The n-Category Café

Jim, thanks for the plug! I think that Alissa Crans, Mohamed Elhamdadi, and Masahico Saito have quite a lot to do with this work. The e.g. makes this implicit, but not explicit.

What we have seen on the higher dimensional algebra side is not as explicit as Ronnie Brown is mentioning but we have a lot of glimpses into this: Take a piece of classical algebra, represent it as diagrams, interpret the diagrams as a categorical internalization, for example, now watch the diagram move to express the axiom. The motion of the diagram can be projected (decategorified), and some extra-structure can be reimposed on the projection of the motion. Then the identities among relations (or similar axioms) can be visualized by watching the singular points of the projection undergo their singularities. The movie move theorem and its smarter brother (C., Rieger, and Saito) are example exploitations of this idea.

The recent stuff enhances this with some cocycles attached at the singular points. I am pretty sure (and I think Masahico agrees) that the whole picture goes through nicely in the Frobenius algebra case because all the identities among relations, and their higher order analogues are controlled by the Stasheff polytopes.

Of course, as a hobby, I try to draw things in as an explicit fashion as possible. And this relates back to Jonathan Van Post’s remark. It takes time and hard work to visualize higher dimensional stuff. I had a conversation with J. H. Conway the other day, and I was surprised at some of the questions that even he could not answer. He mentioned a geometric (4d) proof of Heron’s formula, for example. If higher dimensional problems are hard for him, do the rest of us have a chance? So 4d gaming, 4d fly throughs, and all of these things can help build intuition for higher-dimensional algebra.

Here is a homework problem. Take any suitably short book on Hopf algebras (eg Majid’s big pink) and write all the proofs in diagrammatic form. After finding the diagrammatic proof, then take the kinematics of the proof and elaborate them as “spin foam” pictures. Now are there key axioms, key proofs, etc that manifest themselves in a 3-d branched solid?

In regards to JB’s comment about sending primates into space, I think the original space program did a good thing: It took a fundamentally military competition and turned it into a humanitarian endeavor. Some spin-off technology includes the relays from my keyboard to this forum and back. Maybe not directly moon related, but some organization had to put those tin cans in space. The current proposal to send people to Mars does seem far fetched in light of the overwhelming social problems caused by humanity’s taxing this environment so. Some thought (Asimov?) that our purpose was to get the hell out of Dodge: Space exploration was what we were meant to do. As a species, we tend to foul our homes to the point that they are uninhabitable, and then move on.

NASA has no mandate to address the problems of this planet even if they have the resources to make inroads. We, as scientists, can lobby for that mandate to change.

Scott Carter wrote:

In regards to JB’s comment about sending primates into space, I think the original space program did a good thing: It took a fundamentally military competition and turned it into a humanitarian endeavor.

Just so there’s no confusion: I completely agree!

I also think it was very important for us to see this:

The Apollo mission made good use of technologies available at the time, before computers really took off. My objections are to a manned Mars mission, which at this point would be a huge step backwards. We don’t need to shoot people into space anymore to see these wonderful things, and study them in marvelous detail:

NASA has no mandate to address the problems of this planet even if they have the resources to make inroads. We, as scientists, can lobby for that mandate to change.

For starters, we should lobby to undo some of the damage
already done by the Bush administration’s wrongheaded focus on the Mars mission, combined with their ostrich-like posture when it comes to human-caused global heating:

NASA is canceling or delaying a number of satellites designed to give scientists critical information on the earth’s changing climate and environment.

The space agency has shelved a $200 million satellite mission headed by a Massachusetts Institute of Technology professor that was designed to measure soil moisture – a key factor in helping scientists understand the impact of global warming and predict droughts and floods. The Deep Space Climate Observatory, intended to observe climate factors such as solar radiation, ozone, clouds, and water vapor more comprehensively than existing satellites, also has been canceled.

And in its 2007 budget, NASA proposes significant delays in a global precipitation measuring mission to help with weather predictions, as well as the launch of a satellite designed to increase the timeliness and accuracy of severe weather forecasts and improve climate models.

The changes come as NASA prioritizes its budget to pay for completion of the International Space Station and the return of astronauts to the moon by 2020 – a goal set by President Bush that promises a more distant and arguably less practical scientific payoff. Ultimately, scientists say, the delays and cancellations could make hurricane predictions less accurate, create gaps in long-term monitoring of weather, and result in less clarity about the earth’s hydrological systems, which play an integral part in climate change.

Your analogy is flawed. The monarchs of Europe sent explorers around the globe to take over territory and make money. The practicality of the endeavor was obvious to them, though it certainly involved risks.

And the reason is: they knew Europeans could easily survive on other parts of the Earth. They also soon learned they could subdue the locals with superior weaponry.

Here is a quote from Columbus on his journey: “Many of the men I have seen have scars on their bodies, and when I made signs to them to find out how this happened, they indicated that people from other nearby islands come to San Salvador to capture them; they defend themselves the best they can. I believe that people from the mainland come here to take them as slaves. They ought to make good and skilled servants, for they repeat very quickly whatever we say to them… I could conquer the whole of them with 50 men, and govern them as I pleased.”

Will we see any folks on the Mars mission making similar remarks? No: they’ll be struggling to survive on a chilly airless ball of rock, for no particular good reason.

Until something changes — like genetically engineering ‘men’ that’ll enjoy life on other planets — manned space missions will remain a tired old 1950’s-science-fiction-style vision of the future. There are much cooler things to do now.

I’ve often thought this, and that the distance between escape fantasies and eschatology isn’t necessarily all that large…