A Categorical Manifesto

November 2, 2006

Posted by John Baez

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A while back Gina asked why computer scientists should be interested in categories. Maybe you categorical computer scientists out there have your own favorite answers to this? I’d be glad to hear them. To get you going, here’s one man’s answer:

Joseph Goguen, A categorical manifesto, Mathematical Structures in Computer Science 1 (1991), 49-67.

Abstract: This paper tries to explain why and how category theory is useful in computing science, by giving guidelines for applying seven basic categorical concepts: category, functor, natural transformation, limit, adjoint, colimit and comma category. Some examples, intuition, and references are given for each concept, but completeness is not attempted. Some additional categorical concepts and some suggestions for further research are also mentioned. The paper concludes with some philosophical discussion.

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Joseph Goguen was a professor of computer science at U. C. San Diego. He died on July 3rd, 2006, shortly after his 65th birthday.

Posted at November 2, 2006 4:46 PM UTC

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74 Comments & 1 Trackback

Re: A Categorical Manifesto

For me, two of the most interesting aspects of category theory in computer science have been monads and generalised folds/unfolds.

If M is a functor that happens to be monad, then given an arrow (ie. a function because we’re working in the category of types and functions) A->MB and an arrow B->MC we can compose them to make an arrow A->MC, even though the tail of the first arrow is incompatible with the head of the second. This pattern is incredibly common when programming, especially in functional programming languages. Rather than have to write the glue to compose these functions over and over again, monads give a really nice way to express the glue and simplify the composition. And to grasp monads it really helps to understand some category theory, after all, they are categorical constructs.

Folds are a ubiquitous operation in computing. A fold is easiest to understand by example. Given an ordered list of numbers [1,2,3,4] we can turn it into an expression 1+2+3+4+0. We put a + between each pair of values and a value at the end (often the identity for +). Clearly we can generalise to any binary operator and any ‘end’ value. This is a fold and clearly summing a list is an example of one (and you have to admit, summing a list is a pretty ubiquitious operation). Many (all?) functions of lists can be expressed as folds so understanding folds gives a nice way to abstract your code to see commonality between different algorithms. Additionally, folds generalise to many datastructures besides lists. To fully grasp this needs a little category theory and the notion of an F-algebra.

And the fun thing is that as soon as we view this stuff through category theory it becomes clear that these notions have duals. On the one hand we find comonads, and on the other we find F-coalgebras and unfolds. It turns out that many algorithms are naturally structured via comands and that unfolds are as ubiquitous as folds.

And of course the whole Curry-Howard isomorphism is really about category theory and I’ve found that to be quite illuminating.

A nice readable account on fold/unfold is contained in here. There’s no shortage of tutorials on monads in computer science so I won’t bother with a link to one of those.

Posted by: Dan Piponi on November 2, 2006 10:31 PM | Permalink | Reply to this

Re: A Categorical Manifesto

A Neighborhood of Infinity has had many posts on monads, including this one with diagrams. There are also posts on other topics of interest to us: synthetic differential geometry, Euler characteristic, etc.

Posted by: David Corfield on November 3, 2006 10:07 AM | Permalink | Reply to this

Re: A Categorical Manifesto

Goguen’s description of ‘Foundations’ on page 15 is very relevant to our discussions on ‘causal’ ordering in the Foundations thread:

Foundations should provide general concepts and tools that reveal the structures and interrelations of various areas of mathematics and its applications, and that help in doing and using mathematics.

Posted by: David Corfield on November 3, 2006 9:46 AM | Permalink | Reply to this

Re: A Categorical Manifesto

I have a more philosophical answer that abstracts away even from the gory details of category theory.

If we remember that most mathematics arose from generalizing the notion of counting, set theory (and the mathematical edifice built up from it) can be seen
as formalizing the primitive intuitions about counting things. It is a natural basis for numerical and numerically-originating ideas.

In contrast, computation is mainly about mapping inputs to outputs. The study of mappings and their composition, which gives rise to category theory, is a natural way to formulate the particular concerns of computational thinking.

The fact that the two approaches are dual allows us to build upon the rich legacy of numerically-founded work, while still allowing new insights into the fundamental mechanism of our scientific concern.

Posted by: Marc Hamann on November 3, 2006 5:01 PM | Permalink | Reply to this

Re: A Categorical Manifesto

The study of mappings and their composition, which gives rise to category theory, is a natural way to formulate the particular concerns of computational thinking.

But then any physical operation, preparation or measurement also gives rise to category theory:

a category is the exact mathematical structure of practicing physics! Bob Coecke

And isn’t gathering things together a physical operation, and counting a form of measurement?

Posted by: David Corfield on November 3, 2006 5:36 PM | Permalink | Reply to this

Re: A Categorical Manifesto

And isn’t gathering things together a physical operation, and counting a form of measurement?

Yes, but that’s why the two formulations are dual: they look at the same underlying structures from different starting points.

Posted by: Marc Hamann on November 3, 2006 6:19 PM | Permalink | Reply to this

Re: A Categorical Manifesto

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Can anyone comment on section 2.7, Programs and Program Schemes, and what connection, if any, it has to programs as morphisms in a CCC?

Posted by: Mike Stay on November 4, 2006 7:53 AM | Permalink | Reply to this

Re: A Categorical Manifesto

Can anyone comment on section 2.7, Programs and Program Schemes, and what connection, if any, it has to programs as morphisms in a CCC?

My first idea on this is that there is a functor from the category of program graphs C(G) to the CCC, in the sense that each morphism in the CCC is a meta-shape of shapes of programs. This leaves open the possiblity of an adjuction with a free functor that maps each morphism in CCC to its “Holodeck game” solution; we would have to do a lot more work fill that in.

In thinking about this, I find myself hampered by some important details left out of Goguen’s description. For example, is G an object in a category of like program graphs (and what are the morphisms?). Or is it the free category for the graph G? This seems possible if you interpret/allow the free indentity morphism to serve as an “empty” operation similar to that in non-deterministic finite automata.

While it is a strength that one can find so many different categorical formulations of particular phenomena, it is a drawback that working out the relationship between them can require quite a bit of work.;-)

Posted by: Marc Hamann on November 4, 2006 9:10 PM | Permalink | Reply to this

Re: A Categorical Manifesto

“A while back Gina asked why computer scientists should be interested in categories. …”

Wow! how sweet to have a whole (interesting) post related to my question. Thanks a lot!

Posted by: Gina on November 7, 2006 2:42 AM | Permalink | Reply to this

Ulam on Banach; Re: A Categorical Manifesto

Stanisław Marcin Ulam, another mathematician of the Lwów School of Mathematics, in his autobiography, attributes this to Banach:

“Good mathematicians see analogies. Great mathematicians see analogies between analogies.”

Is this a prescient hint of the campaign to categorize, re-categorize, and/or n-categorize mathematical and Physics foundations?

Anyone think that the quote can be made axiomatical? That “Very Great mathematicians see analogies between analogies between analogies between analogies.”

And what the omega-closure of this? Or should it be embedded in a universe including strongly unreachable analogies?

Just wondering, as an admirer of Ulam who was to have coauthored with him, but suffered when he died before we could agree on the abstract.

Posted by: Jonathan Vos Post on November 11, 2006 3:09 AM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

“Good mathematicians see analogies. Great mathematicians see analogies between analogies.”

Is this a prescient hint of the campaign to categorize, re-categorize, and/or n-categorize mathematical and Physics foundations?

There’s no doubt in my mind that that’s so. My advisor was fond of quoting that line, usually right after noting that “a functor is an analogy”.

Posted by: John Armstrong on November 11, 2006 4:35 AM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

Thank you for that supportive example. Could it be that Ulam or Banach was influenced by Hilbert’s syzygy theorem? This theorem, as summarized on Wikipedia, which also gives a more modern statement, but not so modern as category Theory: “is a result of commutative algebra, first proved by David Hilbert (1890) in connection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it explains how far one has to go to reach a clarified situation. It is now considered to be an early result of homological algebra, and through the depth concept, to be a measure of the non-singularity of affine space.”

Posted by: Jonathan Vos Post on November 11, 2006 5:57 AM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

I mentioned that idea of an infinite chain of higher-order analogies at the n-categories meeting in Minneapolis 2004 (bottom of p.3 of these slides), inspired by what I think was John’s “Behind every analogy there lies a functor.”

Posted by: David Corfield on November 11, 2006 9:11 AM | Permalink | Reply to this

Reality Check

“a functor is an analogy”.

Unfortunately, I’m starting to feel that in the rush to congratulate n-categories as the “next big thing”, we may have got the cart before the horse.

Analogies, which are the foundation of all abstract thinking, and have certainly been at the heart of mathematical inquiry the last few centuries, are at their core mappings: “A is to X as B is to Y”. When group theorists investigated the commonalities between rotations, permutations and the integers under addition, they were implicitly (and often explicitly) investigating mappings between structures.

The innovation of Category Theory was to invert the process: instead of building structures then extracting how they mapped onto each other, CT takes the mapping as the primitive and examines what sorts of structure arise.

So yes, a functor is an analogy, since a functor is a kind of mapping, and a mapping is a kind of analogy. But it would be the height of hubris to assume that this knowledge was nothing but a vague notion before Mac Lane, Eilenberg, Lawvere and others saved the day.

CT and its inventors have accomplished some incredible things, however I’m pretty confident they can’t take the credit for inventing analogy.

Posted by: Marc Hamann on November 11, 2006 6:40 PM | Permalink | Reply to this

Re: Reality Check

You misunderstand the point of his statement. “A functor is an analogy” is not a philosophical slogan, but rather a pedagogical one.

How should a student understand the language of categories and functors, especially in an era where that language is still marginalized among some mathematicians? It’s an analogy. This is even clearer when you consider (Sets,Relations) as a motivation instead of (Sets,Functions).

Posted by: John Armstrong on November 11, 2006 9:51 PM | Permalink | Reply to this

Re: Reality Check

You misunderstand the point of his statement. “A functor is an analogy” is not a philosophical slogan, but rather a pedagogical one.

I think you are misunderstanding the point of my statement ;-). I’m not critiquing the pedagogical intuition that a functor is an analogy: I have explicitly agreed that this is the intuitive interpretation of a functor.

What I’m critiquing is the unspoken assumption that has crept into this thread that Ulam/Banach et al. are really just Old Testament prophets dimly prophesying the coming of the true saviour: Category Theory / n-categories.

Abstract and analogical thinking have long been at the heart of the mathematical enterprise. CT / n-Cat just focus on the analogizing rather than on the component objects of the analogies, and thus make it easier to see a whole bunch of analogies at once.

Posted by: Marc Hamann on November 11, 2006 10:56 PM | Permalink | Reply to this

Re: Reality Check

Really it seems to come down to the real fundamental divide in philosophy among practicing mathematicians: does mathematics seek to understand structures in order to solve problems, or does it solve problems to help understand structures?

If problem-solving is the ultimate goal, the component objects are the important parts. If understanding is the goal, the analogies are the important parts.

Of course, both tasks are important, but the pendulum swings both ways alternately. For a very long time, mathematics has been expanding its studies to solve more and more problems. As I see it, we are entering (have entered?) a contracting phase where we see more and more of these studies to be different sides of the same things. The great theorems tend more and more often to be deep analogies between disparate areas of mathematics.

If anything, this emphasizes Corfield’s view that how mathematics is done is an essentially social and evolving thing. Also, I don’t think anyone seriously thinks that category theory is a “savior” of mathematics. It’s just the best language for a contracting, analogy-heavy phase of mathematical society while set theory was an excellent background language for the previous expanding, problem-solving-heavy phase. Eventually the pendulum will reverse again and we will spend more time solving problems and expanding mathematical studies again, possibly with an even newer language to inform the new phase.

Posted by: John Armstrong on November 11, 2006 11:33 PM | Permalink | Reply to this

Re: Reality Check

I would say you and I are in overall agreement, John, with a few qualifications.

I think the contraction/expansion pendulum setting isn’t global, but varies across sub-disciplines.

CT may be particularly good for some kinds of consolidation, but I wouldn’t want to ghettoize it to just that: it can be used to solve new problems too.

We probably have our greatest disagreement here:

I don’t think anyone seriously thinks that category theory is a “savior” of mathematics.

In other fora over the years, I have frequently served as the defender of CT against skeptics, who, to my confusion, tell me that they often find CT practioners arrogant in the assumption that all other mathematical formalisms have been rendered obsolete by CT.

Until I began frequenting this site, I can’t say that I had ever actually witnessed this phenomenon, but there have been more than a few posts here that have come awfully close to exhibiting this kind of attitude.

Perhaps it is just the exuberant bravado of a vibrant field, but I think it bears pointing out that it might be mistaken for reckless hubris by those who observe from outside.

Don’t forget: being taken seriously is a two-way street.

Posted by: Marc Hamann on November 12, 2006 12:03 AM | Permalink | Reply to this

Re: Reality Check

In other fora over the years, I have frequently served as the defender of CT against skeptics, who, to my confusion, tell me that they often find CT practioners arrogant in the assumption that all other mathematical formalisms have been rendered obsolete by CT.

I think this sort of thing may be a lack of perspective – of not “zooming out” (to hit another leitmotif). It’s analogous to the way physicists thought that everything was more or less figured out at the end of the 19th century.

Actually, physics provides a great parallel to illustrate this pendulum of expansion and contraction, especially since it takes place a lot more quickly than in mathematics. In the last hundred years they’ve gone through at least the full cycle.

I would go so far as to say that CT is the language of choice for the new phase, and I see newer evidence almost every day. I understand, though, that it won’t last forever.

Posted by: John Armstrong on November 12, 2006 2:06 AM | Permalink | Reply to this

Re: Reality Check

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Marc Hamann wrote:

In other fora over the years, I have frequently served as the defender of CT against skeptics, who, to my confusion, tell me that they often find CT practioners arrogant in the assumption that all other mathematical formalisms have been rendered obsolete by CT.

I don’t think category theory “makes all other mathematical formalisms obsolete”. I doubt anyone on this forum thinks that.

I believe that in the present epoch, any approach to the foundations of mathematics that recognizes the importance of $n$ -categories only for $n = 0$ is a needlessly crippled one.

But, I also think that it would be silly to say the study of 1-categories makes the study of 0-categories “obsolete”. It would be sort of like saying the number 1 made the number 0 obsolete.

Posted by: John Baez on November 12, 2006 5:15 AM | Permalink | Reply to this

Re: Reality Check

But, I also think that it would be silly to say the study of 1-categories makes the study of 0-categories “obsolete”.

I imagine some set theorists would find this phrasing tendentious, and would resist being seen as 0-category theorists. For one thing it suggests that there ought to be insights emerging from their work about the kinds of 1-category that can be formed from collections of sets and functions.

That different communities have different goals and different preferred means to achieve these goals is nothing to fret about. In fact, I take it to be a good thing, in that if their respective activity comes into close enough contact it provides an opportunity to gauge what is partial about each party’s viewpoint. There’s no place for arrogance. One should expect that it is no easy task to get into a position to understand the other party’s motivations.

I’d be very interested to hear a set theorist’s response to Angus MacIntyre’s piece here (note also the interesting paper by Lawvere), which begins:

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory.

Perhaps they disagree with this judgement. Perhaps they agree but don’t think it matters with respect to their goals.

Posted by: David Corfield on November 12, 2006 8:51 AM | Permalink | Reply to this

Re: Reality Check

I don’t think category theory “makes all other mathematical formalisms obsolete”. I doubt anyone on this forum thinks that.

As a literal statement meaning “set-theoretic truths no longer apply”, or “set-theoretic objects are of no interest” I can see that is probably true.

As a statement meaning “Strictly set-theoretic approaches are no longer productive avenues of investigation”, I’m much less sure.

Let’s say you met a prospective grad to student who said “I like set theory, and I want to investigate problem X within that framework. I recognize that there is ongoing investigation in CT on problem X, and I plan to follow those developments, but I’m more comfortable formalizing my own work set-theoretically”.

What would you say to him/her? What would you think about that person’s academic and intellectual future?

Posted by: Marc Hamann on November 12, 2006 5:22 PM | Permalink | Reply to this

Re: Reality Check

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Marc Hamann wrote:

Let’s say you met a prospective grad to student who said “I like set theory, and I want to investigate problem X within that framework. I recognize that there is ongoing investigation in CT on problem X, and I plan to follow those developments, but I’m more comfortable formalizing my own work set-theoretically”.

What would you say to him/her? What would you think about that person’s academic and intellectual future?

It completely depends on what problem X is. I can imagine various cases:

For many topics I would expect them to work under the de facto umbrella of set theory. A typical example might be studying the scattering operator for the nonlinear Klein-Gordon equation. This is what folks call “hard analysis”: it’s all about manipulating inequalities, so I don’t see much need for category theory here. Another example is my student Miguel Carrión Álvarez’s work on quantizing p-form electromagnetism. Here I wanted him to relate the subject to n-categories, but he wound up focusing on other aspects - and that was fine.

But in such cases, I wouldn’t expect the student to emphasize that they “wanted to formalize their work set-theoretically”. This would be like emphasizing you wanted to get around LA using a car. It’s just business as usual - not worth mentioning!
If they wanted to use the existence of a Woodin cardinal to get new results in descriptive set theory, I’d admit that I don’t know much about this sort of hard-core set theory. So, I’d have to say I am unable to advise them on this topic. And if I were feeling mentorly, I’d add that there aren’t many jobs in hard-core set theory, so that having a successful career depends crucially on finding an advisor who is a bigshot in this subject - someone whose letter of recommendation would impress the few other experts in the subject. There’s nobody like that here at UCR.

Personally I find this sort of research uninteresting, but if a student is all fired up about it, and seems knowledgeable about it, it would be silly to try to talk them out of it. All good research comes from people who are burning with desire to do something new - often something that few others care about. The trick is making sure these burning ambitions get tackled in a realistic way, with a good chance of success. This usually requires finding the right advisor - and later, a job.
If they wanted to work on categories or n-categories, I might become their advisor, since I know about this stuff. But in this case, I would force them to work on n-categories applied to a problem of independent interest. This is what I did with Alissa Crans and Toby Bartels, and it’s what I’m doing with Jeff Morton.

There are two reasons for this. First, if you get branded as a “category theorist”, your chances of getting a good job drop significantly - especially in the US. Second, it’s all too easy for youngsters to work on self-contained issues in n-category theory and not learn how their work is related to the rest of math. I don’t want my students to become specialist idiot savants: I want them to get a sense of the big picture, and have more choices after they graduate.

So, I always tell such students to become “an algebraic topologist who uses n-categories in her work”, or “a mathematical physicist who uses n-categories in his work”, or something like that, rather than a “pure n-category theorist”.

Posted by: John Baez on November 12, 2006 7:51 PM | Permalink | Reply to this

Re: Reality Check

First, if you get branded as a “category theorist”, your chances of getting a good job drop significantly - especially in the US.

Thanks, John: this post clarified a lot for me.

I think I have identified what different assumptions I’ve been making that have coloured my interpretation of some of the other comments.

It seems many people here perceive CT to still be the underdog. My perception was that it has been the big dog on the block for some time now.

My perception may be warped by various pecularities of my relationship with CT.

First of all, I’m not a professional academic, so I’m not familiar with the hiring politics surrounding CT specifically. (Though I understand the importance of these issues for those who ARE professional academics).

Secondly, I BEGAN my serious study of higher mathematics with CT (as a result of its relevance to programming language theory), and the early advice I received was “don’t try to understand it set-theoretically; you’ll confuse yourself and won’t get the full benefit of its value”.

However, I repeatedly found that I couldn’t understand particular examples in the literature, so I started methodically learning abstract algebra, set theory, foundations, topology, etc. so that I could understand those examples of particular CT constructions. ;-)

I developed an independent interest and respect for these fields, and, from flipping back and forth between them, came to see them as expressing different aspects of the same ideas.

So you are starting with the assumption that “everybody” already knows “business as usual” and needs to hear about the fresh contributions that CT brings to the table, whereas I started with the notion that “everybody” wants to work in the exciting new field of CT but needs to learn to understand and respect the “traditional” approach.

So once again the principle of duality shows that we are in complete agreement (up to isomorphism). ;-)

Posted by: Marc Hamann on November 12, 2006 8:41 PM | Permalink | Reply to this

Re: Reality Check

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Category theory isn’t the same as n-category theory. Category theory is old, and in lots of mathematics it is very well established. (I agree that, pedagogically, it often makes sense to come to category theory after having learned a lot of concrete examples set-theoretically, so that you can be surprised and grateful that it unifies things so nicely. :-) For that matter, it makes sense to come to set theory after having learned a lot of concrete examples of, e.g. functions, and relations, in the context of numbers, elementary algebra and analysis, Euclidean and other geometries, etc.)

However, n-category theory is not old, and certainly not widely established and recognised as a basis for other parts of mathematics. It is just starting to be known in a few particularly suitable areas. Moreover, the whole area of n-category theory is not very well developed as a branch of mathematics. Lots of very basic theorems and concepts are missing or only partly formulated. So I think that accounts for the ghetto mentality, the proselytising and the sense of a young, exciting new field.

(OK, maybe I’m exaggerating a bit for dramatic effect.)

It’s true that the n-categorical community grew out of the categorical community in a way that the latter did not grow out of the community that created set theory, but to mathematicians (and mathematical physicists) outside these communities, this is not a very important distinction.

(Disclaimer: I don’t know the real history tremendously well. This is just my impression from the bits I have read.)

Posted by: Tim Silverman on November 12, 2006 9:11 PM | Permalink | Reply to this

Re: Reality Check

Category theory is old, and in lots of mathematics it is very well established.

Well established or not, it can still be a bit of a black mark. My dissertation is extremely concrete, especially considering that the nearest relative is Khovanov homology. However I still had many people warning me that merely having the word “category” in the title would halve my job offers. That may be overstating the case somewhat, but sadly not by too much.

Posted by: John Armstrong on November 12, 2006 9:46 PM | Permalink | Reply to this

Re: Reality Check

Marc wrote:

So once again the principle of duality shows that we are in complete agreement (up to isomorphism).

Great! That’s a relief. It’s nice when that happens.

I realized how much category theory and especially n-category theory still have underdog status in the USA when the topologist Peter May and I applied for an NSF grant on the subject.

It turned out that:

Nobody had gotten a National Science Foundation grant for work on pure category theory for many decades. According to May, saying you’re a category theorist is the kiss of death. You can use category theory in your work, but don’t say you are a category theorist!

Subsequently, two students associated with May got NSF postdocs for work on pure n-category theory. So, things may be starting to change.
I’m the most senior n-category theorist in the USA. If you count the whole world, this is far from true - compared to the bigshots in Canada, Britain, Europe and Australia, I’m still a kid. In the USA, I win the competition by default. There’s just barely anyone here seriously doing n-categories except me, my pal James Dolan, my students, and Peter May’s students.

(May doesn’t consider himself an n-category theorist. Dolan doesn’t have an academic job. Otherwise, either of them would outrank me as the “senior expert on n-categories” for the purposes of writing grant proposals.)

Our grant proposal was rejected. Twice. Luckily we were able to convert it into a proposal for an IMA workshop that got accepted.

So, I feel entitled to drum up interest in n-categories without worrying about the extinction of traditional set-theoretic mathematics. I find the most effective way to do this is to ignore the old prejudices against category theory and dive in, applying n-categories to everything in sight. But, the old prejudices still exist.

Posted by: John Baez on November 12, 2006 9:46 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

“a functor is an analogy”

But Jim Dolan once convinced me that an analogy is actually a span.

Judging from Google, the paradigmatic example of an analogy is

graceful : clumsy :: late : early,

and this is an analogy (according to the same web page), because both are examples of antonyms.

The source of the span here is the abstract notion of antonym; one leg maps this to the (relatively) concrete notion of movement, while the other maps to the concrete notion of time.

The point: In an analogy, you don’t just have the two concrete examples; you also have abstract pattern that appears in both of them.

Posted by: Toby Bartels on November 16, 2006 11:51 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

graceful : clumsy :: late : early

Seems an odd choice to me and no better than

graceful : clumsy :: early : late.

Isn’t a better example:

morning : evening :: childhood : old age ?

But the span idea’s interesting. A common source - a directed line - mapping into a day and a lifetime respectively.

The next stage to consider is Turner and Fauconnier’s Conceptual Blending. Goguen tried to represent this with what he called 3/2-pushouts. I have some slides of a talk I once gave around this theme.

Posted by: David Corfield on November 17, 2006 1:03 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

So, I guess what seems a bit lacking in the analogy you give is the lack of structure in the source, although it includes a symmetry. At what point of removing structure does the result fail to be an analogy? If the source were merely a pair of points, that wouldn’t do.

Posted by: David Corfield on November 17, 2006 1:28 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

I’m sorry, but I’m lost now, even with threaded comments. To which analogy (and which “you”) are you referring here?

Posted by: John Armstrong on November 17, 2006 2:31 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

I was referring to the analogy Toby mentioned:

graceful : clumsy :: late : early

Posted by: David Corfield on November 17, 2006 4:08 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

graceful : clumsy :: late : early

I wanted an example from ‘real life’ (as they say), so I searched Google for ‘analogy’. This was the first permanent example on the first hit, which is the only reason that I chose it.

But it works. We are in the category of sets equipped with involution, with the intended interpretation that each set is a set of adjectives describing something and the involution takes each adjective to its antonym; morphisms are functions that preserve (equivalently, reflect) the involution. The abstract source object doesn’t fit this interpretation, except in the most general and abstract way; it is an ordered 2-point set (maybe {‘first’, ‘last’}) equipped with the nontrivial involution. The first concrete target object consists of adjectives describing styles of movement; it includes ‘graceful’ and ‘clumsy’, which are antonyms and thereby linked by that sets involution. The second concrete target object consists of adjectives describing times of an event (relative to an expected time); it includes ‘late’ and ‘early’, which are also antonyms. The actual morphisms comprising the legs of the span are obvious.

(Of course, one can quibble over exactly which objects appear as the targets, even over exactly which category is involved. And it’s hard to believe that the objects that I’ve given are really well defined, since antonyms don’t always come in neat pairs. Then again, this example was probably never intended to work with mathematical precision.)

This isn’t a very deep analogy, which is why it exists within a single category. Deeper analogies (such as the analogies which are functors) should be spans in some 2category. The deepest analogies will be spans in an ωcategory.

Posted by: Toby Bartels on November 17, 2006 10:10 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

And a Goguenian pushout creatively throws up an image of graceful lateness and clumsy earliness. I guess it doesn’t work in baseball, but you’ll know you’ve understood cricket when you learn to appreciate an elegantly controlled late cut.

Posted by: David Corfield on November 17, 2006 10:43 PM | Permalink | Reply to this

Re: Ulam on Banach; Re: A Categorical Manifesto

I think there’s room for both views, especially when you take into account the inherent directionality of the “is a” relation.

So given these two slogans, what’s a metaphor?

Posted by: John Armstrong on November 17, 2006 1:33 PM | Permalink | Reply to this

Re: A Categorical Manifesto

Banach (according to Ulam): “Good mathematicians see analogies. Great mathematicians see analogies between analogies.”

Jonathan asked: “Anyone think that the quote can be made axiomatical? That ‘very great mathematicians see analogies between analogies between analogies between analogies’?”

Actually Banach’s alleged saying sounds appealing but rather suspicious as shown by Jonathan’s suggestion that seems to push it towards being absurd.

We can check this by examples:I am sure there are many examples of useful analogies in mathematics (what are the best such examples, in your opinion) but can you give a few of analogies between analogies? And can you give a single example of an analogy between analogies between analogies??

(We can also check it by generalizing: Does Banach’s saying (or Jonathan’s) make sense if we replace “mathematician” by “scientists”,”police detectives”, “medical doctors”??)

So maybe Banach’s statement is not much better when scrutinized than the common public perception:

“Good mathematicians can multiply 2 digit numbers in their heads and great mathematicians can multiply 5-digit numbers in their heads!”

(And what about computer science?)

Posted by: Gina on November 13, 2006 1:23 PM | Permalink | Reply to this

Re: A Categorical Manifesto

The analogy between number fields and function fields treated in TWF205 has surely got to be in anyone’s Top 10.

For a meta-level analogy, three steps up the ladder are modifications of pseudonatural transformations. Anyone got a pretty example to show us?

Posted by: David Corfield on November 13, 2006 1:50 PM | Permalink | Reply to this

Re: A Categorical Manifesto

We can check this by examples:I am sure there are many examples of useful analogies in mathematics (what are the best such examples, in your opinion) but can you give a few of analogies between analogies? And can you give a single example of an analogy between analogies between analogies??

Analogy: A topological space X is to a covering space Y of X as the fundamental group of X is to the fundamental group of Y (a functor is an analogy). Better: The collection of covering spaces (up to homeomorphism) is like the collection of subgroups of the fundamental group (up to conjugacy).

Analogy: A field K is to an extension field K’ as the group of symmetries of K is to the group of K’. Better: the collection of extension fields is like the collection of subgroups.

Analogy between analogies: Both of these situations express an adjointness, and the analogies between covering spaces and subgroups of the fundamental group and between extension fields and subgroups of the symmetry group are analogous.

Posted by: John Armstrong on November 13, 2006 1:58 PM | Permalink | Reply to this

High-level Analogies and endofunctions; Re: A Categorical Manifesto

Re: High-level Analogies and endofunctions; Re: A Categorical Manifesto

An analogy is a parallelogram in semantic space.

I’m sure you could. I’d start with comparing V.16 (“If four magnitudes are proportional, then they are also proportional alternately.”) with I.33 (“Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.”).

Oh wait, if alternate ratios and parallelograms are line segments then I’ve just drawn an analogy between analogies, haven’t I? Gosh, if there are so few of them around how come I can’t help but trip over them?

Posted by: John Armstrong on November 13, 2006 11:37 PM | Permalink | Reply to this

Re: High-level Analogies and endofunctions; Re: A Categorical Manifesto

Oops, I even missed a level.

Proportionalities are analogies between quantities. Moving to an alternate proportion is an analogy between the analogies.

Equal and parallel line segments are analogies between points. Making a parallelogram is an analogy between analogies.

So the connection between V.16 and I.33 is an analogy between analogies between analogies, as requested.

Posted by: John Armstrong on November 13, 2006 11:43 PM | Permalink | Reply to this

Re: A Categorical Manifesto

Thanks, John, for the two very nice analogies which ringed a bell for me, and also the analogy between them. (This marvelous adjoint and how it relates the two examples sounds like a great invitation to category theory.) I could not follow the later stuff and, in particular, your analogies between analogies between analogies example. (I’d love to see more examples that I can understand or at least relate to.)

Now we can empirically test Banach’s statement. Is it true to say that the mathematicians who found the first and second analogies you mentioned are good mathematicians while the mathematician who found the analogy between the analogies is an excellent mathematician. (Who are these mathematicians?)

Posted by: gina on November 14, 2006 12:36 AM | Permalink | Reply to this

Re: A Categorical Manifesto

I’ll try to explain a bit more thoroughly.

Okay, a proportionality is a statement about four quantities of the same type (four lengths, say). It says that the ratio of length A to length B is equal to that of length C to length D – A:B::C:D. It’s pretty much the ur-example of an analogy.

V.33 says that if A:B::C:D then A:C::B:D. Do it with fractions if you like. Euclid’s method is a bit ugly. Anyhow, there’s a definite way that the analogies A:B::C:D and A:C::B:D are related. Given another proportionality E:F::G:H and its alternate E:G::F:H, we can see an analogy

(A:B::C:D):(A:C::B:D)::(E:F::G:H):(E:G::F:H)

Now equal and parallel line segments we can think of as displacement vectors. That is, given points A, B, C, and D we may have the segment AB and the segment CD equal and parallel, which basically means that B is displaced from A the same as D is from C. But if that’s true, then C is displaced from A the same as D is from B (the points are the corners of a parallelogram. The pair of segments (AB,CD) is an analogy, as is the pair (AC,BD), and these two are related in a specific way. In fact, given another set of points forming a parallelogram we have two analogies, (EF,GH) and (EG,FH). These are also related in the same way, so

(AB,CD):(AC,BD)::(EF,GH):(EG,FH)

The top of this pyramid is subtler. The idea is that each of the above analogies is true because of a commutativity. The parallelogram analogy essentially comes down to the commutativity of translations in Euclidean space, while the proportionality analogy comes down to the commutativity of multiplication of ratios of quantities.

As for who did what, both of those results are in Euclid, but we have little idea for sure who discovered them – many results in Euclid are known to not be original. Someone may have seen them as analogies (between analogies) and seen the analogy between them before, but I actually just concocted it on the fly in response to JVP’s comment.

So I suppose the question comes down: am I (John Armstrong) a great mathematician? If I am, surely it’s for reasons unknown (but time will tell).

Posted by: John Armstrong on November 14, 2006 1:10 AM | Permalink | Reply to this

Re: A Categorical Manifesto

John, I was not referring to the geometry stuff which while elementary does not speak to me. I was referring to (your great examples of) the analogies between subgroups of the fundamental groups and coveing spaces, and the analogy between extensions of a field and subgroups of a group; and the analogy between these two analogies.

According to Banach’s principle, good mathematicians make analogies and excellent mathematicians make analogies between analogies so we can test it here. The mathematician behind the second analogy you mention is Galois, right? As for the first example, perhaps Poincare? (I am not sure; he discovered the fundamental group, I think.) They were, for sure, good mathematicians. Who is the excellent mathematician behind the analogy between the analogies?

Posted by: Gina on November 14, 2006 2:02 AM | Permalink | Reply to this

Re: A Categorical Manifesto

The analogy I mentioned between function fields and number fields (related to John’s Galoisian/Poincarean one) could easily be cast as an analogy between analogies. Then we’d have Riemann responsible for the first, Kummer (followed by Dedekind) responsible for the second, and Dedekind-Weber responsible for the analogy between them (see chap 4 of my book). Later Artin and Weil got involved with a third analogy, which they related to the other two, by analogy.

I suppose that as particular analogies become part of the furniture of mathematics, it’s not so clever to see an analogy between them. It’s when one or both of the base analogies isn’t so well established, and yet still one sees the meta-analogy that we should gasp.

Posted by: David Corfield on November 14, 2006 8:41 AM | Permalink | Reply to this

Re: A Categorical Manifesto

Thanks, David, this is very interesting. I suppose Banach’s saying does make sense.
(He could just say: “good and great mathematician spend theit sweet lives by seeking analogies and analogies between analogies, among various other things.” But this would have been much less impressive.)

I question the third level form of analogies, also because naively it looks that you can express “analogies between analogies betwwen analogies” already by repeating twice somehow “analogies between analogies”.

But apropos Banach’s saying, what about “greatness” is it THE moving force behind it all? Greatness and the seek of greatness can also be questioned.

Posted by: Gina on November 14, 2006 3:19 PM | Permalink | Reply to this

Re: A Categorical Manifesto

I question the third level form of analogies, also because naively it looks that you can express “analogies between analogies betwwen analogies” already by repeating twice somehow “analogies between analogies”.

I know that my example wasn’t what you’re most interested in, but go back and read it as an illustration anyhow. I’m not just repeating “analogies between analogies”, but showing how those two are specifically analogous.

But apropos Banach’s saying, what about “greatness” is it THE moving force behind it all?

I think that the “greatness” is best understood in the context of the above-mentioned pendulum between problem-solving and intrinsic nature in mathematical fashion. I read Banach as exhorting a shift to searching for the hidden connections – the intrinsic nature – and noting that this took a great mathematician in a time when the focus was on solving specific problems.

On the flip side, nowadays I find that algebraic geometry has swung away from concrete problems to the point that it would take a phenomenal mathematician indeed to tell me what I’m supposed to do to solve an actual polynomial equation rather than just tell me how it’s like that equation over there.

Posted by: John Armstrong on November 14, 2006 3:53 PM | Permalink | Reply to this

Re: A Categorical Manifesto

Sure John, I will be happy to go back and read the geometry examples. (They are quite elementary.)

Posted by: Gina on November 15, 2006 6:00 AM | Permalink | Reply to this

Re: A Categorical Manifesto

One of the highest forms of meta-analogy we could mention is the Langlands Program. According to Frenkel:

Langlands Program has emerged in recent years as a kind of Grand Unified Theory of Mathematics. Conceived initially as a bridge between Number Theory and Automorphic Representations, it has now expanded into such areas as Geometry and Quantum Field Theory, tying together seemingly unrelated disciplines into a web of tantalizing conjectures. The Langlands Correspondence manifests itself in a variety of ways in these diverse areas of mathematics and physics, but the same salient features, such as the appearance of the Langlands dual group, are always present. This points to something deeply mysterious and elusive, and that is what makes this correspondence so fascinating.

But then we’d need to consider Kapranov’s ‘Analogies between the Langlands correspondence and topological quantum field theory’, which as Urs points out involves a categorification of a part of Langlands.

Posted by: David Corfield on November 15, 2006 12:34 PM | Permalink | Reply to this

Re: A Categorical Manifesto

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But then we’d need to consider Kapranov’s ‘Analogies between the Langlands correspondence and topological quantum field theory’, which as Urs points out involves a categorification of a part of Langlands.

At that link I was complaining that experts I met did not try to see (geometric) Langlands itself as a categorification of something (namely of certain concepts in linear algebra #).

But now I see a comment by Ben-Zvi, which says, among other things:

After all, geometric Langlands is a very general categorification program in representation theory, so one would expect it to relate to the kinds of categorifications that give rise to Khovanov homology.

That sounds more like what I had in mind.

Posted by: urs on November 15, 2006 12:49 PM | Permalink | Reply to this

Re: A Categorical Manifesto

Shouldn’t something Langlandsian come out of the 2-representations of the general linear 2-groups we’ve been considering in Klein 2-geometry? Aside from Crane-Sheppeard and Freidel at al. on the Poincaré 2-group, have the 2-reps of any other 2-group been calculated?

Posted by: David Corfield on November 15, 2006 1:09 PM | Permalink | Reply to this

Re: A Categorical Manifesto

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David wrote:

Shouldn’t something Langlandsian come out of the 2-representations of the general linear 2-groups we’ve been considering in Klein 2-geometry?

If only I knew… I’d have even one more project I’m too busy to actually do.

Aside from Crane-Sheppeard and Freidel at al. on the Poincaré 2-group, have the 2-reps of any other 2-group been calculated?

Yes, though with certain annoying “strictness” assumptions on the representations. Back in January 2005, I listed the papers I knew on this subject in week210 :

Magnus Forrester-Barker, Representations of crossed modules and cat¹-groups, Ph.D. thesis, Department of Mathematics, University of Wales, Bangor, 2004. Available at http://www.informatics.bangor.ac.uk/public/mathematics/research/ftp/theses/forrester-barker.pdf
John Barrett and Marco Mackaay, Categorical representations of categorical groups, available as math.CT/0407463.
Josep Elgueta, Representation theory of 2-groups on finite dimensional 2-vector spaces, available as math.CT/0408120.
Louis Crane and David Yetter, Measurable categories and 2-groups, available as math.QA/0305176.
Louis Crane and Marnie D. Sheppeard, 2-categorical Poincaré representations and state sum applications, available as math.QA/0306440.

By now, Baratin, Freidel and coauthors possibly including myself have done a lot more work on representations of the Poincaré 2-group on 2-Hilbert spaces of Crane-Yetter type, following up on the work of Crane, Yetter, and Sheppeard.

Also, Danny Stevenson, Alissa Crans and myself are writing a paper where we classify representations of the Lie 2-algebra $𝔤_{k}$ on 2-vector spaces of Baez-Crans type. This is equivalent to the Lie 2-algebra of the Lie 2-group that Urs likes to call ${String}_{G}$ .

Maybe something related to “geometric Langlands” should appear here, since ${String}_{G}$ is built using the Kac-Moody central extension of the loop group of $G$ , and that’s a key player in geometric Langlands theory.

I’m not sure what I should be keeping an eye out for, though.

Posted by: John Baez on November 16, 2006 2:04 AM | Permalink | Reply to this

Re: A Categorical Manifesto

It would be good to see someone give a comparison of 2-reps for a particular 2-group in different 2-vector spaces, according to Urs’ big picture (August 07, 11.40am).

I presume people are wondering about which equivalents of features of group representation theory might be found? Are there ‘locally compact 2-groups’, and if so, are there Haar measure and Peter-Weyl theorem equivalents? For finite 2-groups, is there an equivalent of orthogonality in the character table? In general, is there an equivalent of the adjunction between the restricing and inducing functors? What about the branching rules? There must be dozens more questions like these.

Posted by: David Corfield on November 16, 2006 9:00 AM | Permalink | Reply to this

Re: A Categorical Manifesto

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a paper where we classify representations of the Lie 2-algebra $g_{k}$ on 2-vector spaces of Baez-Crans type.

Do these reps induce reps of the strict version of $g_{k}$ ? They should, I guess.

This is very interesting. As I certainly said somewhere before, the String-connection defined by Stolz-Teichner should be nothing but a 2-vector transport associated to a principal 2-transport for the String-2-group.

To see this, take the strict ${String}_{G} = (\hat{Ω} G \to P G)$ and form, for any highest weight rep

(1)

ρ : Σ (\hat{Ω} G) \to Hilb

of the loop group the corresponding canonical 2-rep

(2)

\tilde{ρ} : Σ ({String}_{G}) \to Bim \overset{\subset}{\to}_{Hilb} Mod

on 2-Hilbert spaces. This rep sends the single object of $Σ ({String}_{G})$ to the (von Neumann) algebra $A$ generated by the image of $ρ$ , sends elements in $g \in P G$ to the $A$ - $A$ bimodule which is $A$ as an object, with the right action twisted by the image of $g$ in automorphisms of $A$ - and so on.

The construction is the same as in principle for any finite dimensional strict Lie 2-group. Only difference here is that the tensor product of these loop group algebra bimodules involves Connes fusion. But Andreas Thom assured me that the construction goes through as expected nevertheless.

So, you see, a $\tilde{ρ}$ -associated 2-vector transport yields a 2-connection on String-bundles, and locally it manifestly looks like the String-connection as defined by Stolz-Teichner.

This makes use of the fact that it is easy to represent 2-groups on 2-vector spaces in $Bim (Hilb) \subset_{Hilb} Mod$ .

Now, I am wondering, if we have nontrivial reps of $g_{k}$ on Baez-Crans 2-vector spaces, which should be in

(3)

_{Disc (K)} Mod,

for $K$ the ground field, which notion of String 2-vector bundle connection that would yield.

Posted by: urs on November 16, 2006 9:46 AM | Permalink | Reply to this

Re: A Categorical Manifesto

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David Corfield wrote:

It would be good to see someone give a comparison of 2-reps for a particular 2-group in different [i.e., differently defined types of] 2-vector spaces, according to Urs’ big picture (August 07, 11.40am).

Yes, it would. Until that happens, people will be restrained by not knowing which types of 2-vector spaces are the most interesting for representing different types of 2-groups - or more generally, what the really interesting questions are.

I presume people are wondering about which equivalents of features of group representation theory might be found? Are there ‘locally compact 2-groups’, and if so, are there Haar measure and Peter-Weyl theorem equivalents?

Good questions! I’ve never heard anyone talk about them until you did just now. Some of them seem straightforward. For example: yes, one can define 2-groups in any 2-category with finite limits (see HDA5), so in particular one can define topological 2-groups, locally compact topological 2-groups and so on. A strict locally compact topological 2-group will have a Haar measure on the group of objects and a Haar measure on the group of morphisms, which one can use to “average over the group action”. So, one can try start up the Peter-Weyl machinery in this case.

Weak 2-groups would take more thought, since they don’t have a group of objects - and for weak topological 2-groups, one should really be using Henriques’ definition instead of the obvious definition given in HDA5.

If one had an indefinite supply of grad students, and the energy to advise them, one could set them to work on all these issues.

But, right now there are very few people working on 2-groups. There are no “2-group theorists”: all these people are doing lots of other interesting things too. So, instead of trying to set up a vast edifice of machinery, the right approach is to find some really exciting examples, with connections to other branches of math, which will get more people interested in the subject. Then the machinery will practically build itself.

That’s why I’m working on specific examples, only developing enough general machinery to exhibit these examples. The string 2-group is the best one so far, since it hooks on to loop groups, affine Lie algebras, the WZW model - in short, the whole apparatus of postmodern Lie theory. The Poincare 2-group may also be really interesting - we’ll see.

As you know, various fields of mathematics tends to switch between periods of rapid expansion and periods of consolidation - periods of exploring examples and periods of systematizing what they’ve found.

People tend to typecast category theory as being all about consolidation and systematization, so you might think $n$ -category theory was even more like that. But in fact, right now $n$ -category theory is all about expansion and exploration. There’s gold in them there hills! If we find the nuggets, later people will set up efficient mines.

Posted by: John Baez on November 16, 2006 4:55 PM | Permalink | Reply to this

2-vector spaces

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David and John were talking about the

big picture

of 2-vector spaces.

The idea behind that is that like an ordinary vector space is a module for a sufficiently nice ring, a 2-vector space should be a module category for a sufficently nice (in particular abelian) monoidal category.

For instance Kapranaov-Voevodsky 2-vector spaces are $Vect$ module categories.

I think the term “big picture” was supposed to allude to the fact that also the Baez-Crans 2-vector spaces are examples for that general idea.

A BC 2-vector space is, by definition, a category internal to $K$ -vector spaces.

Now let $Disc (K)$ be the discrete category on the field $K$ . It is monoidal and abelian.

It is clear that every BC 2-vector space is a $Disc (K)$ -module category.

I’d think that, conversely, every $Disc (K)$ -module category is a BC 2-vector space.

All right, but now I have a question:

John has also invented yet another notion of 2-vector space, namely 2-Hilbert spaces as defined in HDA II.

Stripped of the Hilbert space structure, according to this definition we would probably say that a 2-vector space is an abelian and $Vect$ -enriched category.

Now: how exactly are abelian $Vect$ -categories related to module categories over abelian monoidal categories?

I’d think that for a suitably well behaved notion of module category, every $Vect$ module category is in fact also $Vect$ -enriched.

This is certainly true for the image in $_{Vect} Mod$ of the embedding

(1)

Bim (Vect) \overset{\subset}{\to}_{Vect} Mod .

So, in as far as this image is all I really care about anyway, I guess all $Vect$ module categories are also 2-vector spaces in the sense of HDA II.

But is there a converse to this statement?

Posted by: urs on November 16, 2006 5:42 PM | Permalink | Reply to this

Re: A Categorical Manifesto

$MathML-enabled post (click for more details).$

David Ben-Zvi seems to be the kind of Langlands expert I was looking for when I lonesomely wrote those SCT entries on Langlands (like this one).

D. Ben-Zvi:

the Langlands duality is […] a kind of nonabelian generalization of the Fourier transform #

me:

Is it a nonabelian generalization, or isn’t it rather a categorification of the Fourier transform?

D. Ben-Zvi:

well it’s both.. […] Categorifications of the Fourier transform have been used for almost 30 years […] and the geometric Langlands program suggests that one can extend this to nonabelian settings […] #

That’s all I wanted to hear.

Posted by: urs on November 15, 2006 3:20 PM | Permalink | Reply to this

Godel numbering endofunctions; Re: A Categorical Manifesto

Good examples a la Euclid, who may indeed have been an anthology editor, as suggested for Homer.

If Archimedes and Euler and Gauss are the greatest mathematicians, what analogies between analogies… are each best known for? Was Grothendieck or Turing or Godel or von Neumann the deepest into analogies of analogies of analogies…?

So what is the answer to my query on combinatorially enumerating the endofunctions on endofunctions on n points?

Posted by: Jonathan Vos Post on November 14, 2006 4:43 PM | Permalink | Reply to this

Re: Godel numbering endofunctions; Re: A Categorical Manifesto

There’s no such thing as an “endofunction of an endofunction”. Nobody has defined functions between functions.

Posted by: John Baez on November 23, 2006 1:19 AM | Permalink | Reply to this

Re: Godel numbering endofunctions; Re: A Categorical Manifesto

There’s no such thing as an “endofunction of an endofunction”. Nobody has defined functions between functions.

Rather, to make sense of this, we need to categorify! So while a set has a monoid of endofunctions, a category has a monoidal category of endo-functors; each endofunctor has a commutative monoid of endo-natural transformations.

I must confess that I’m not sure exactly what Jonathan has been counting (surely not all endofunctions of an nelement set), but to go higher, he should look at finite categories (or perhaps just finite groupoids to start with); but he’ll need to look at more than just the number of elements (objects or morphisms) to specify one of these.

Posted by: Toby Bartels on November 23, 2006 10:41 PM | Permalink | Reply to this

Graphs of morphisms of graphs; Re: A Categorical Manifesto

Has this been generalized to 2-categories?

BROWN, R., MORRIS, I., SHRIMPTON, J. and WENSLEY, C.D.
Graphs of morphisms of graphs

Abstract: This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton’s (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely `bands’ and `loops’. Such edges are distinguished by the behaviour of morphisms with respect to these elements

And I’ll ask the simplest way that I know: how many 2-categories are there on n objects, where the n objects are mapped to themselves by all possible endofunctions, and the n functions mapped to themselves by all possible endofunctionals?

Posted by: Jonathan Vos Post on November 14, 2006 9:19 PM | Permalink | Reply to this

Re: Graphs of morphisms of graphs; Re: A Categorical Manifesto

how many 2-catgories are there on n objects, where the n objects are mapped to themselves by all possible endofunctions, and the n functions mapped to themselves by all possible endofunctionals?

I’m not sure this is a meaningful question in general: the number of morphisms in a category is a characteristic property of that category rather than predictable from the number of objects. (Looking at this from the inside out, this means that the number of possible morphisms depends on the internal structure of the objects).

Posted by: Marc Hamann on November 14, 2006 9:57 PM | Permalink | Reply to this

Re: Graphs of morphisms of graphs; Re: A Categorical Manifesto

Looking at this from the inside out, this means that the number of possible morphisms depends on the internal structure of the objects

And of course we shouldn’t think about internal structure of objects. Instead we should study such structure by the morphisms between objects rather like scattering theory.

I’ll have to look through that linked paper to see what JVP is talking about at all, since the question is already nonsense for 1-categories. How many categories are there on one object? One for each monoid. There’s got to be some restriction of the kinds of categories we’re talking about in that paper that JVP is glossing over.

Posted by: John Armstrong on November 14, 2006 11:19 PM | Permalink | Reply to this

Re: Graphs of morphisms of graphs; Re: A Categorical Manifesto

And of course we shouldn’t think about internal structure of objects.

Of course not… unless, of course, it helps you to understand a particular category. ;-)

There’s got to be some restriction of the kinds of categories we’re talking about in that paper that JVP is glossing over.

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November 2, 2006