Oz and the Wizard -

Defeat and Triumph

Oz and John Baez

It was night. Oz had pushed his final bunch of answers under the Wizard's door. He had put down any old stuff from the 'blatantly obvious' to the 'not even wrong', but his confidence had cracked. There was only one thing left to be done and he went back to his cave to pack. He was beaten. He couldn't answer the Wizard's final question. He would have to be gone in the morning before the Wizard awoke.

Oz, his shoulders hunched despondently, slowly pushed his few meagre belongings into his old patched bag. Finally all that was left was the few tattered and worn scraps of paper he had made his notes on. He looked at them one by one. One by on he carefully and reverently placed them on the fire. Ah, well, what did it matter now. Slowly the pile grew smaller and the fire brighter till only one paper was left. It was the Curse Notes that had materialised out of the air long ago. On it he had written "Number Two". Ah, those hopeful enthusiastic days when there were no difficult questions. Oz sighed deeply. He went to throw them on the burning pyre, but couldn't bring himself to. With a nervous glance at his door he hurriedly thrust them to the bottom of his old bag and rose to go.

He walked slowly over the white quartzite flags and glanced for the last time with longing at the Wizard's curtain. Now he would never have the faintest idea what wonders lay in the books behind it. He turned sadly round as he took his final look at the Wizards cave. Then he walked over to the entrance and looked out into the night, at the heavens covered in bright stars. The snowdrops had come out over the last day or so and made a cheery and springlike display by the cave, but Oz was too sorrowful to notice. With his shoulders hunched and with a doleful pace he set off for the village. He slowly faded into the darkness as he stepped out into the night, alone, away from the Wizard's Keep to a future, who could tell.

A while later, but still very early in the morning, the wizard rose from his bed, yawned, and stretched. He had fallen asleep thinking about theory-model duality. His most promising student (who does not figure in this tale) had explained this to him earlier in the evening: in the category-theoretic approach to logic, a wide class of theories can be described as complete categories, and a model of such a theory is simply a limit-preserving functor from it to the category of sets... "It's a kind of incarnation of the abstract theory in the concrete world of sets," thought G. Wiz as he absent-mindedly strolled into his little kitchen.

He filled a kettle with water and carried it over to the fire, pushed aside some alchemical apparatus, and hung the kettle over the flames. "But then, as it turns out, the category of models is just the opposite category of the category of theories! At least given some mild assumptions which I don't remember." He picked up a mortar and pestle from the floor... hmm, it was full of some evil-smelling substance. He frowned. What was that stuff, again? He shook it out into the sink, washed the pestle, and shook some coffee beans into it. Grinding them slowly, he thought "So in some sense, the category of incarnations of an abstract theory is just like the category of things which figure in the theory, only backwards. But then..." Now came the vague part, the part he hadn't figured it out yet. He let out another loud yawn. "But then... if our original category was a star-category, so it's isomorphic to its own opposite, this means the category of theories is equivalent to the category of models!"

After the beans were ground, he poured them into his mug. He really should send the Courier to Mundania to get some more filters. "Let's see, though. We never really found many complete star-categories. Hmm. But this self-duality reminds me so much of Hilbert spaces. It's just got to mean something! Damn!" The kettle was letting out a piercing shrill whistle that made it impossible to think. He went over to take it off the fire. "Now what was I thinking. Yes, right, that if we stuck with 2-Hilbert spaces as theories, replacing limit-preserving functors by 2-Hilbert space morphisms, and represent them, in, say, Hilb...."

He poured the steaming water in the mug and smelled the wonderful fragrance of fresh coffee. "Yes, then we see that with this notion of theory and model, we'll get equivalence between the theory and the model! And of course, something like this might well hold in further categorified versions, too...." What an inspiring thought! If it all hung together, maybe the thing about topological quantum field theory was really just this... that the theory of the world was equivalent to the n-category of models of that very theory! He had no idea what *that* implied, but surely it must mean something. Sipping his coffee, he strolled over to his study to write down the idea in his big book of cool ideas.

He noticed a wad of papers stuffed under the door. "What's that?" he asked. Pulling it out and looking at it, he immediately realized it was written by Oz. He wondered if the apprentice had finally answered the black hole question correctly... the fellow had been floundering around for a week now, doing everything except the simple, obvious, right thing. He shook his head in annoyance and read the rambling notes, starting in the middle where Oz had written:

"It's obviously got to be something to do with curvature."

"I sometimes wonder if we should have 'done' a simple example first, like a satellite orbiting a massive body for example. Probably a mite trivial from your lofty position but ......."

"I am beginning to get the really horrible feeling that the answer you wants is so straightforward that one almost wouldn't deign to mention it." The wizard smiled. "Either that or we are looking in the wrong direction. How about the other terms. You have studiously avoided even making yes-no comments, so it's a mite hard to see where to go without positive or negative reinforcement. We derived R11 for the big bang, at least I think we did, I don't know that anyone said if it was right or not. Did it tend to zero, or -infinity? Nobody said either way."

"OK, let's stick with curvature. You doesn't want any maths, and he doesn't want it rigourous, that's helpful I suppose. We know ideally he wants us to show that at some point nothing can escape, but just saying that the curvature eventually gets to oo isn't good enough. In any case we know that the curvature at the event horizon for a massive black hole can be rather small. What distinguishes it from anywhere else? Well if you were there you would see the universe outside infinitely blue shifted, but if you were a point that's about all, I think. Oh, hang on a minute you are talking about *singularities* NOT event horizons. You want to know about singularities."

"So what do we know about singularities, or more important (I suppose) things heading that way? Well, I suppose this E+3P should have something to do with it. The only thing that springs to mind is that the more E & P you keep shoving into a volume of space to get it to prevent collapse, the more curvature you produce that needs more P to resist it but this produces more curvature."

"YES!" shouted the wizard. "He's done it! By golly, he's done it! Yes! PRESSURE CREATES GRAVITY, that's the point - so in certain situations, no matter how hard the star tries to resist collapse by means of internal pressure, it's a no win situation! The pressure only makes it collapse all the more!" He grinned widely and took another swig of coffee. "I knew he had it in him. Brilliant!" He read on:

"This is trivial and I've said it before. How to show that you are guaranteed to fail is not so obvious without bringing in photons, but I got the impression that using photons as a limit was not part of the answer."

"What is he talking about?" the wizard asked the air rhetorically. "Yes, it's trivial, but no, he never said it before! It's exactly what I wanted! I'm sure I would've noticed. Wouldn't I?" He frowned. Could he have ignored it, perhaps momentarily spacing out while Oz had been chattering away?

He read on:

"So, like a poor little flower deprived of light and water, I am about ready to give up. I'm not getting anywhere, and my knowledge of the subject is peripheral at best. I think perhaps I ought to stick with shattering glasses. Disappointing, though."

"As a final guess, since there's nothing to lose, the temptation to use R11=E-3P and just guess that this eventually results in a negative curvature (despite Ed & I asking what it meant) even if we did derive it in the BB scenario. Now what this would mean one might guess is that a little vector would end up pointing backwards. I suppose this would mean that no matter how it tried, it would always end up pointing back the way it came. The turnround would occur at E=3P."

"Anyway. I admit defeat. I have stopped learning. The fun has gone. I preferred tensors. Great disappointment. I am leaving this night."

"WHAT!" shouted the wizard. "Leaving? The fool! Leaving!? How can he have given up just when he figured out the right answer?" He cursed loudly and strode back to the bedroom, hurriedly tossing on his cape and putting on his hat. He would have to make good time to catch the fellow. Had he been too hard on Oz? Not enough encouragement? Or was Oz simply too hard on himself? He'd made so much progress in such a short time, and here he was giving up in defeat at his very moment of triumph! "The absolute ninny! Just when I was getting used to having him around!"

Picking up his staff, he rushed out. Luckily the village was small; there were not many places Oz could be, assuming he hadn't left town. Oz had occaisionally mentioned a certain wench by the name of Rosie at the village pub... he might have headed there to drink away his sorrows... or maybe he'd just spent the night at one of the cheap inns. In any case, it should not hard to be find him.