorifies to 3Hilb, and so on. Jamie Vicary has some interesting thoughts on the hypothesized limit of this process, $\infty$ Hilb.)

Posted by: John Baez on December 2, 2008 5:04 PM | Permalink | Reply to this

Re: What is Categorification?

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Ah! I just understood. Urs wrote in nLab:

Vertical categorification refers roughly to a process in which ordinary categories are replaced by higher categories.

This is the $\mathbb{C} \to Vect \to 2Vect$ process you were referring to. I was asking about horizontal categorification:

“Horizontal categorification” describes the process by which

a concept is realized to be equivalent to a certain type of category with a single object;

and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.

You never horizontally categorify a particular instance of a gadget: there’s no horizontal categorification of $\mathbb{Z}_n$ , but there is a horizontal categorification of a group, namely a groupoid. So I answered my own question: the horizontal categorification of a monoidal category is a bicategory, full stop.

On the other hand, if there’s some property that uniquely specifies a one-object category and that property sensibly extends to many-object categories, then we could consider that to be a horizontal categorification of an instance. For example, $\mathbb{N}$ is the free rig on one generator; I guess you could say a horizontal categorification of this is, for each n, the free rigoid on a directed graph with n nodes and for any pair of nodes A, B, exactly one edge $A\to B$ . We could do something similar with finite-dimensional vector spaces to get something that’s a vertical-then-horizontal categorification of $\mathbb{N}$ .

Posted by: Mike Stay on December 2, 2008 11:46 PM | Permalink | Reply to this

Re: What is Categorification?

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Hi Mike,

I was thinking that in this case it is a “coincidence” that the horizontal categorification of $Vect$ is also related to its vertical categorification. That’s why I didn’t mention the 2-vector space aspect in my reply. John mentioned that.

You never horizontally categorify a particular instance of a gadget:

That’s true. But you can still ask for a particular one-object gadget if it appears as $End_C(a)$ for some object $a$ in some ( $n$ -)category $c$ .

the horizontal categorification of a monoidal category is a bicategory

Right, exactly.

But now you can still ask: which bicategories $C$ with object $a$ are there such that $Vect \simeq End_C(a)$ .

There are many possible answers. For one, it is possible to cook up simple dumb examples by hand. And the more natural examples there are have plenty of further generalizations. For instance the 2-category that John pointed out is a sub-2-category of the one I mentioned.

That these “horizontal categorifications” (or maybe we should say, following your emphasis: “embeddings of $Vect$ into a horizontally categorfied category” or the like) here also have to do with “vertical categorification” has to do with some enrichment phenomenon, I suppose. It is the analogue of noticing that in $Vect$ itself, we recover the ground field $k$ (the 0-category of -1 vector spaces) as $k \simeq End_{Vect}(k) \,.$

By the way: if you or anyone figures out a good answer to the following question, I’d be grateful:

what is the direct limit (after one makes sense of it) of the sequence $k \hookrightarrow Vect_k \simeq k-Mod \hookrightarrow 2Vect \simeq Vect_k-Mod \hookrightarrow 3Vect \simeq (Vect_k-Mod)-Mod \hookrightarrow \cdots \,.$

The answer may contain the word “spectrum” if that helps. And if the answer is “K-theory spectrum” then I am asking: why in detail?! :-)

Posted by: Urs Schreiber on December 3, 2008 7:43 AM | Permalink | Reply to this

Re: What is Categorification?

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“embeddings of Vect into a horizontally categorfied category”

‘embeddings of Vect into a bicategory’

Jargon should not be multiplied beyond necessity.

Posted by: Toby Bartels on December 4, 2008 1:20 AM | Permalink | Reply to this

Re: What is Categorification?

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So, I asked what the horizontal categorification (“oidification”) of Hilb was. Restricting to the one-object case, we should get (for a proper choice of object x) Hilb back again as a monoidal category whose objects are the endomorphisms of x.

Urs replied with a bicategory where

objects are von Neumann algebras
morphisms are bimodules between them
2-morphisms are bimodule homomorphisms since Hilb is the case with the one object $\mathbb{C}$ .

You said it was 2-Hilb, where we’ve got

objects are 2-Hilbert spaces–roughly, vectors whose elements are powers of $\mathbb{C}$ instead of elements of $\mathbb{C}$
morphisms are roughly matrices with elements that are powers of $\mathbb{C}$
2-morphisms are roughly matrices with elements that are linear transformations, applied pointwise

If we restrict this to a single object $K^n$ –i.e. a “vector” of dimension $n$ whose elements are powers of $\mathbb{C}$ –then the morphisms are the $n \times n$ “square matrices”, and I can’t see how to get Hilb from that.

Can you expound a little?

Posted by: Mike Stay on January 12, 2009 3:30 AM | Permalink | Reply to this

Re: What is Categorification?

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Oh! If I choose $n=1$ , then the $1\times 1$ “square matrices” are all the powers of $\mathbb{C}$ , which are the objects of Hilb (or at least, FinDimHilb).

Posted by: Mike Stay on January 12, 2009 3:37 AM | Permalink | Reply to this

Re: What is Categorification?

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Mike wrote:

Oh! If I choose $n=1$ , then the $1 \times 1$ “square matrices” are all the powers of $\mathbb{C}$ , which are the objects of $Hilb$ …

Right. Just as you can find

$0 Hilb = \mathbb{C}$

as a hom-set inside

$1 Hilb = Hilb,$

namely

$\mathbb{C} \cong hom(\mathbb{C}, \mathbb{C}),$

you can find $1 Hilb$ as a hom-category inside $2 Hilb,$ namely

$Hilb \simeq \hom(Hilb, Hilb)$

just as you described.

…or at least, $FinDimHilb$ .

Indeed, right now I’m using $n Hilb$ to mean the $(n+1)$ -category of finite-dimensional $n$ -Hilbert spaces for $n = 0, 1, 2$ .

Why? First, because infinite-dimensional 2-Hilbert spaces are just beginning to be understood.

Second, because even though we know a lot about infinite-dimensional Hilbert spaces, the category of them has very different formal properties than the category of finite-dimensional ones, and these formal properties are not deeply understood.

These two problems are closely linked, of course, because $Hilb$ sits inside $2 Hilb$ (both as an object and as a hom-category). Since the category-theoretic aspects of infinite-dimensional Hilbert spaces remain a bit mysterious, a lot of research on category theory in quantum mechanics and quantum computing focuses on the finite-dimensional case. So, experts on quantum field theory find such research a bit childish. But we’ll eventually get around to the infinite-dimensional stuff.

Posted by: John Baez on January 16, 2009 6:29 PM | Permalink | Reply to this

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October 22, 2008