Quantum Physics and Logic at Oxford

February 3, 2010

Quantum Physics and Logic at Oxford

Posted by John Baez

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I’m trying to cut back on jetting about, with some success — but I couldn’t resist going to this:

Quantum Physics and Logic (QPL7), Saturday/Sunday May 29-30, 2010, Oxford. Organized by Bob Coecke, Prakash Panangaden, and Peter Selinger. Before the workshop itself there will be a school from May 24th to 28th.

It’s the seventh of the QPL series. But the meaning of the abbreviation “QPL” has changed. For the first four workshops of this name, it meant “Quantum Programming Languages”. Now it means “Quantum Physics and Logic”. That’s because the scope has broadened to cover everything about “the interaction between modern computer science logic, quantum computation and information, models of spatio-temporal causality, and quantum foundations.”

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So far all I know about the speakers is that there will be 1-hour talks by Antonio Acin, Louis Crane and myself. Antonio Acin works at the Institute of Photonic Sciences in Barcelona. Louis Crane was working on 2-categories and extended topological quantum field theories long, long before they became fashionable. He was the one who got me interested in that stuff!

I plan to give a talk on “Duality in logic and physics”, which will go from categories where the morphisms are matrices taking values in a rig (as we’ve often discussed here, to categories of profunctors and spans, to the way star-autonomous categories act like commutative Frobenius algebras in the world of profunctors, to some $n$ -categorical and/or “derived” analogues of these ideas if the audience is still awake.

Posted at February 3, 2010 4:08 PM UTC

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Re: Quantum Physics and Logic at Oxford

I’m trying to cut back on jetting about…

I take it then that your conviction in AGW is as strong as ever. You wrote back here:

I would hope that everyone of good will, whether they believe humans are causing global warming or not, would agree that this issue is worth understanding and discussing – and that physicists can help.

I think it would be very helpful if scientists, especially those with no career-related stake in climate science such as yourself, could talk us through the case for AGW from their own perspective.

I must say my own confidence in it has reduced over the past few years. But I am no scientist. Perhaps the best things philosophers have achieved throughout the centuries is to point out how the passions interfere with our reasoning. Popper’s major point was to warn us to guard against a psychological tendency we all have to seek only confirmation of our beliefs. It is not hard to imagine passions interfering on both sides of a theory predicting a catastrophic future. The Cambridge philosopher Simon Blackburn makes a brief point about this in his lecture Does Relativism Matter?

A useful case study could be made of the Y2K problem. Perhaps one already has been made.

Posted by: David Corfield on February 4, 2010 12:21 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

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AGW = “Anthropogenic Global Warming”?

It took me a while to come up with that.

Posted by: Simon Willerton on February 4, 2010 1:05 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Yes, that’s right. It seems to have morphed into ‘climate change’ now.

Posted by: David Corfield on February 4, 2010 1:21 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Yes, the anthropogenic is added to emphasise it’s the “hypothesis” that climate level change is not only ocurring but that the dominant is human activity (rather than say variable solar output).

I’d also point out that it’s not a purely scientific question but a decision theoretic question (given some probability density over the believed accuracy of climate change models and possible consequences and mitigation, what’s the “best” actions to take). I think I’ve said before that my “belief” in significant AGW is about 25 percent, but unless you can get my belief below about 5 percent I think AGW is worthing spending time and resources on.

Not being a climate scientist I never read much of the literature (on the grounds that they could write “true but misleading” stuff I wouldn’t have the background to notice, just as I’m sure I could about some areas of computer science), but recently I’ve stopped doing that. Instead what little reading I do now is on the side of what effects possible actions can acheive, which AFAICS is the much more vauge side of the equation. (It’s interesting that a lot of people who would consider reducing their flying for environmental reasons to be beyond the pale have been saying that they won’t fly anymore because of the introduction of milimeter wave scanning (ie, virtually naked imaging). A lot of the comment is probably just bombast, but it’ll be interesting if it does turn out to be a more “effective” action.) So if there are any climatologists I’d be much, much more interested in the extent to which there are technically viable (and behaviourally possible) actions that could deal with a putative global warming. After all, if there ain’t nothing effective that can be done, why worry about it?

Posted by: bane on February 4, 2010 2:30 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

John already has.

Posted by: Todd Trimble on February 4, 2010 2:29 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Ah I hadn’t seen that. Any new thoughts since 2006? Has the credibility of the projections changed for you? Any worries about the accuracy of the temperature record and of the reconstruction of the past millennium’s temperature? Any credence to give to the thought that natural oscillations (Pacific Decadal Oscillation, Atlantic Multidecadal Oscillation, Southern Oscillation) explain much of the variation? Or other periodic forcings?

Posted by: David Corfield on February 4, 2010 4:41 PM | Permalink | Reply to this

Emotions in Science

David Corfield wrote:

A useful case study could be made of the Y2K problem.

There is a little hint that the Y2K was not a hoax: German EC cards hit by 2010 bug.

If you are looking for a case study that relates the public fear with the money that was spent on the Y2K bug, I’m very sceptical: That seems to be untractable to me.

But I can relate a little story from my bread-and-butter job: There are about 100 microcontrollers in a modern car (little computers), each one needs specific software to operate (in fact some cars have a bigger acceleration than others because the controllers of the engine operate with a better tuned software, while having identical hardware). There are different versions of the software for every microcontroller. Only certain configurations are tested before a model series hits the streets. If you have a problem with your car and bring it to a garage, the mechanic may solve the problem by upgrading the software for one or more of the microcontrollers. After that, you drive a car that operates with a software configuration that has never been tested before! Why is that? Because it works most of the time, and it would be very expensive to do otherwise, and no customer who buys a car thinks about the software that is used to operate it, all the emotions are located elswhere. (Can you imagine someone bragging like: “Hey Jonny, just upgraded my engine software to v.3.1, let’s see who is faster now!”).

Posted by: Tim van Beek on February 4, 2010 3:00 PM | Permalink | Reply to this

Re: Emotions in Science

For me, the important question about Y2K is not that there was software that would give wrong results. The difficult to analyse bit is whether, had the vast majority of those problems not been fixed, they would have resulted in significant “real world” problems or not. The 2010 bug may provide a means for estimating the Y2K counterfactual.

Posted by: bane on February 4, 2010 3:28 PM | Permalink | Reply to this

Re: Emotions in Science

There may be a chance to get some data if you concentrate on one big company and software systems that were developed specifically for it - but it’s hopeless to try to answer that for the Y2K bug, IMHO. The damage to reputation is especially hard to estimate: If someone is killed by a car accident caused by a software bug, the company that builds these cars is dead (the rumor alone would suffice, I guess).

But I think that David is mostly interested in the role that irrational emotions play when a large group is faced with a danger that is hard to evaluate.

Think of the famous pictures from outer space of earth: This beautiful, white-blue ball, precious, small, fragile, utterly alone in a black, cold and inconceivable vast space.
Question: Would AGW stand a chance to get public attention if humans weren’t sensitized to the vulnerability of earth by these pictures? Wouldn’t you ridicule the idea that anything that you do could harm the whole planet?

Posted by: Tim van Beek on February 4, 2010 3:44 PM | Permalink | Reply to this

Re: Emotions in Science

But I think that David is mostly interested in the role that irrational emotions play when a large group is faced with a danger that is hard to evaluate.

Think of the famous pictures from outer space of earth: This beautiful, white-blue ball, precious, small, fragile, utterly alone in a black, cold and inconceivable vast space.

Question: Would AGW stand a chance to get public attention if humans weren’t sensitized to the vulnerability of earth by these pictures? Wouldn’t you ridicule the idea that anything that you do could harm the whole planet?

Something about the way this is framed is somewhat off-putting: it suggests that those who worry about a significant threat posed by AGW are the ones falling prey to fear, and the skeptics are the ones behaving more rationally. I expect there is plenty of irrational emotion on both sides of the aisle, as well as rational argument and counterargument.

This is particularly clear in the tendentious framing of the question. I think my own answers would be, respectively, “I don’t see why not” (working climatologists and ecologists who study the matter and sound a warning about AGW generally have no need to and don’t invoke these images, AFAIK) and “no, not necessarily” (if “you” is taken in a collective sense, and over a good span of time).

I will admit that I personally have not made a deep study of the issue. But I’m not sure the present discussion is moving in a fruitful direction. I would rather be interested in hearing more from David: do you think you generally detect irrationality in those who are scientifically arguing for AGW?

Posted by: Todd Trimble on February 4, 2010 5:03 PM | Permalink | Reply to this

Re: Emotions in Science

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Todd wrote:

…it suggests that those who worry about a significant threat posed by AGW are the ones falling prey to fear, and the skeptics are the ones behaving more rationally.

Oh, sorry, that was not what I meant - I like to explore the reasons why I react in certain ways to certain inputs and compare that to others, residing entirely on the meta-level in this instance. If you pin me down, I will say that I’m glad that these images of the earth exist, that I hope that they have a similar effect on others as they have on me, and that I consider the AGW to be an important issue. Obviously I managed to deliver the opposite message.

But I’m not sure the present discussion is moving in a fruitful direction.

In that case I won’t pursue it, but allow me to apologize and to try to clarify my point.

Posted by: Tim van Beek on February 4, 2010 5:24 PM | Permalink | Reply to this

Re: Emotions in Science

Thanks for clarifying, Tim!

Posted by: Todd Trimble on February 4, 2010 10:02 PM | Permalink | Reply to this

Re: Emotions in Science

Perhaps it’s understandable that that the debate is not heading in a fruitful direction as I introduced a tangle of issues in my own comment. There’s the question of the rationality or otherwise displayed amongst the experts, and then the rationality of the uptake of their claims by the media, politicians, scientists from other fields and the public. As for the latter, all kinds of passions enter, including in some cases, as Blackburn suggested, feelings of guilt. In my own case I can perceive in a younger me a wish for capitalism to show itself to fail. Perhaps parenthood put a stop to that. But that’s just a new passion.

As for the question you actually asked about irrationality in scientific AGW promoters, I must say I have been disturbed by some of the revelations of the past months. The climate science community shows deviations from an optimal arrangement where a discipline openly shares its data and methods and is open to criticism from their fellow scientists, especially here where results are very sensitive to the choice of statistical techniques. The desire to be proved wrong, never the most frequently appearing of the intellectual virtues, seems to be in very short supply.

Posted by: David Corfield on February 4, 2010 5:54 PM | Permalink | Reply to this

Re: Emotions in Science

As for the question you actually asked about irrationality in scientific AGW promoters, I must say I have been disturbed by some of the revelations of the past months.

Yes, it is very disheartening to see in those who call themselves scientists. But these revelations are more recent; you said that over the past few years, your confidence in AGW has waned. So it sounds like an ongoing thing in your case, and I’d be interested to hear more about what’s behind that.

Despite those revelations (which are not quite black and white, I don’t think), I for my part have a hard time in general believing that the overwhelming consensus in the peer-reviewed academic journals, that AGW is real and significant, is due to ‘liberal bias’, as often suggested here in the US. I like to think of science as more robust than that. In any case, I would be interested in other case studies where liberal bias in science is said to exist, and what sorts of sociological conditions prevail in those cases.

Posted by: Todd Trimble on February 4, 2010 7:04 PM | Permalink | Reply to this

Re: Emotions in Science

I’ve read Steve McIntyre’s blog for quite a while now, and the deficiencies of practice exposed in the leaked e-mails were apparent way before the leak. One large problem seems to be the use of questionable statistical techniques to extract a signal from noisy climate data and especially temperature proxies. This seems to be a danger in many fields where practitioners might not be expected to be expert in statistics and yet rely heavily on statistical techniques, as in medical research and psychology.

The issue of AGW is not one like continental drift or puerperal fever where someone is either right or wrong. The question here is one of degree, so there is a spectrum of belief from a catastrophic version to a very mild warming version. I don’t think there’s a consensus around Al Gore’s catastrophic end as is sometimes portrayed.

If it turns out that sub-optimal scientific practice has led to overestimates of warming, it would strike me as odd to offer a single and vague factor such as ‘liberal bias’ as the reason. Even trying with hindsight to understand the reasoning which led people to reject Semmelweis and Wegener is no easy matter, but we should not underestimate forms of institutional irrationality which discourages proper questioning of beliefs.

Posted by: David Corfield on February 5, 2010 10:25 AM | Permalink | Reply to this

Re: Emotions in Science

David,

“One large problem seems to be the use of questionable statistical techniques to extract a signal from noisy climate data and especially temperature proxies.”

In my opinion, too much emphasis has been focused on the reconstruction statistical methods as “questionable”. Not all statistical methods currently in use share the same flaws of some early reconstructions, and there is a nice new crop of more elegant methods coming out in cooperation with professional statisticians which, from preliminary results, I don’t expect to differ dramatically in their final reconstructions. (The error bars, however, will likely get larger.)

The more serious problem with paleotemperature reconstructions, in my opinion, is in the choice of which proxy data to use in the first place. It is reasonable to suppose that some sources are better proxies for temperatures than others, that some paleo sites aren’t reporting useful information about temperature at all and should be ignored, and there are physical/physiological reasons to believe this could be the case. But it is difficult to go from this supposition to an actual test to determine which proxies should be included in the reconstruction and which should be discarded as “outliers”. I personally am not yet convinced that proxy selection has been done in a sufficiently objective manner in these big multiproxy studies.

This is especially relevant in light of the notorious “divergence problem”, which has not yet been definitively accounted for. If some of the proxies demonstrably don’t always behave as temperature proxies, how much can we trust them in the past without knowing the exact reason why? Again, there may be plausible hypotheses about what could cause them to deviate from temperature proxies, but they have not yet been demonstrated.

Posted by: Nathan Urban on February 5, 2010 7:05 PM | Permalink | Reply to this

Re: Emotions in Science

David,

“I don’t think there’s a consensus around Al Gore’s catastrophic end as is sometimes portrayed.”

I’m not sure what you mean by that, but I thought I should point out that Al Gore is usually a moderate, as far as the spectrum of scientific opinion on climate change is concerned.

Posted by: Nathan Urban on February 5, 2010 7:19 PM | Permalink | Reply to this

Re: Emotions in Science

I think we can agree that

1. CO2 is a greenhouse gas;

2. Humans have been adding CO2 (and other green house gases) to the atmosphere at a significant rate since the beginning of the Industrial Revolution (8.7 Billion tons of CO2 in 2008 alone according to the Global Carbon Project).

3. The concentrations of CO2 and other greenhouse gases have been increasing in the atmosphere (We can argue if this is directly related to 2 or not, but it is hard for me to imagine that 2 is not related to 3).

4. Increased concentration of greenhouse gases in the atmosphere is bound to affect climate. That’s basic physics.

Everything else is details that I won’t pretend to understand.

Posted by: Eugene Lerman on February 4, 2010 9:00 PM | Permalink | Reply to this

Re: Emotions in Science

In response to a comment of David’s elsewhere (which I think was addressed to John), I was about to break down my understanding of the basic argument in a similar way; I’m glad to see it already done here.

The reason is that I wanted to see where the main objections to AGW lie. AFAICT, most of David’s objections have to do either with the reliability of temperature measurement, or whether any significant temperature variation that is indeed there has less to do with greenhouse gas emissions than with other factors that we have no control over anyway. The first has more to do with scientific methodology, and the second has more to do with point (4) and its relative significance.

David, is that a fair summary?

Posted by: Todd Trimble on February 5, 2010 11:58 AM | Permalink | Reply to this

Re: Emotions in Science

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About the warming due to increased $CO_2$ , I believe there’s little dispute. As far as I know, the suggested warming is 1.2 $^{\circ}$ C per doubling. A reasonable estimate is that a doubling of pre-industrial levels to 580 parts per million will have occurred by the end of the century, if we carry on as usual. We should be about half way through that increase by now.

All the dispute, I believe, is around the feedback effects, e.g., will there be increased cloud cover and at what height, and to predict what these are one needs climate models. There are even those who believe in negative feedback from temperature increase, such as Robert Lindzen at MIT. Reconstructions of earlier temperatures matter because models ought to capture the kind of variation shown by say the mediaeval warm period when man had emitted little $CO_2$ . However even temperature data for the twentieth century is disputed.

Posted by: David Corfield on February 5, 2010 1:40 PM | Permalink | Reply to this

Re: Emotions in Science

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I’m going to quickly bow out of this thread (although I find it interesting), because I don’t want to further irritate people who want to get back to the n-categorical topics this blog should be concerned with. I just have one further question for David, regarding

Any credence to give to the thought that natural oscillations (Pacific Decadal Oscillation, Atlantic Multidecadal Oscillation, Southern Oscillation) explain much of the variation? Or other periodic forcings?

I am not quite sure how this squares with “About the warming due to increased $CO_2$ , I believe there’s little dispute.” Should I conclude from this last sentence that you essentially have no problem with the summary statement from the IPCC, that “[most] of the observed increase in global average temperatures since the mid-20th century is very likely due to the observed increase in [human greenhouse gas] concentrations”? So that for you the dispute really centers on predicting the severity of effects, and the methodology being used for that?

Posted by: Todd Trimble on February 5, 2010 4:55 PM | Permalink | Reply to this

Re: Emotions in Science

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Ok, let’s wrap it up then. From what I know (and keep remembering I’m no expert) the thesis that the rise of half a degree of so in the last century is down to human emitted $CO_2$ is reasonably well supported. The trouble is predicting what comes next. The temperature bobbles up and down from month to month and shows no upward trend in the past decade as some models predicted. But is that down to the heating being ‘masked’ by some other process?

Posted by: David Corfield on February 5, 2010 5:49 PM | Permalink | Reply to this

Re: Emotions in Science

There are US senators who proudly say that the threat of catastrophic global warming is the “greatest hoax ever perpetrated on the American people”. The arguments about global warming in the US are all too often like the arguments about Evolution — they are political.

I feel sorry for climatologists: all too many people have all too many reasons to question their motives.

Posted by: Eugene Lerman on February 5, 2010 5:17 PM | Permalink | Reply to this

Re: Emotions in Science

David,

“All the dispute, I believe, is around the feedback effects”

Well, most of it.

“Reconstructions of earlier temperatures matter because models ought to capture the kind of variation shown by say the mediaeval warm period when man had emitted little CO2.”

It’s not clear how much reconstructions of earlier temperatures can inform estimates of feedback effects. So it’s debatable how much these reconstructions actually matter in the first place, despite the great furor over them.

Suppose we learned that, say, temperatures in the Medieval Warm Period were larger than is currently supposed based on commonly accepted reconstructions. What does this imply about the climate system? There are several possibilities:

1. Natural variability in temperature may be larger than currently believed.

What does this imply for climate sensitivity to CO2? Unclear. You can probably keep climate sensitivity estimates the same and increase natural variability and still reproduce the paleo temperatures (just greater fluctuations about the mean, although you might be able to constrain it we don’t learn that other events, say the Little Ice Age, also had greater variability). You can also increase or decrease the feedbacks and get the same paleo temperatures, depending on what you assume about the natural variability (e.g. its amplitude and phase).

We could suppose, say, that during the MWP there was some oscillation in ocean temperatures which caused significant warming beyond what is attributable to external forcings, but that says little about whether such an oscillation also affected 20th century temperatures, or even whether such an oscillation today ought to be in a warming phase or a cooling phase. To determine that, you have to look at the instrumental record, because the paleo record is unlikely to give you sufficient spatial or temporal resolution to tell you detailed information about ocean circulation.

2. Natural climate forcings (e.g., solar irradiance) may have been different than is currently believed.

What does this imply for climate sensitivity to CO2? Unclear, unless you can pin down how the forcings were different. For example, it could be that a larger MWP could be explained by a larger solar forcing, keeping feedbacks the same. But unless you accurately know how much larger the forcing was than currently believed, you could end up explaining it with either larger or smaller feedbacks, too.

3. Feedbacks to natural climate forcings (e.g., solar irradiance) may have been larger than is currently believed.

What does this imply for climate sensitivity to CO2? That it could be larger than currently believed (if the feedbacks operate equally on CO2 and other natural forcings). But if the feedbacks operate on different forcings differently, in some unknown way, then it remains inconclusive.

In short, millennial scale temperature reconstructions are of only limited use in determining climate sensitivity to CO2, because of limitations in our ability to resolve the confounding between climate responses to feedbacks, natural variability, and natural forcings. At best you can probably use such reconstructions to somewhat downweight sensitivities on the very low or high sides of the IPCC range. They are a weaker line of evidence (either for weak or strong AGW) than is commonly assumed.

I’ve focused my response on the millennial-scale “hockey stick” reconstructions which you seem to be emphasizing (e.g. by reference to the MWP). The situation is somewhat different for temperature variations on different time scales, and/or with larger climate signals, such as the Quaternary glacial-interglacial cycle or the Eocene hyperthermal events. That would be a matter for a separate (and lengthy) discussion!

Posted by: Nathan Urban on February 5, 2010 7:08 PM | Permalink | Reply to this

Re: Emotions in Science

I don’t know either way what a study might show of the Y2K bug. I know some people who think a crisis was averted. Money was well spent. Of course, there are others who think it was a scam to earn some people a lot of money. There are other options, naturally – an honest mistake, and so on.

But you’re suggesting it would be difficult to get to the truth of what happened. If we can’t get to the bottom of the effects of our interventions on a manmade system for a limited period in the past, I can’t hold out much hope for predicting possible future effects of open-ended interventions on a natural complex system.

Why would it be so hard? Weren’t there countries at very similar levels of computer use, some of which spent many times more money on the problem than others? If so, can’t we see if the lower spending country fared worse?

Posted by: David Corfield on February 4, 2010 4:53 PM | Permalink | Reply to this

Re: Emotions in Science

There are many different aspects and I will certainly fail to do them all justice, but I’ll try:

It’s a fact that many software systems had the Y2K bug, and it’s a fact that some software systems would have behaved very badly if this bug were not fixed in time.

As an outsider it is most of the time not possible to estimate the costs of a bugfix, for various reasons: What impact on the whole software system will the change have? How many users (that includes other software systems) will be affected? What skill level do the programmers need to have? Are people capable to do this even available? (Side remark: programmers vary in skill and cost by a factor of 50, easily). Do we need an extra release or can we package the bugfix with one we already planned to do?

I have to participate in cost estimates all the time, and even for small scale systems with few users to be off by a factor of 20 or more happens (not too often, but nevertheless).

If we can’t get to the bottom of the effects of our interventions on a manmade system for a limited period in the past…

Maybe we could do it, but at what costs and with what precision? Maybe it could have been done if it had been monitored from the very beginning, in a similar fashion that the US government collects reports on the usage of the money spent for it’s latest economic stimulus act.

…I can’t hold out much hope for predicting possible future effects of open-ended interventions on a natural complex system.

That’s certainly warranted, but there seems to be one crucial difference between nature and software systems: Software systems are prone to the butterfly effect. One false character in one line of code can bring the whole system down. A backup won’t help, because it will suffer from the same error. Inspection by several software engineers won’t always help because humans tend to make similar mistakes - if two did not spot the error, the chance that ten do is marginal. The Y2K bug is a good example of this. If we are lucky then my prejudice that nature seems to have some magic self healing power that we don’t understand yet, and that software systems certainly do not have, is not totally wrong.

Posted by: Tim van Beek on February 4, 2010 5:47 PM | Permalink | Reply to this

Re: Emotions in Science

I think you mean companies rather than countries? (Most of the problems that lead to real-world effects, eg, loss of customers or failing contract terms, would be in a company environment rather than home users.)

I also think that you need to bear in mind that this is a system involving huge numbers of human beings whilst, if you assume that some chosen actions can be made to happen, climatology is overwhelmingly a system of physics and animals, which should be something where the envelope of predictions should be more tractable. (Of course, if you take the view – as I do that – that trying to get people’s behaviour to actually match the technically derived actions is non-trivial then you’re back to the same problem.)

Posted by: bane on February 4, 2010 6:00 PM | Permalink | Reply to this

Re: Emotions in car maintenance

Can you imagine someone bragging like: “Hey Jonny, just upgraded my engine software to v.3.1, let’s see who is faster now!”

Yes, definitely. I don't think that it's entered the public consciousness yet that this is something that one can say, but I expect that people will be saying it eventually.

If someone is killed by a car accident caused by a software bug, the company that builds these cars is dead (the rumor alone would suffice, I guess).

I don't buy this. The acceleration problems that have sparked Toyota’s current recall have killed four people (acknowledged by the company), and the company has been hit pretty hard, but I don't think that anybody's predicting its demise. Why would death due to a software bug cause a stronger reaction than death due to a hardware bug? (Incidentally, there are already some reports that the underlying cause here is electronic. This is still short of a rumour that the problem is a software bug, so we'll see how the company is doing when those start.)

Posted by: Toby Bartels on February 4, 2010 8:22 PM | Permalink | Reply to this

Re: Emotions in car maintenance

Tobey Bartels wrote:

I don’t think that it’s entered the public consciousness yet that this is something that one can say, but I expect that people will be saying it eventually.

Absolutly, that’s my point - it has not happened yet, but it will happen eventually.

The acceleration problems that have sparked Toyota’s current recall have killed four people …

Good point, I may have exaggerated - and I’m really surprised that Toyota seems to get off cheaply (compared to my expectations), but let’s see how the story unfolds.

Why would death due to a software bug cause a stronger reaction than death due to a hardware bug?

A hardware bug is less scary to most people than a software bug: They can understand it, touch it, see for themselves that it has been fixed, the feeling of control is stronger. People tend to accept greater risks if they have the feeling that they control the situation, if they can influence it actively. I heard the “software is so abstract, you can’t see it, you can’t touch it” complaint actually from someone who used to be responsible for hardware parts (as a kind of materials requirements planner) and switched to software.

Posted by: Tim van Beek on February 4, 2010 9:41 PM | Permalink | Reply to this

Emotions on duality

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We have seen discussion of Emotion in science and Emotion in car maintenance . Does anyone have on-topic emotions on quantum physics and logic, and duality in $n$ -categories?

I have the emotion of frustration that what used to be my favorite blog on fascinating topics in math, physics and philosophy is exhibiting as of late a curious pull away from all the interesting on-topics. There were times at which a message by John saying he would fly to a workshop to give a talk on dualities in $n$ -categories would have ignited a mighty discussion on dualities in $n$ -categories. Not about the wheather.

Why is that? We are experiencing amazing times for a blog that ought to be about $n$ -categories in math and phyics. It’s delightful what is going on in that direction currently. One would think here’d be a meeting place for people fascinated by these topics who would share their observations, questions and ideas. And from time to time some do, but I am afraid they are driven away when they see their RSS reader filled to the top with discussion about the wheather. I myself feel driven away by that! It’s sad that I have to go to MathOverflow to get a reaction on an $n$ -categorical question that receives at best silence on the $n$ -category Café. That shows me that something went wrong along the way.

That was my bit on emotion. Now to make up for this, something on-topic:

John says he might talk about dualities in $n$ -categories. The following question on this has come up before here, but never led to a discussion. It looks like a very interesting question to think about:

Question. One reason the original cobordism hypothesis was lacking a formalization was that it had been unclear how to say adjoint in an $n$ -category. It is clear how the pattern should start, but there was no and – as far as I can see – is no proposal for what coherently weakened adjunctions in an $n$ -category should be. (But I’d be more than glad to be pointed to work where this is done). So when Jacob Lurie’s formalization came out, the first two major question were:

how does he manage to define the $n$ -category $Bord_n$ ;
how does he manage to say “adjunction” and “duality” in an $(\infty,n)$ -categorical way.

The answer to the first question is very simple and very pleasing. (The simple idea is described here).

But surprisingly, his answer to the second question is also pretty simple: he just says an adjunction in an $(\infty,n)$ -category is an incoherent weak adjunction, i.e. one where the zig-zag-law holds up to some equivalence/homotopy, which is not required to satisfy any further conditions. I.e. an adjunction in the homotopy 2-category of the $(\infty,n)$ -category, (def 2.3.13 and the paragraph below that).

This is surprising, because a priori one would have thought that this is too simple! This goes against the grain that categorification needs coherent weakening to make good sense.

On the other hand, the result that is then proven using this definition certainly looks like things did go right. Even though also here there is room for wondering a bit: originally the idea was that the cobordisms hypothesis says that $Bord_n$ is the free $(\infty,n)$ -category with all dualy on a single object. But what Jacob Lurie actually shows, theorem 2.4.6 – mighty impressive and useful as it is! – is something much weaker, it seems: one would expect that a “free” something on the point is giving by acting with a left adjoint functor on the point. Lurie’s theorem shows that, yes, it seems that the evaluaton on the point of that free functor exists. But it doesn’t say what that functor would do away from the point. What’s the free $(\infty,n)$ -category with all duals on a nontrivial given $(\infty,n)$ -category?

Don’t get me wrong, I am not trying to find fault with the achievements here. I am just trying to understand why the somewhat surprisingly simple definition of duals in $n$ -categories used here still works.

It looks to me like this is a pretty interesting question concerning the topic of John’s talk, on dualities in higher categories.

P.S.

I am grateful to Eugenia Cheng and Nick Gurski for some chat about this in Sheffield the week before. Eugenia had posted the same kind of question here recently, but nobody reacted. So I thought I’d highlight it again here, given that this entry is about dualities in $n$ -categories.

Posted by: Urs Schreiber on February 5, 2010 1:32 PM | Permalink | Reply to this

Re: Emotions on duality

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Perhaps we’ll have to resort to colour-coded discussions to sort wheat from chaff, as Wittgenstein suggested should happen with Russell’s books when he strayed into ethics. Or take John’s more radical step.

Part of the trouble for me is that you’re heading rapidly out of sight now that you’ve absorbed Lurie, Toën, model categories, etc. It was a source of unexpected joy when I came across John’s work in the mid-90s and found I could understand something of contemporary maths. Even more amazing to me, from time to time we could have mathematical chats, such as on fundamental categories with duals. But somehow I doubt I’ll be able to keep up the pace with all this $(\infty, 1)$ -stuff.

Anyway, that conversation I mentioned started with an observation I made about swallowtails. Now I see lax 2-adjunctions has swallowtail identities. Is it to be expected that we’ll see all of singularity theory emerge from duality in n-categories?

Posted by: David Corfield on February 5, 2010 2:58 PM | Permalink | Reply to this

Re: duality and singularities

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David wrote:

Is it to be expected that we’ll see all of singularity theory emerge from duality in $n$ -categories?

Yes, though the classification of singularities involves continuous parameters when the singularities get complicated enough, and I don’t have any intuition as to how $n$ -categories will handle this.

Posted by: John Baez on February 5, 2010 4:34 PM | Permalink | Reply to this

Re: Emotions on duality

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I doubt I’ll be able to keep up the pace with all this (∞,1)-stuff.

If I may, I’ll bet, on the contrary, that if we actually start talking about it, you’ll easily follow.

All of math is trivial, the nice math in particular, if you just don’t let yourself be scared away by new terminology.

See, many mathematicians out there are, or used to be, happily admitting that they will never be able to understand category theory, because it is too abstract. You once, helped by John maybe, decdided to give it a try, and then – you understood it.

But one needs to talk about things. Once we start talking about it here the way we used to talk about, say doctrines or other fancy categorical stuff, every kid will be able to follows. There is nothing in the concept of, say, a derived $\infty$ -stack, that can’t be easily explained and understood. It’s not an esoteric science. The only thing is that it is maybe new and unfamiliar. There used to be times when people thought they would never understand complex numbers of differential forms or the like. Today we laugh about this. Because today we just happily talk about these things, and that makes them become trivial.

Should we start a thread: derived $\infty$ -stacks for the inclined philosopher of math (and everyone who wants to joint in) ?

Posted by: Urs Schreiber on February 5, 2010 5:04 PM | Permalink | Reply to this

Re: Emotions on duality

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It is certainly true that a lot of scary sounding things really aren’t so when you come to look at them (and if they are explained).

Mind you,

There used to be times when people thought they would never understand complex numbers of differential forms or the like. Today we laugh about this.

I was talking to some philosophy graduate students about the absurdity of this philosopher’s claim

Stated in realist terms, the extended number system [of the complex numbers – DC] is presumed in effect to stake out a ‘natural kind’ of reality. Far from ‘carving reality at the joints’, however, the system can be shown to feature a flagrantly gerrymandered fragment of heterogeneous reality that is hardly suited to enshrinement at the centre of a serious science like physics, not to mention a rigorous one like pure mathematics. Couched in these ultra-realist terms, the puzzle might be thought to be one that someone with more pragmatic leanings – the system works, doesn’t it? – need not fret over; and in fact such a one might even look forward to exploiting it to the discomfort of the realist. Fair enough. I should be happy to have my discussion of this Rube Goldberg contraption (as the extended number system pretty much turns out to be) serve as a contribution to the quarrel between anti-realist and realist that is being waged on a broad front today (Bernadete, Metaphysics: The Logical Approach 1989: 106).

Yes, that’s 1989 not 1889 or 1789! How we’ve moved on since Leibniz’s description of a complex number as “an amphibious monster living in an ideal world in between being and not being.”

I thought I’d gently introduce the complex numbers with that nice story about how the real solution to $x^3 = 15 x + 4$ required taking the cube root of an expression with a root of a negative number. Just believe in it and it works.

So I thought I’d work up to this with some earlier, Babylonian maths (c. 2000 BC). If 11 times the area of a field added to 7 times its length is 6 and 15 sixtieths, what is its length? Plenty of blank stares.

So let’s push back the bounds of ignorance and start derived $\infty$ -stacks for the inclined philosopher of math (and everyone who wants to joint in). Of course, ‘the’ is being used in the general sense, rather than to designate an individual, so all you lurking philosophers of math should chip in.

Posted by: David Corfield on February 5, 2010 6:08 PM | Permalink | Reply to this

Re: Emotions on duality

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So let’s push back the bounds of ignorance and start derived ∞-stacks for the inclined philosopher of math (and everyone who wants to joint in) . Of course, ‘the’ is being used in the general sense, rather than to designate an individual, so all you lurking philosophers of math should chip in.

All right! Thanks. That will be fun.

Let me just collect my senses, then I’ll post something…

While I am thinking, maybe you can give me a hint where you would like to be picked up by the Derived Mystery Tour Bus . One route the bus goes has as main stops

9:00 simplicial sets – 10:15 simplcially enriched categories – 11:35 simplicial model categories – 12:55 simplicial presheaves east – 13:35 simplicial presheaves downtown – 14:15 simplicial presheaves west – 16:00 $\infty$ -stacks – 17:00 derived $\infty$ -stacks – 18:2o examples and applications I: Lie $\infty$ -groupoids – 19:00 ex and app II: derived loop spaces – 20:15 ex anmd app III : BV-BRST quantizaton .

But then the bus returns and goes the other direction. We can also take that reverse trip.

There are more routes. Just let the company know where you are and we’ll tell you where we can pick you up.

Posted by: Urs Schreiber on February 5, 2010 7:26 PM | Permalink | Reply to this

Re: Emotions on duality

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Rereading the bus schedule I just wrote:

this may come across too unleisurely. That’s not how I mean it. We can walk the course by foot. We can start elsewhere and whatnot. This bus route was just the quick result of clicking on “find route” in my internal GoogleMaps. Just to get an overview.

Posted by: Urs Schreiber on February 5, 2010 7:51 PM | Permalink | Reply to this

Re: Emotions on duality

All of math is trivial, the nice math in particular, if you just don’t let yourself be scared away by new terminology.

Either you forgot to include “modulo details” or “modulo most of any paper written by Jean Bourgain”, or else I have failed to understand words like “math” and “nice” and “trivial”.

Or both.

Posted by: Yemon Choi on February 5, 2010 11:03 PM | Permalink | Reply to this

Re: Emotions on duality

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Hi everybody :). I agree with Urs, it’s a shame the conversations here have changed nature over the years. And I also agree with David, that $(\infty,n)$ -related topics are less immediately accessible than what we used to talk about here.

Urs: to respond to your point, I don’t understand why these questions need the added complexity of $\infty$ -category theory. Why must we ask what the free $(\infty,3)$ -category with all duals is, when the free 3-category with all duals is hard enough? I’m certainly interested in understanding these sorts of topics, and I’m slowly writing up a paper with Bruce Bartlett which might be related to a tiny corner of it.

Maybe I should make the leap to the $(\infty,n)$ way of thinking, and would be better off for it — but I’m not quite sure why I would be better off, and even if I wanted to make the leap, I’m not sure where I would start without investing a lot of time reading a lot of literature containing an awful lot of rather difficult topology!

Posted by: Jamie Vicary on February 5, 2010 3:57 PM | Permalink | Reply to this

Re: Emotions on duality

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And I also agree with David, that $(\infty,n)$ -related topics are less immediately accessible than what we used to talk about here.

They may be less familiar. But not less accesible. The “ $\infty$ “-sign in there is just supposed to indicate everything will be good now, relax and not huh, this will be scary . Don’t fall for making that mistake.

That said, I want to make the following point, concerning the topics that we used to talk about here:

several of the topics and questions that were discussed here over the years happened to have nice and good answers – but only in $(\infty,1)$ -category theory. (tangent categories, inner automorphism groups, quantization and cohomology, obstruction theory, generalized smooth geometry, covering spaces, etc, etc.)

That this wasn’t discussed back then is not something to be fond of, unfortunaly. Instead of thinking “oh how cosy was it”, we should be thinking “oh, how we missed to see the obvious”.

Posted by: Urs Schreiber on February 5, 2010 5:11 PM | Permalink | Reply to this

Re: Emotions on duality

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Should we have done Klein 2-geometry the $(\infty, 1)$ -way too?

Posted by: David Corfield on February 8, 2010 2:07 PM | Permalink | Reply to this

Re: Emotions on duality

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Should we have done Klein 2-geometry the (∞,1)-way too?

I might not remember the details of the discussion to a suitable extent. One aspect I do remember was that there was some discussion about how to say normal subgroup for higher 2-groups. Maybe what was missing from the discussion back then was the notion of regular epimorphism of higher groups and of their homotopy kernels? I forget what was said and what not.

So I am not sure if it helps, but I could remark that the notion of regular epimorphism and (of course) of its homotopy kernel makes sense for 2-groups just as well as for $\infty$ -groups.

At least there is this notion of (regular) $\infty$ -epimorphism. And a notion of regular $\infty$ -monomorphism. And of course the notion of homotopy kernel.

But I am not sure what to check this against. Did we have any motivating examples to check the formalization against?

What comes to mind is that generally one may look at morphisms of $\infty$ -groupoids

$\rho : \mathbf{B}H \to \mathbf{B}G$

and obtain a fibration sequence to the left

$G//_{\rho}H \to \mathbf{B}H \to \mathbf{B}G$

where $G//H$ is the action groupoid of $H$ acting on $G$ , the groupoid version of the bundle that is $\rho$ -associated to the universal $H$ -principal $\infty$ -bundle.

Maybe one wants to require $\rho$ to be a regular $\infty$ -monomorphism here, in order to speak of higher Klein geometry. (?)

Posted by: Urs Schreiber on February 8, 2010 6:18 PM | Permalink | Reply to this

Re: Emotions on duality

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Suddenly it all feels like an age ago. I guess this thread was the most recent activity. Mike summed up infinitely categorified Klein geometry as

Klein $(\infty, 1)$ -geometry is the homotopy theory of pairs of connected based spaces and fibrations between them – the corresponding homogeneous space then being the fiber of the fibration.

This is perhaps me being dim, but in $(\infty, -)$ -land presumably there’s no place for distinguishing between 1-groups, 2-groups, …

Posted by: David Corfield on February 9, 2010 9:19 PM | Permalink | Reply to this

Re: Emotions on duality

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David wrote:

Mike summed up infinitely categorified Klein geometry as:

Klein (∞,1)-geometry is the homotopy theory of pairs of connected based spaces and fibrations between them – the corresponding homogeneous space then being the fiber of the fibration.

This is perhaps me being dim, but in (∞,−)-land presumably there’s no place for distinguishing between 1-groups, 2-groups, …

Forgive me if you already know all this stuff:

A “connected based space” is just another name for “the classifying space $B G$ of some topological group $G$ ”. So, Klein $(\infty,1)$ -geometry as Mike summarized can be seen as a subtle way of studying topological groups.

By the homotopy hypothesis, a topological space is just another way of thinking about an $\infty$ -groupoid. Similarly, a topological group is just another way of thinking about an $\infty$ -group.

Starting from this, a topological group with $\pi_n$ trivial for $n \gt 0$ is just another way of thinking about a 1-group. A topological group with $\pi_n$ trivial for $n \gt 1$ is just another way of thinking about a 2-group. And so on. So, there is a way to distinguish between 1-groups, 2-groups and so on, if we think of them as special cases of $\infty$ -groups (or topological groups).

The place where things get a bit subtler is when you start trying to understand topological $n$ -groups.

I’ll also warn you that some things I just said are a bit oversimplified: for example, the maps between connected based spaces, say $B G \to B H$ , are more general than than homomorphisms $G \to H$ between topological groups. But they’re better! You can think of them as homomorphisms ‘up to coherent homotopy’.

Posted by: John Baez on February 10, 2010 7:13 AM | Permalink | Reply to this

Re: Emotions on duality

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A “connected based space” is just another name for “the classifying space $B G$ of some topological group G”.

Just beware that usually (or sometimes? might depend on who one talks to) “topological group” refers to strict topological groups, whereas this here is about weak $\infty$ -groups.

A quick and potentially insightful way to say it is that under the equivalence between $Top$ and $\infty Grpd$ a “connected based space” is a one-object $\infty$ -groupoid $\mathbf{B}G$ . And morphisms $\mathbf{B}H \to \mathbf{B}G$ are those of (weak! of course) $\infty$ -groupoids.

Posted by: Urs Schreiber on February 10, 2010 9:40 AM | Permalink | Reply to this

Re: Emotions on duality

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Urs wrote:

Just beware that usually (or sometimes? might depend on who one talks to) “topological group” refers to strict topological groups, whereas this here is about weak ∞-groups.

Right. Of course I was trying to discuss technicalities as little as possible, but I was hinting at a story like this. We can think of a weak $\infty$ -group as a Kan complex with a single vertex. By geometric realization, we can turn this into a connected pointed topological space. Then any such space is homotopy equivalent to $B G$ for an ordinary (strict) topological group.

And with more work we I bet we can show these three concepts give equivalent model categories, any of which deserves to be call the model category of weak $\infty$ -groups.

But of course for this to work, the morphisms in the category of topological groups are not the strict topological group homomorphisms. I believe my real sin lay in not emphasizing that. That’s why, to assuage my sense of guilt, I added this remark at the end:

I’ll also warn you that some things I just said are a bit oversimplified: for example, the maps between connected based spaces, say $B G \to B H$ , are more general than than homomorphisms $G \to H$ between topological groups. But they’re better! You can think of them as homomorphisms ‘up to coherent homotopy’.

Posted by: John Baez on February 11, 2010 7:19 AM | Permalink | Reply to this

Re: Emotions on duality

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But of course for this to work, the morphisms in the category of topological groups are not the strict topological group homomorphisms.

I don’t know about topological groups, but there is a model structure on the category of simplicial groups, with strict simplicial group homomorphisms, which is Quillen equivalent to a model structure on the category of simplicial sets with exactly one vertex. This is Prop. V.6.3 in Goerss&Jardine. And of course the geometric realization of a simplicial group or group homomorphism is a topological one.

Posted by: Mike Shulman on February 11, 2010 7:27 PM | Permalink | Reply to this

Re: Emotions on duality

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And with more work we I bet we can show these three concepts give equivalent model categories,

I think there are known model structures that do this, if you need a model structure. Can dig out the references for you if you like.

But I also think the construction of the $(\infty,1)$ -category of group objects $Grp(C)$ in an $(\infty,1)$ -catgeory $C$ is entirely $\infty$ -functorial and hence obviously respects equivalence of $(\infty,1)$ -categories, such as that of $Top_{cg,wH}$ and $\infty Grpd$ .

See for instance the def of $Grpd(C)$ with the condition in prop. 6.1.2.4 (4”) in HTT and then of $Grp(C)$ in prop 7.2.2.4. This all evidently respects equivalences, I would say.

Posted by: Urs Schreiber on February 11, 2010 9:14 AM | Permalink | Reply to this

Re: Emotions on duality

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Can dig out the references for you if you like.

Luckily Mike dug for me. I briefly archived it here:

Models for group objects in $\infty Grpd$

Posted by: Urs Schreiber on February 11, 2010 10:25 PM | Permalink | Reply to this

Galois theory

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I guess this thread was the most recent activity.

Oh, I see, so we said what I just said before. All the better.

Looking back, I notice one other thing: $\infty$ -Galois theory is mentioned here for instance:

when $A$ is connected, spaces with an action of $\pi_\infty(A)$ can be identified with fibrations over $A$ .

For whatever it’s worth, I’ll remark that this is just the $\infty$ -Grothendieck construction in action, it seems:

let me think of $A$ and everything else here as an $\infty$ -groupoid to make it more vivid, so that the connected $A$ is identified with $\mathbf{B} \pi_\infty(A)$ , then this says that $\infty$ -functors

$\mathbf{B}\pi_\infty(A) \to \infty Grpd$

are equivalent to fibered $(\infty,1)$ -groupoids

$X//\pi_\infty(A) \to \mathbf{B}\pi_\infty(A)$

sitting in a fibration sequence

$X \to X//_{\pi_\infty(A)} \to \mathbf{B}\pi_\infty(A) \to \infty Grpd$

identifying the second term from the left as the action groupoid of $\pi_\infty(A)$ acting on $X$ .

By the way, over on the $n$ Lab we are currently talking about the $\infty$ -Galois theory of objects in an $(\infty,1)$ -topos more general than $Top$ . See homotopy group of an $\infty$ -stack.

I’d be interested in comments. Especially on the formulation of the Idea that the bare fundamental $\infty$ -groupoid of an $\infty$ -stack is the image under the left adjoint to the functor that assigns constant $\infty$ -stacks of locally constant $\infty$ -stacks with values in a given $\infty$ -groupoids.

From this the notion of $\infty$ -monodromy and Galois theory seems to follow from a simple argument of $\infty$ -Tannaka duality, hence just an application of the $(\infty,1)$ -Yoneda lemma.

As the entry discusses in detail, there is a result by Toën and similar reslts by Polesello and Waschkies and by Mike Shulman and Denis-Charles Cisinski and Jacob Lurie on $\infty$ -Galois theory, that with a tiny little bit tweaking support the point of view that $\Pi(-) : \infty Stacks \to \infty Grpds$ is to be thought of as the left adjoint to $LConst : \infty Grpd \to \infty Stacks$ .

P.S. I am grateful to Richard Williamson for alerting me of the Toën and the Cisinski-reference and of their relevance, and then to Mike Shulman for pointing me to page and verse in Johnstone’s Topos theory for the $\pi_1$ -case and to the whole $n$ Lab team for Grothendieck’s Galois theory and then notably to David Roberts for pointing out the Polesello-Waschkies reference and finally to Zoran Škoda for highlighting that Lurie also has a result of this kind (as discussed at the entry). While Polesello-Waschkies work explicitly only up to 2-stacks, they present the abstract picture, as I see it now, in the clearest fashion. Notably their reference is the one that notices expliccitly that locally constant $(n-1)$ -stacks are precisely the sections of the constant $n$ -stack constant on $(n-1)Grpd$ , so that for $X$ an object in an $(\infty,1)$ -topos $\mathbf{H}$

$LConst(X) := \mathbf{H}(X, LConst_{\infty Grpd})$

may be thought of as the $\infty$ -groupoid of locally constant $\infty$ -stacks on $X$ . This nice reformulation then makes immediate that we are searching a left adjoint $\Pi(-)$ to $LConst : \infty Grpd \to \mathbf{H}$ in order to write

$\cdots \simeq \infty Func(\Pi(X), \infty Grpd) \,.$

This way it all becomes a bit of abstract nonsense, which however can be checked to be consistent with what one wants it to be consistent. Notably the references listed above demonstrate, slightly implicitly maybe, that when $\mathbf{H} = Sh_{(\infty,1)}(Top)$ and $X$ is a representable, i.e. a topological space, then this abstract nonsense does reproduce the right results, in that it does reproduce the ordinary fundamental $\infty$ -groupoid and homotopy groups of $X$ .

Posted by: Urs Schreiber on February 10, 2010 9:35 AM | Permalink | Reply to this

Re: Galois theory

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We certainly reached an interesting point, but I guess I never felt I’d seen the answer to my original question which was:

Klein recasts familiar geometries as pairs consisting of a Lie group, $G$ , and a (closed) subgroup, $H$ , so that the geometric space is the coset space $G/H$ , upon which $G$ acts with $H$ as the stabilizer of a point, e.g., $G$ is Euclidean transformations, and $H$ rotations about a point, then $G/H$ is the Euclidean plane. Then you could look for other subgroups of $G$ and see which figures they fix, and then consider incidence relations via double cosets. So what’s the story for Lie 2-groups?

Posted by: David Corfield on February 10, 2010 10:54 AM | Permalink | Reply to this

Re: Galois theory

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but I guess I never felt I’d seen the answer to my original question

Yes, so that’s why I wondered if we had any motivating examples to check the formalism against.

What we have now is a good guess what $\infty$ -Klein theory should be, as a formalism. What is needed next is some candidate objects from the literature that we could point to and say: see, this may be thought of as an example of a Klein 3-geometry.

There must be many such examples. Quotients $G//H$ and their higher analogs play a central role in geometric representation theory, of course. In the derived way that this is done these days, these quotients will be higher Klein geometries.

But I can’t point to a specific article just this second. Have to run…

Posted by: Urs Schreiber on February 10, 2010 12:20 PM | Permalink | Reply to this

Re: Galois theory

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No, we never had motivating examples, and that was no doubt the major problem all along. From a different perspective though, it always seemed to me that there would be something reasonably easy to find once we had set up the right way of thinking of Lie sub-2-groups. Lie 2-groups are ten a penny, so find a sub-2-group and turn the handle. John spoke of it being as ‘easy as finding sand in the Sahara’.

Perhaps another major issue is that in Klein geometry one tends to begin with rather special Lie groups, i.e., ones for which there are subgroups which may be construed as fixing points, lines, etc. I believe I asked of John once how one knew just looking at a Lie group that it had this geometric flavour, and I think he mentioned Dynkin diagrams. Is this something to do with maximal parabolic subgroups?

So if I show you a Lie 2-group what might give you the expectation of categorified points and lines being present? Was there a categorified Dynkin story?

I think I used to hope that categorified points would turn out to be points with internal symmetries.

A final thing, about this counting business $(\infty, 1), (\infty, 2),...$ , once we have gone to $(\infty, 1)$ -geometry, is it that there’s no need to go to $(\infty, 2)$ -geometry as we’re only dealing with invertible things in the first place?

Posted by: David Corfield on February 10, 2010 1:10 PM | Permalink | Reply to this

Re: Galois theory

Is anyone else having problems posting to the n-café? I have twice typed in longish messages and then on trying to preview I get an error.

Posted by: Tim Porter on February 10, 2010 3:05 PM | Permalink | Reply to this

Re: Galois theory

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…if I show you a Lie 2-group…

Ah, and by the way, just for the record:

the definition of $\infty$ -Klein geometries as homotopy fibers of (regular mono?)morphisms of the form $\mathbf{B}H \to \mathbf{B}G$ of course makes sense in every $(\infty,1)$ -topos, hence in particular for Lie $\infty$ -groups $H$ and $G$ .

A final thing, about this counting business (∞,1),(∞,2),…, once we have gone to (∞,1)-geometry, is it that there’s no need to go to (∞,2)-geometry as we’re only dealing with invertible things in the first place?

At least the passage to the non-invertible case would be a whole different generalization all by itself. Already before we categorify: you are asking effectively about the situation where we have $A \hookrightarrow B$ two (Lie-) monoids with qotient $B/A$ and the generalization of that to $\infty$ -monoids. By itself I see no reason why one couldn’t look into this as a model for generalized geometries, but the question of motivating examples does not seem to get better (which may of course still be more a matter of our ignorance than of principle),

Also the number of choices to be made in setting up the definitions then increases, for instance for $\mathbf{B}A \to \mathbf{B}B$ given, what do we want to take as its fiber? It should be a weak pullback

$\array{ B//A &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}A &\to& \mathbf{B}B }$

but for $\mathbf{B}A$ etc $(\infty,1)$ -categories there are several notions of that: homotopy pullback, lax pullback, oplax pullback, comma pullback. The number of choices increases as $n$ in $(\infty,n)$ increases.

So I think before putting much energy into this, let’s put energy into getting good examples in our hands. For instance, maybe we should start looking at the geometry induced by

$\mathbf{B}String(n) \to \mathbf{B} \hat {ISO}(n)$

i.e. the quotient of the cover of the Euclidean group not just by the Spin group but by the String-2-group.

I haven’t thought about this. Just someting that comes to mind.

Posted by: Urs Schreiber on February 10, 2010 2:39 PM | Permalink | Reply to this

Re: Galois theory

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I have been looking at two situations (for use with HQFTs) that may provide some intuition on Klein 2-geometries.

(i) is the classical $BSpin(n)\to BSO(n)$ relating to spin structure.

(ii) is the $BPL\to BTop$ map that comes into the triangulations and smoothings stuff and the Hauptvermutung problem (Kirby Siebenmann et al) from the 1970s.

Both are possible examples for motivation, but I do not understand how to unpack something like that exactly to get something like a 2-geometry, (or perhaps I misunderstand 2-geometries).

Example (ii) is very rich as there are variants about smooth structure overlying a given topological one and so on. the homotopy fibre has been intensively studied and has a lot of geometric significance I believe. (I’m no expert, more an interested bystander.)

I will add this to Urs’s latest. He mentions String(n) which is one step further than (i)

Posted by: Tim Porter on February 10, 2010 3:18 PM | Permalink | Reply to this

Re: Galois theory

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Is the cover of $ISO(n)$ a semidirect product of $Spin(n)$ with the translation group, perhaps to be called $ISpin(n)$ ?

You’d expect the base space of the quotient to be Euclidean space. How does the fiber reflect the co-killing of the third homotopy group?

And what structure should the whole thing inherit? It ought to know it’s coming from Lie 2-groups.

Posted by: David Corfield on February 11, 2010 9:36 AM | Permalink | Reply to this

Re: Galois theory

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You’d expect the base space of the quotient to be Euclidean space. How does the fiber reflect the co-killing of the third homotopy group?

The homotopy fiber of

$\mathbf{B}String(n) \to \mathbf{B}ISO(n)$

is (equivalent to) the strict Lie 2-groupoid $X$ with

$X_0 = ISO(n)$ ;
$X_1 = ISO(n) \times P_* Spin(n)$
$X_2 = ISO(n) \times P_* Spin(n) \times \hat \Omega_* Spin(n)$

where the source $X_1 \to X_0$ is projection on the first factor, the coresponding target is endpoint evaluation followed by taking product and where the source $X_2 \to X_1$ is projection on the first two factors and the corresponding target is the identity on the first factor and the action of loops on paths in the second two.

Compositions are the evident ones.

Now to better understand this you want to squash this down to something equivalent but smaller. I won’t try to make this fully precise just now, but essentially what happens is that the $P_* Spin(n)$ -part in $X_1$ quotients out the $SO(n)$ -part in $ISO(n)$ leaving just $\mathbb{R}^n$ in degree 0, as in the usual case.

But then some $\pi_2$ -stuff will be left. At least locally the thing will look like $\mathbb{R}^n \times\mathbf{B}^2 U(1)$ , but probably not globally. I need to think about this in more leisure than I have right now. Am supposed to be doing something else! :-)

Posted by: Urs Schreiber on February 11, 2010 10:12 AM | Permalink | Reply to this

Re: Galois theory

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Looks promising! I wonder if there will be a story in the Lie 2-group case for when you consider you’ve got hold of the basic geometric shapes.

John wrote once about complex simple Lie groups:

We say a subgroup $P$ of $G$ is “parabolic” if it contains some Borel subgroup $B$ . We say $G/P$ is the corresponding space of “flags”. The smallest parabolic subgroup is $B$ itself, and $G/B$ is the space of “maximal flags”. But we’re really interested in the “maximal” parabolic subgroups: the biggest possible ones apart from G itself. If $P$ is maximal parabolic, $G/P$ will be a space of minimal flags. These minimal flags are the “fundamental” types of figure, from which fancier ones can be built.

I won’t explain it here, but it turns out that after fixing a Borel subgroup of $G$ , you get parabolics from subsets of dots in the Dynkin diagram of $G$ . The dots themselves correspond to maximal parabolics, and these give fundamental types of figure in an incidence geometry. Similarly, the edges give fundamental incidence relations!

This apparently ties in with some representation theory:

…the parabolic subgroups are precisely those subgroups such that $G/P$ is compact. In fact, all compact simply-connected Kähler manifolds with a transitive action of $G$ are of this form. So, they’re really just another way of talking about the “coadjoint orbits” of the compact real form of $G$ . You can apply geometric quantization to these manifolds to get all the unitary irreducible representations of the compact real form of $G$ ; the maximal parabolics give the so-called “fundamental representations”, which generate the representation ring.

But then again perhaps 2-geometry will just be about points, lines, … with different kinds of internal symmetry.

Posted by: David Corfield on February 11, 2010 10:49 AM | Permalink | Reply to this

Re: Galois theory

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In a comment above I indicated a way to define geometric homotopy groups of objects in an $\infty$ -stack $(\infty,1)$ -topos.

Since maybe this discussion shouldn’t fill up the thread here but eventually get its own post, I’ll just briefly note that it turns out that what I said in terms of $\infty$ -stack language happens to implicitly be known.

Up to the evident re-identification of concepts and terminology, the definition of the fundamental bare $\infty$ -groupoid $\Pi(X) \in \infty Grpd$ of an $\infty$ -stack $X \in Sh_{(\infty,1)}(C)$ is already there on p. 18 of

Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture notes in math 161, Springer (1995).

That’s all I am gonna say in this thread here about this. Further discussion will be here on the $n$ -forum for the time being.

Posted by: Urs Schreiber on February 10, 2010 3:28 PM | Permalink | Reply to this

Re: Galois theory

I think that was Lecture notes in math 1616?

Posted by: Tim Porter on February 10, 2010 3:43 PM | Permalink | Reply to this

Re: Galois theory

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I think that was Lecture notes in math 1616?

Yes, thanks.

Posted by: Urs Schreiber on February 10, 2010 4:30 PM | Permalink | Reply to this

Re: Emotions on duality

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I don’t understand why these questions need the added complexity of ∞-category theory. Why must we ask what the free (∞,3)-category with all duals is, when the free 3-category with all duals is hard enough?

But it’s the other way round!: $n$ -categories are hard to handle. $(\infty,n)$ -categories are comparatively easy to handle!

think about it: you can barely write down even the definition of a tricategory, let alone do anything nontrivial with it. But there are now half a dozen or so definitions of $(\infty,n)$ -category that fit neatly on half a page and that one can actually use for something. For higher category theory, notably!

it is no coincidence that nobody managed to write down the $n$ -category of $n$ -cobordisms. Think about this. It is not true that this would be easier: many years people tried. Then when the solution came, it was a 5-line definition in terms of $(\infty,n)$ -categories. 5-lines! Have a look, if you don’t believe it, it is very simple.

This is the general pattern: everything becomes easier and is beginning to make sense in category theory if we first let all the equivalences run to $\infty$ . Everything that is hard in higher category theory is higher coherences. These are all about equivalences hence $\infty$ -groupoids. Once these are under control, everyting else follows easily. That’s why it is much much easier to go $0, 1, (\infty,1), (\infty,2), (\infty,3), \cdots$ than to go $1, 2, 3, 4, 5, , \cdots$

Posted by: Urs Schreiber on February 5, 2010 5:28 PM | Permalink | Reply to this

Re: Emotions on duality

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Urs, thanks for that comment. I’ve had a look at that definition, and those five lines require more knowledge than I have to understand without an awful lot of time and head-scratching.

I’m not saying I’m not prepared to take the time, or do the head-scratching — but if it’s so important, maybe somebody in the know could write a really accessible, step-by-step guide appropriate for people like me who aren’t so hot on their topology?

But there’s something I’m still confused about. Urs, your earlier comment expressed surprise about the lack of ‘interesting’ algebraic conditions arising out of the $(\infty,n)$ approach to $\mathrm{Bord}_n$ . So, I presume that Lurie’s definition doesn’t let us calculate these. Let’s take $\mathrm{Bord}_2$ , the only one which is well-understood in a conventional sense; hidden away in this, we would expect to find all the axioms for a commutative Frobenius algebra. Are you saying that these aren’t present in Lurie’s $(\infty,n)$ version of $\mathrm{Bord}_2$ ?

Posted by: Jamie Vicary on February 5, 2010 6:42 PM | Permalink | Reply to this

Re: Emotions on duality

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$0, 1, (\infty, 1), (\infty, 2), (\infty, 3),...$

seems a funny way to count. Why not

$(\infty, 0), (\infty, 1), (\infty, 2), (\infty, 3),... ?$

Posted by: David Corfield on February 8, 2010 2:22 PM | Permalink | Reply to this

Re: Emotions on duality

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seems a funny way to count.

Yes you are right. I noticed after I had hit “submit” that this was maybe not the most evident counting.

Posted by: Urs Schreiber on February 8, 2010 5:33 PM | Permalink | Reply to this

Re: Emotions on duality

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Would you agree with the following reformulation of the $(\infty,-)$ phenomenon?: to make progress in higher category theory, we needed to realize what the homotopy theorists, algebraic geometers, and type theorists (to name just a few) have known for a long time: that when instead of just a set of things, nature gives you a moduli space of things, you should remember and work with that whole space rather than squashing it down to a set.

Posted by: Mike Shulman on February 8, 2010 10:35 PM | Permalink | Reply to this

Re: Emotions on duality

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I’m not sure where I would start without investing a lot of time reading a lot of literature containing an awful lot of rather difficult topology!

Okay, I see. I’ll tell you.

First, you don’t mean topology, you mean homotopy theory .

(Which happened to be first discovered by topologists. But those who worked on it then were renamed homotopy theorists .)

After a little more development, it turns out that homotopy theory is precisely the higher category theory of $(\infty,1)$ -categories.

So if you feel you are a category theorists, or in parts a category theorist, then you shouldn’t be thinking: “Ah, why do I also have to study homotopy theory.” You should think: “Homotopy theory is the historical name for a toolset for understanding $(\infty,1)$ -category theory, the most fundamental part of higher category theory. I should have learned this in kindergarten! Unfortunately they didn’t tell me about it. So now I should learn it as soon as possible.”

Today you have the luck that you can entirely turn this around: you can study pure higher category theory and at the end look back and notice that as you did, you have implicitly learned everything people call homotopy theory, too.

The best way to go about the fact that there is a bit to be learned here is maybe not to think: “Oh, how long it will take me to learn this.” If you are a category theorists, in parts, then the attitude is: “Every minute that passes in which I don’t know this stuff is a minute spent as an incomplete category theorist. I should drop everything that I am doing and on the spot open Joyal’s lecture notes.”

André Joyal is in general the figure to follow here, especially if your heart is more categorical than homotopical. And lucky as we all are, right now he is in the process of beginning to write a presentation of $(\infty,1)$ -category theory – on the $n$ Lab. You can look over his shoulder while he is exposing it all, at Joyal’s CatLab.

He is asking for comments there. So have a look and see what happens. Maybe you have the unique historical chance to converse with one of the greatest figures of category theory as he writes the fundamental exposition of it.

Posted by: Urs Schreiber on February 5, 2010 6:48 PM | Permalink | Reply to this

Re: Emotions on duality

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Thanks Urs, I’m very grateful for that useful and excellent perspective, that’s just the sort of guidance I’ve been looking for. Those notes of Joyal look like a great resource.

I am still confused about something that I asked about earlier. You suggested here that Lurie’s technology doesn’t generate increasingly rich notions of adjunction for functors between $(\infty,n)$ -categories as $n$ gets larger, but rather relies heavily on the standard notion of adjunction between categories. I’m sure this is useful perspective, but it seems to clash loudly with other strong intuitions.

For example, we already have a good idea of what the correct higher coherences are for adjunctions between 2-categories — is this information now useless in the $(\infty,n)$ -world? We could use these higher coherences to construct the free fully-dualizable 3-category on a 3-category with one object and one nontrivial 1-cell — do we not expect the result to have something to do with $\mathrm{Cob}_3$ ? Because, since it uses these higher coherences, it would surely be an irreconcilably different beast to the free fully-dualizable $(\infty,3)$ -category with one object and one nontrivial 1-cell.

Posted by: Jamie Vicary on February 8, 2010 12:15 PM | Permalink | Reply to this

Re: Emotions on duality

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You suggested here that Lurie’s technology doesn’t generate increasingly rich notions of adjunction for functors between (∞,n)-categories as n gets larger, but rather relies heavily on the standard notion of adjunction between categories.

Yes, to be precise: in his work on the cobordism hypothesis he defines – unless I am dreaming – an $(\infty,n)$ -category to have “all adjoints” if each $k$ -morphism has an adjoint up to incoherent homotopy.

I might well be missing something here, I am not really concentrating on the cobordism stuff as much but rather follow it with half an eye. I was originally thinking that possibly in the given context, if we have adjoints for all k-morphisms for all k below some degree, that’s as good as having coherently weakened adjunctions. Or something. I might be saying nonsense. Don’t really have the leisure to concentrate on that aspect at the moment. I hope you and Bruce and others who professionally think about this will look into this and explain it to me! :-)

For example, we already have a good idea of what the correct higher coherences are for adjunctions between 2-categories — is this information now useless in the (∞,n)-world?

No, wait, in the fully developed (or fully exposed) part of the work there is a notion of adjoint $\infty$ -functor between $(\infty,1)$ -categories that has all the expected properties of an adjunction, hence is a coherently weakened $\infty$ -adjunction. That’s not quite the notion that appears in the Cobordism Hypothesis proof, though.

We could use these higher coherences to construct the free fully-dualizable 3-category on a 3-category with one object and one nontrivial 1-cell — do we not expect the result to have something to do with Cob 3? Because, since it uses these higher coherences, it would surely be an irreconcilably different beast to the free fully-dualizable (∞,3)-category with one object and one nontrivial 1-cell.

Right, so that was basically the question that I meant to raise in that comment that you linked to. I don’t really know the answer. I know that Eugenia Cheng and Nick Gurski started looking into this, though. Maybe meanwhile they know more.

Posted by: Urs Schreiber on February 8, 2010 1:31 PM | Permalink | Reply to this

Re: Emotions on duality

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Hi Urs and Jamie, yes I was also bothered by Lurie’s too-simple-looking definition of an $(\infty, n)$ -category with duals. But it seems Eugenia has looked a bit more closely at the definition now and perhaps it does do the job, though she is asking if anyone sees the coherence conditions (anyone?).

Mike wrote:

When Eugenia was here in Chicago the week before last, we thought about this some, and eventually convinced ourselves that any incoherent 2-adjunction can be rectified into a coherent one! This surprised both of us, and we wondered whether any of the Australians had noticed it before.

I also ran into that doozer at one point when I was writing my thesis. I engaged Eugenia and Nick Gurski, and Nick sketched out a proof, though I couldn’t quite understand it because it used a certain way of thinking about higher categories which I haven’t mastered. I needed it because I needed the precise sequence of moves which would unbraid a bradied pair of pants! That is, I needed to show:

$pic of braided pair of pants$

Basically, what I mean is, if one draws out all the gismos involved in a 2-adjunction, then the coherence laws become equalities between “movies”. (I hope I’m not completely embarasssing myself here by confusing what Mike and Eugenia were talking about with something else. Perhaps I’m thinking at the wrong categorical level, terror!).

This is where I asked Scott Carter for help, as he seems to be the kind of these movie moves in higher-categories. He pointed out an important coherence equation, the “horizontal cusp” move:

$horizontal cusp$

Then he showed me painstakingly (I just couldn’t get it, he needed to hold my hand through it) how to untwist the pair of pants using these moves. It’s a sequence of 15 equations, involving two applications of the “swallowtail” move and two applications of “horizontal cusp”. Here is a sample of the first four equalities:

$First four equalities$

The full sequence is in Appendix C of my thesis. I know that Chris Schommer-Pries has also thought about this, because this problem of “unbraiding the pair of pants” using the coherence laws (except now the coherence laws are in his cobordism higher category) is precisely slide 52 of this talk about his thesis work he gave last year!

We should ask Chris to give us his thoughts on Lurie’s definition of an $(\infty, n)$ -category with duals.

Posted by: Bruce Bartlett on February 11, 2010 4:31 PM | Permalink | Reply to this

Re: Emotions on duality

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though she is asking if anyone sees the coherence conditions (anyone?).

This is exactly what I meant by

any incoherent 2-adjunction can be rectified into a coherent one!

The coherence conditions are not there in the definition, but if the given “incoherent” definition is satisfied, you can always make a choice (some particular choice, not just any choice) of the higher cells which do satisfy the coherence conditions.

I don’t really understand what this has to do with unbraiding a braided pair of pants, maybe you can edify me?

Posted by: Mike Shulman on February 11, 2010 7:29 PM | Permalink | Reply to this

Re: Emotions on duality

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The coherence conditions are not there in the definition, but if the given incoherent definition is satisfied, you can always make a choice (some particular choice, not just any choice) of the higher cells which do satisfy the coherence conditions.

Okay, thanks. But it seems a bit weird, as if there isn’t any information in the particular coherent choice made, whereas there is usually information in that stuff. Like in the associators of a monoidal category: there are different inequivalent choices one could make.

For the unbraiding of a pair of pants, I mean this as follows (sorry for this looong post). One can draw out the data of what one means by a ‘coherent adjoint equivalence’ between two 2-categories $A$ and $B$ in 3d string diagrams. For instance, you start with a pair of weak 2-functors $F: A \rightarrow B$ and $G: B \rightarrow A$ , which one might draw in 3d as:

$pic$

So we’re thinking of the objects $A$ and $B$ as being 3d cubes, and the morphisms $F$ and $G$ as being surfaces between these cubes. My composition of 1-morphisms is going from right to left, (sorry about this convention if you don’t like it!)

Then you’ll need transformation 2-morphisms $\eta : id_A \Rightarrow GF$ , $\eta^* \colon GF \Rightarrow id_A$ , $\epsilon \colon FG \Rightarrow id_B$ and $\epsilon^* \colon id_B \Rightarrow FG$ . (By the way, $\eta^*$ doesn’t mean the adjoint or anything, it’s just some other 2-morphism which I’m calling $\eta^*$ for want of a better notation). We’ll draw these as follows:

$pic$

I’m drawing 2-morphisms from front to back, into the paper. Is it clear? For instance, $eta$ goes from the identity (i.e. “nothing”) to the composite of $G$ after $F$ .

Okay, now usually we would say that the $\eta$ ’s and $\epsilon$ ’s must satisfy the triangle (i.e. “snake”) equation. But now we’re working one categorical level up. So they’ll need to satisfy the triangle equation up to the “triangulators”!. In other words, we’ll need some 3-morphisms, namely we’ll need invertible modifications which I’m calling $\Delta_1, \Delta_1', \Delta_2$ and $\Delta_2'$ which go from the various identity 2-morphisms to the various snake 2-morphisms composed out of the $\eta$ and $\epsilon$ ’s:

$pic$

To be clear: these are pictures of 3-morphisms. In my pictures, a 3-morphism runs from top to bottom (down the screen).

Okay, now because we want this to be an “adjoint equivalence” we’re going to want those transformation 2-morphisms $\eta$ and $\epsilon$ to be “invertible”. Which means we’re going to need invertible 3-morphisms (modifications) $b \colon id_{id_A} \rightarrow \eta' \eta$ (for ‘birth of a circle’), $s \colon \eta \eta' \rightarrow id_{GF}$ (for ‘saddle’), and starred versions, drawn as:

$pic$

Okay, that’s the data. Now for the coherence equations. Firstly, we’ll need the inverse laws for the modifications (because we said they were “invertible”). Then we’ll need the swallowtail rule, which couples the $\eta$ ’s and the $\Delta$ ’s together. We’re going to draw it as an equality between movies (because I can’t visualize properly in 3d!):

$pic$

Let’s make this movie thing clear. Each frame in the movie is a certain 2-morphism, and going between each frame is a 3-morphism. So the movie as a whole stands for a certain composite of 3-morphisms.

For instance, the top-left-hand frame above is the 2-morphism $\eta$ . The right leg of that $\eta$ is $id_F$ , the identity 2-morphism on the 1-morphism $F$ . And $\Delta_1^{-1}$ is a 3-morphism which goes from the 2-morphism $id_F$ to the snake 2-morphism. So between frames 1 and frames 2 in the left hand movie there is a 3-morphism, which is the identity on the other “stuff” and $\Delta_1^{-1}$ on that right leg.

The movies are nicer to work with because you don’t have to imagine the full surface in 3d, you just watch how slices of it evolve.

The other coherence equations we’ll need is the horizontal cusp rules:

$pic$

(You also need the 3 other variants of this equation).

Anyhow, we want to check if we can use these coherence laws to unbraid the pair of pants!In terms of 3d surfaces, we want to prove the following:

$pic$

Sorry about those “S” and “T” dots, they were a technical requirement for what I was doing but aren’t needed here! So ignore them :-(. Anyhow, in terms of movies, we want to show that we can use the coherence equations above to prove that the following two movies are equal:

$pic$ $pic$

Those moves which make the circles “go around” correspond to the interchange 3-morphisms in the 3-category 2Cat. But in 2Cat, the interchange 3-morphisms are identities. So you can freely move things round like that. In a general 3-category, presumably you could still do this (this would presumably be the “coherence theorem” for tricategories).

Anyhow, that’s the summary of this unbraiding the pair-of-pants thing. The sequence of steps can be found at this link (pg 230).

As I say, the interesting thing is that these data-and-coherence laws for making an adjoint equivalence into a coherent adjoint equivalence seem to be basically the same generators-and-relations which Chris Schommer-Pries used to build his 3-category of cobordisms. Which isn’t weird or anything, that’s what HDA0 said was going to happen. But it does underline that the coherence laws are topologically important somehow.

Posted by: Bruce Bartlett on February 12, 2010 2:59 PM | Permalink | Reply to this

Re: Emotions on duality

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But it seems a bit weird, as if there isn’t any information in the particular coherent choice made, whereas there is usually information in that stuff.

Interesting point. I think the key is that you always have to be careful with what you regard as structure and what you regard as property. For instance, in a 2-category:

Given a single morphism $f$ , saying that it is an equivalence is a property, since the category of inverse adjoint equivalences to f is contractible.
Given a pair of morphisms $f$ and $g$ which are inverse equivalences, specifying a particular η and ε making them into an adjoint equivalence is structure: you can do it in multiple inequivalent ways.
But given a pair of morphisms $f$ and $g$ which are inverse equivalences and a particular isomorphism $\eta\colon 1\cong g f$ , there exists a unique isomorphism $\varepsilon\colon f g \cong 1$ satisfying one, and hence also the other, triangle identity.

An analogue for biequivalences in a Gray-category can be found as Prop. 3.1 in this paper. Similarly, given a morphism $f$ , saying that it has a right adjoint is a property, whereas saying that a particular morphism $g$ is the adjoint of $f$ is a structure. (In particular, saying that an endomorphism $f$ is its own adjoint is a structure, rather than a property!) If we just know that $f$ has a right adjoint, then the category of all its right adjoints is contractible.

I think the situation for ( $(\infty,n)$ -)categories with duals should be analogous: since we are just asserting duals to exist, rather than choosing them as part of the structure, there is no data carried in the higher morphisms, because the space of choices is contractible. But if you start adding extra conditions/structure, such as requiring morphisms to be their own adjoints, then you get the more general sort of “ $G$ -structured duality” which is a structure, not just a property.

Posted by: Mike Shulman on February 15, 2010 8:31 PM | Permalink | PGP Sig | Reply to this

Re: Emotions on duality

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Oh, and thanks for the detailed explanation of the unbraiding. But I still didn’t quite follow what unbraiding the pair of pants, assuming a coherent adjunction, has to do with the question of coherentifying an incoherent adjunction.

Posted by: Mike Shulman on February 15, 2010 8:57 PM | Permalink | Reply to this

Re: Emotions on duality

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Thanks for the reference to Prop 3.1 in Steven Lack’s paper; I didn’t know about that. Nice.

But I still didn’t quite follow what unbraiding the pair of pants, assuming a coherent adjunction, has to do with the question of coherentifying an incoherent adjunction.

You’re right, it doesn’t have anything to do with coherentification. I was just pointing out that the coherent versions are important, you actually need them to perform topological computations.

Posted by: Bruce Bartlett on February 16, 2010 1:11 PM | Permalink | Reply to this

Re: Emotions on duality

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Urs wrote:

This is the general pattern: everything becomes easier and is beginning to make sense in category theory if we first let all the equivalences run to $\infty$ .

And it’s worth pointing out that this was to be expected from the general philosophy behind higher categories. Equations are bad, equivalences are good.

In the definition of $n$ -category we try to follow this philosophy up to the $n$ th stage. But then we abruptly quit: we talk about $n$ -morphisms being equal or not, instead of going on and talking about equivalences between them. So, it’s like being ethical and moral until you get tired of it.

In the definition of $(\infty,n)$ -category, we really follow the philosophy whole-heartedly. So, perhaps it’s not surprising that $(\infty,n)$ -categories actually work better and more simply.

This is especially true when it comes to duality. The really tricky part about duality is defining duals for $n$ -morphisms in an $n$ -category. People who read the category theory mailing list will remember the big discussion about $\dagger$ -categories being ‘evil’. Here we saw that defining duals for 1-morphisms in a 1-category has the nasty side-effect of imposing equations between objects!

I only know two principled ways out of this dilemma. We talked about one on the category theory mailing list. This is to cease thinking of $\dagger$ -categories as ‘categories with extra stuff’, and instead think of them as a wholly new kind of thing. Another is to work with $(\infty,1)$ -categories instead of categories.

Posted by: John Baez on February 5, 2010 6:51 PM | Permalink | Reply to this

Re: Emotions on duality

cease thinking of †-categories as ‘categories with extra stuff’, and instead think of them as a wholly new kind of thing

I keep meaning to write about this at [[evil]], but I haven't gotten around to it. But I want to note for the record: while it doesn't work well to define †-categories as categories with extra stuff, it works just fine to define them as groupoids with extra stuff. Note that the underyling groupoid here is the groupoid of unitary isomorphisms.

Posted by: Toby Bartels on February 5, 2010 7:46 PM | Permalink | Reply to this

Re: Emotions on duality

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Good point, Toby. I hope you do write it up.

For me, an approach to $\dagger$ -categories or $n$ -categories with duals is only good if it does something nice in two key examples, namely cobordisms between manifolds and operators between Hilbert spaces.

It’s very natural to think about the groupoid of Hilbert spaces and unitary operators equipped with extra stuff — since unitary operators play a special role as symmetries in quantum physics.

It’s natural to ask what happens in the $\dagger$ -category $n$ Cob, where the morphisms are $n$ -dimensional cobordisms. Any diffeomorphism between $(n-1)$ -dimensional manifolds gives a unitary morphism in $n$ Cob. So, you might hope that all unitary morphisms arise this way. After all, diffeomorphisms of manifolds sound like very natural ‘symmetries’. However, I’m pretty sure that not every unitary morphism in $n$ Cob comes from a diffeomorphism when $n \ge 4$ .

This fact doesn’t mean your idea is no good! But it’s a bit of a bummer.

Posted by: John Baez on February 5, 2010 9:49 PM | Permalink | Reply to this

Incoherent adjunctions

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This is a really interesteng question. When Eugenia was here in Chicago the week before last, we thought about this some, and eventually convinced ourselves that any incoherent 2-adjunction can be rectified into a coherent one! This surprised both of us, and we wondered whether any of the Australians had noticed it before.

The argument is pretty easy and seems like it should generalize to the $n$ or $(\infty,n)$ case. Namely: suppose $f: C \rightleftarrows D: g$ is an incoherent adjunction, i.e. we have 2-cells $\eta\colon 1\to g f$ and $\varepsilon\colon f g \to 1$ such that the triangle identities hold up to invertible 3-cells, with no further coherence conditions imposed. Suppose that we are working in the (weak) 3-category $2 Cat$ ; the general case should follow from this by representability. Then by the Yoneda lemma, $\eta$ supplies us with a transformation $\overline{\eta}\colon D(f x, y) \to C(x, g y)$ natural in $x\in C$ and $y\in D$ , and likewise $\varepsilon$ supplies us with a transformation $\overline{\varepsilon}\colon C(x,g y) \to D(f x, y).$ The fact that the triangle identities hold up to equivalence means that $\overline{\varepsilon}\circ\overline{\eta} \cong 1$ and $\overline{\eta}\circ\overline{\varepsilon} \cong 1$ , i.e. they are inverse equivalences in the 2-category of pseudonatural transformations and modifications. But we know that any equivalence in a 2-category can be rectified to an adjoint equivalence. Performing this rectification, and then passing backwards across what we just did, we find that we have rectified our incoherent adjunction into a coherent one.

Posted by: Mike Shulman on February 8, 2010 10:29 PM | Permalink | PGP Sig | Reply to this

Re: Incoherent adjunctions

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Mike exclaimed:

any incoherent 2-adjunction can be rectified into a coherent one!

Very good.

Now assume an $(\infty,n)$ -category with all adjoints, i.e. each $k$ -morphism of low enough $k$ has an incoherent 2-adjoint in its hom- $(n-k)$ -category.

Do you see a way to use the above result to inductively coherify the higher adjunctions then?

I am thinking vaguely of something like choosing an incoherent adjoint for a 1-morphisms. Then choosing incoherent adjoints for the 2-morphisms involved and so on, and then recursively making things coherent by applying the above statement repeatedly.

Something like that, i don’t know. Does anyone see if such an iterative coherification might follow now?

Posted by: Urs Schreiber on February 11, 2010 10:42 PM | Permalink | Reply to this

Re: Incoherent adjunctions

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Do you see a way to use the above result to inductively coherify the higher adjunctions then?

That might work, but my $(\infty,n)$ -category-fu isn’t up to actually verifying it. Another approach, which is what I was thinking of, would be to do an argument like the 2-dimensional version in the $(\infty,n)$ -category $(\infty,n-1)Cat$ (which would require proving that any equivalence in an $(\infty,n-1)$ -category can be rectified to an adjoint equivalence, and possibly there could be some induction going on here) and then argue representably. Is the technology of $(\infty,n)$ -categories sufficiently advanced to enable such an argument?

Posted by: Mike Shulman on February 12, 2010 1:33 AM | Permalink | PGP Sig | Reply to this

Re: Incoherent adjunctions

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Is the technology of (∞,n)-categories sufficiently advanced to enable such an argument?

At least I am not aware that the required tools would have been discussed in the literature beyond $(\infty,1)$ . But I’ll keep my eyes open.

Posted by: Urs Schreiber on February 12, 2010 5:13 PM | Permalink | Reply to this

Re: Emotions on duality

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Urs wrote:

This is surprising, because a priori one would have thought that this is too simple!

Jacob Lurie pointed out another important issue here. His ‘free braided monoidal category on one fully dualizable generator’ turns out to be different from my hopes regarding the ‘free braided monoidal category with duals on one object’. Namely, he’s not getting framed oriented tangles.

I hope the people working on such questions think about this! It should shed some light on the relation between $n$ -categorical duality and Thom spaces, and thus the way $n$ -categories with duals arise from stratified spaces. It would be good to compare Lurie’s approach to Jonathan Woolf’s work. Woolf gets the category of framed oriented tangles, just as I’d expect. But there’s a lot of work to be done to relate his approach to Lurie’s. Probably they’ll both turn out to be ‘right’, just two aspects of a bigger story.

Posted by: John Baez on February 12, 2010 6:38 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

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John wrote:

I’m trying to cut back on jetting about…

David wrote:

I take it then that your conviction in AGW is as strong as ever.

Yes. But let me introduce a couple of irrelevant personal side notes, just to be done with them.

First, I’ve been getting too many invitations to speak. I realized I was happier back when I had more time to stay home and write. And I’ve decided to change my research direction — so I don’t even want to give talks about topological quantum field theory, $n$ -categories and that kind of stuff. I’d rather think about new things. So there are plenty of reasons to stop jetting about even apart from anthropogenic global warming.

Second, you’ve probably noticed that I want to switch to working on things that ‘help save the planet’. But there are lots of reasons why. For example, I think we’re in the midst of a mass extinction event due to habitat loss. So, even without anthropogenic global warming, I think my change in direction would be a worthwhile thing.

In short, I don’t think I’m doing anything (yet) that would seem like a colossal mistake if anthropogenic global warming turned out not to exist.

But anyway: you seem to want me to start presenting the case for anthropogenic global warming. You’ve convinced me that I need to do this — in part, just as a way to learn more about the issue. So, I’ll do it soon. But not in this blog entry.

Posted by: John Baez on February 4, 2010 9:11 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Say hello to Alex Wilce for me.

Posted by: Scott Carter on February 5, 2010 12:37 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Oh, and ask him to tell you about his encounter with Sarah Miles and her dog.

Posted by: Scott Carter on February 5, 2010 12:40 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

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More information about the conference:

Description: This event has as its goal to bring together researchers working on mathematical foundations of quantum physics, quantum computing and spatio-temporal causal structures, and in particular those that use logical tools, ordered algebraic and category-theoretic structures, formal languages, semantical methods and other computer science methods for the study physical behaviour in general. Over the past couple of years there has been a growing activity in these foundational approaches together with a renewed interest in the foundations of quantum theory, which complement the more mainstream research in quantum computation. A predecessor of this event, with the same acronym, called Quantum Programming Languages, was held in Ottawa (2003), Turku (2004), Chicago (2005) and Oxford (2006). The first QPL under the new name Quantum Physics and Logic was held in Reykjavik (2008) and the second in Oxford (2009); with the change of name and a new program committee we emphasise the intended much broader scope of this event, aiming to nourish interaction between modern computer science logic, quantum computation and information, models of spatio-temporal causality, and quantum foundations.

Submission: Prospective speakers are invited to submit a 2-5 pages abstract which provides sufficient evidence of results of genuine interest and provides sufficient detail to allows the program committee to assess the merits of the work. Submissions of works in progress are encouraged but must be more substantial than a research proposal. We both encourage submissions of original research as well as research submitted elsewhere. Submissions should be in Postscript or PDF format and should be sent to Bob Coecke by March 28, with as subject line QPL Submission. Receipt of all submissions will be acknowledged by return email. Extended versions of accepted original research contributions will be published as a special issue of a jounal - we are currently still exploring the options.

Deadlines:

March 28: Submission

April 13: Notification of authors

May 16: Corrected papers due

Posted by: John Baez on February 10, 2010 7:17 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Do you know already what you are going to talk about?

What are “quantum programming languages”?

You may assume that I know quantum mechanics, a little about quantum computation - enough to understand why the factorization of big numbers is not hard to do for quantum computers: ‘Shor’s algorithm’ - and the mainstream programming language paradigms, the ones that students of computer science are supposed to write programs with (imperative, functional, procedural, object oriented and declarative).

Posted by: Tim van Beek on February 10, 2010 9:14 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

After reading the following survey…

Simon Gay: Bibliography on Quantum Programming Languages

…I’m able to phrase my question somewhat more precisely: Simon Gay classifies the papers about “quantum programming” and has an item called

2.Semantics, (b) categorical and domain-theoretic techniques

There he mentions some topics that look very interesting, like

Abramsky (2004, 2005), Coecke (2004a; 2004b) and Abramsky & Coecke (2003; 2005; 2004) have developed a categorytheoretic formulation of the axioms of quantum mechanics, in the setting of strongly compact closed categories with biproducts.

Believe it or not but I don’t even know what a strongly compact closed category is. Is there an up to date expository paper about “application of category theory to the design of quantum programming languages”? Or would someone here like to explain what’s it about?

Posted by: Tim van Beek on February 10, 2010 12:00 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Tim wrote:

Do you know already what you are going to talk about?

Pretty much. A lot of stuff I’ve been trying to say in fragmentary form here for a long time… together with the paper I’m writing with Paul-André Melliès.

For example, I mentioned in my blog entry that I want to talk about how “star-autonomous categories act like commutative Frobenius algebras in the world of profunctors”.

This may sound like mysterious gobbledygook, but it boils down to a nice relationship between propositional logic and pictures of 2-dimensional manifolds! I explained it here. It was noticed by Ross Street, but not enough people seem to understand it. I want the world to know about it — because it’s a clue to something, but I don’t know what, and I want someone to figure it out.

Tim wrote:

What are “quantum programming languages”?

Languages for hypothetical quantum computers. Try this:

Peter Selinger, Towards a quantum programming language.

To me the interesting thing is not writing programs for not-yet-existent computers, but the deep analogy between computer programs and quantum processes. I taught a class on this, and you can see extensive notes here:

John Baez, Classical versus quantum computation.

Then Mike Stay and wrote a paper based on some of these ideas:

John Baez and Mike Stay, Physics, Topology, Logic and Computation: a Rosetta Stone.

At the end, this paper describes another quantum programming language.

Tim wrote:

Believe it or not but I don’t even know what a strongly compact closed category is.

Shocking, shocking — when I was a youth we all studied this along with our multiplication tables. What do kids learn in school these days, anyway???

You can learn what it is in my paper with Mike, except we use the more commonly used modern term: “dagger compact category”. This is another expository paper you might like, which will appear in the same book:

Bob Coecke and Eric Paquette, Categories for the practicing physicist.

Posted by: John Baez on February 11, 2010 6:52 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Thanx, I’d like to say that your words are not completly lost on me, I remember reading TWF 268, I bought and read the book

Joachim Kock: “Frobenius Algebras and 2D Topological Quantum Field Theories”

back in 2008, as advised (to be honest: the shipping date of my copy is May 9th 2008, so I must have discovered it elsewhere), and the rosetta stone, too…that the analogy explained there extends to quantum computation did not stick to me, however.

As for my stupid question about “strongly closed compact categories”, I see that Wikipedia explains that this is synonymous to the more modern term dagger, (compact closed), category.

Posted by: Tim van Beek on February 11, 2010 11:18 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

I hope you and Mike are busy trying to invent Rosetta Stone duality.

Posted by: Tom Leinster on April 9, 2010 3:05 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

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You can now see the program for the school taking place from Monday May 24th to Friday May 28th, 2010 — right before the Quantum Physics and Logic workshop announced above.

Posted by: John Baez on April 7, 2010 3:36 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

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In case anyone wants to talk:

With luck, I’ll be arriving at Heathrow on Sunday May 23rd at 10:30 am. I will somehow find my way to Oxford. (What’s the best way? The one-hour bus ride on the Airline?)

I will attend the school, skipping some classes on subjects that I should already know — like the definition of a monoidal category.

I will give a talk at the QPL workshop, which lasts from Saturday May 29th to Sunday May 30th.

I’ll leave on Monday the 31st — catching a flight that departs from Heathrow at 3:45 pm.

If anyone wants to travel from or to Heathrow at about the same time, it might be fun to travel together and talk about stuff.

It seems that Jamie Vicary, Bruce Bartlett and Eugenia Cheng will be around for at least part of this week. So, I want to talk to them. Will any other $n$ -Café regulars be around?

I’m also planning to talk to Dan Ghica (about type theory and hardware synthesis) Thomas Fischbacher (about all sorts of things, including engineering design for a low energy society) — and last but not least, the famous Oz!

Posted by: John Baez on April 7, 2010 3:57 AM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

I’m certainly not asking you to give it away if it’s not meant to be public knowledge, but I’ve been curious for years: do you, John, know who Oz is in real life? Or more to the point, did you when he was under the wizard’s tutelage?

Posted by: Mark Meckes on April 7, 2010 2:55 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

Hi, Mark. Yes, I’ve known for a long long time who Oz is ‘in real life’. He’s a dairy farmer who lives in Upthorpe, near Oxford. He did an undergraduate degree in physics at Cambridge. I’ve visited him and stayed at his house a couple of times. He’s a jolly old fellow.

Posted by: John Baez on April 7, 2010 5:39 PM | Permalink | Reply to this

Re: Quantum Physics and Logic at Oxford

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John said:

I will somehow find my way to Oxford. (What’s the best way? The one-hour bus ride on the Airline?)

Yes, that’s the best way!

I want to talk about how “star-autonomous categories act like commutative Frobenius algebras in the world of profunctors”.

That should set the scene perfectly for the talk that Bruce and I will give — about how compact categories are the same as dagger-Frobenius algebras in the world of profunctors!

Posted by: Jamie Vicary on April 9, 2010 2:49 PM | Permalink | Reply to this

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February 3, 2010