On the first day of CLIQUEmas, irons sent to me:

When you cut into the present, the future leaks out.

—William S. Burroughs

Loch has long since stopped being just a hobby for us; Irons has his writing, Treellama as a programmer has his dead rats and tampons, and I have my cohomology. To illustrate my point, I’ve partially copied some notes over I wrote about using homotopy sheaves to enlarge the category of spaces for which we can build ordinary cohomology. Without further ado:

A recurring theme in the foundations of things connected to topology is an inability to geometrically describe what cohomology “means” above the first few bottom degrees. This problem has also been regularly resolved by introducing homotopy closer to the bottom of the pyramid, so to speak; for instance, Quillen’s +-construction (which gave way to full-blown algebraic K-theory) was built by introducing a kind of homotopy to algebraic geometry, rather than trying to build the algebraic K-theory functors in isolation of their homotopy-theoretic roots, which is basically what had been going on before that. (As reference, look at the Wikipedia article’s subsection on the lower groups.) For exceedingly polite rings, their algebraic K-theory is known for formal reasons — and while this seems like an egregious sin in the context of algebra, the exact same thing is going on in the classical cohomology of spaces.

Namely, for nice spaces, one way to build the cohomology is to produce the singular chain complex of the space, dualize, and find its cohomology; the roots of this construction are in producing maps from the n-simplex into our space, and the amount of such maps depends upon how coarse the topology is on the target space. Namely, the coarser the better. Another definition is to think about maps from our space out into a representing space for cohomology, called an Eilenberg-Maclane space; this approach yields a lot of information when the topology is fine. On CW-complexes (or other similar models for nice spaces), these two definitions agree, but in the general category of spaces they produce quite different results. Furthermore, since neither coarse nor fine topologies are “good”, neither approach seems universally better than the other. This is a problem we ought to rectify.

Another face of cohomology comes from the interpretation in terms of principal G-bundles: a cohomology class [f] in H¹(X; G) corresponds to a map f: X → K(G, 1) = BG, where BG is the “classifying space” of G, used in the sense that BG supports a principal G-bundle EG → BG such that EG is a contractible space. Pulling EG back along f gives an evident map from Hom(X, K(G, 1) to isomorphism classes of principal G-bundles over X. If we consider f up to homotopy, then this association is injective — and because EG is contractible, this association is surjective, so cohomology classes in degree 1 can be interpreted as principal G-bundles over X.

If we assert that G is discrete, then we can also say that these are the same as sheaves over X whose stalks are (coherently) isomorphic to G. The back-and-forth between sheaves and covering spaces is a point of view with incredible clarity, so we’d better take time out to explain. The first general assertion is that to any map of spaces Y → X, we can build a sheaf ΓY over X, called the sheaf of sections, where the elements associated to an open set U in X are the continuous maps U → Y such that the composition U → Y → X is the identity on U. The second general assertion (and this is the incredible one) is that this example is generic — i.e., given any sheaf F over X we can build a space Y over X such that the sheaf of sections of Y is isomorphic to the original sheaf! Y is called the étale space of F, and we denote it as Ét(F).

How do we build such a thing? We must satisfy the key condition that any element of F(U) should correspond to a section of our map Y → X over U. We should start by building the set of points Ét(F) = ∪_{x ∈ X} F_x consisting of all the stalks of F; this comes with a map back down to X by picking a point in Ét(F) and sending it to the point in X that owns its parent stalk. We now need to induce a topology so we can control what sections are continuous. This step is actually kind of obvious, once we’ve made it this far: pick an element f in F(U), and let f_x be the element in the stalk F_x corresponding to the restriction of f. Finally, declare the union ∪_{x ∈ U} f_x to be open in Ét(F), and consider f as the map f: U → Ét(F) that takes x to f_x. f is continuous and a section of the projection Ét(F) → X, and one can show that these are the only sections that this topology admits. So we’re done! In addition, it can be shown that the map Ét(F) → X is a “local homeomorphism,” in the sense that restricting to a small neighborhood in Ét(F) makes the projection into a homeomorphism down to X.

(As an aside, we can build this same object for presheaves, which produces a sheaf that comes with an isomorphism on stalks back to the original presheaf. This process is called “sheafification,” and it’s what powers topos theory as built on top of sheaves.)

So this lets us talk about G-cohomology classes in X of degree 1 as certain kinds of sheaves over X. But what about the other degrees? It is remarkably unclear how to proceed; any K-theory-styled operations that we learn about from studying principal bundles will produce more principal bundles, and we’ll never escape H¹, so they’re of no use. The critical thing to note is that if we’re extremely careful, we can build BG in such a way that it too is an abelian topological group. We can then iterate this construction to produce BBG, which turns out to be a K(G, 2), and it supports a contractible BG-bundle we call EBG. Again, isomorphism classes of BG-bundles over X correspond to second degree G-cohomology classes of X as induced by the pullback of EBG.

But this turns out to be much harder to translate into the language of sheaves. The core problem is that the subspace corresponding to any particular stalk in Ét(F) is necessarily discrete, whereas principal BG-bundles emphatically do not have discrete fibers — their fibers are, of course, isomorphic to BG. This is a direct consequence of the convention that sheaves take values in the category of sets — which we consider here as the subcategory of spaces consisting of homotopy 0-types. If we generalize our notion of “sheaf” to allow them to take on values in the category of homotopy 1-types, then we can perform a very similar construction to the one above that translates G-bundles into sheaves with stalks coherently isomorphic to G — but instead, we translate BG-bundles (i.e., cohomology 2-classes) into certain kinds of stacks, and in particular, BBG is the classifying stack of the topological group BG.

There’s no reason to stop here! If we reformulate our definition of sheaf to allow our sheaves to assign open sets to arbitrary spaces, then we achieve the flexibility that we need to translate Hⁿ(X; G) into the context of sheaves for arbitrary n. This is one of the core motivations for Lurie’s work on “homotopy topoi,” which he graciously took the time to write a book about. A sizable portion of that book is also dedicated to developing a good theory of (∞, 1)-categories, which he calls ∞-categories and more traditional methods call either quasicategories (like Joyal) or weak Kan complexes (like Boardman and Vogt).

The transfer to topoi is part of this general practice of “enlarging” your data. The reason we care about schemes is that they’re like rings with the localization data made explicit; the localization was always there, but now we can handle it somehow explicitly. The reason we care about a category of sheaves over a space (i.e., a topos) is that it’s supposed to contain all the data that can be detected by strictly gluing “things” together — i.e., all the data that (sheaf) cohomology can detect about the space. That data was already there — the only thing the space dictates is how the gluing has to happen, which is encoded in the topos. The reason we care about homotopy topoi is that they contain all the data that general cohomology can detect, i.e., a tool that allows for patching data together up to higher coherent isomorphism. Again, this data is all “in” the space, but transferring to these larger categories where we deal with representations of the data is terribly useful for manipulating it.

There’s a trade-off, of course; namely, these homotopy sheaves don’t have a built-in notion of algebra, and we know that restricting to module-valued cohomology theories produces all kinds of strong representability results. It would be nice to understand the usual algebraic structures we’ve come to expect on ordinary cohomology in this homotopy sheaf context — what procedure can we follow to build the “product” of two G-bundles, itself a BG-bundle? What do the Steenrod operations look like, and how do we produce them? These are — to me, at least — questions with nonobvious answers, though it’s not clear that trying to come up with an answer would yield any kind of valuable information about homotopy sheaves in general, but instead just about these particularly algebraic structures. (And we already understand them classically, so…)

That’s enough dense loch for one post, I think. The point is that you can get paid for such nonsense (though not well). JFO: not for nothing.

Previous installments: part one, part two, part three. Now, prepare for the conclusion. It’s been great fun writing so much crap and having Bleating Lhowon disapprove of it so deeply. Let me know if you want me to keep the Essentials series going, and feel free to give me suggestions, or (if you happen to have an account here), run wild with the concept on your own.

(Try to read between the lines a little bit more)

Now that MARARTHON has officially taken its own life, there are a few things I’d like to get off my chest. I mean, they’ve bothered me for something like a year. That’s a long time if you think about it. (Also, I have the gift of Bourbon, so truth is coming more naturally to me at the moment.)

(Try to read between the lines a little bit more)

In this community, as in most others—well, never mind; I’ve never even seen others—we have a broad spectrum of members. There’s the CLIQUE personality, a grating know-it-all whose only redeeming quality is that he does actually contribute something worthwhile, even if it’s only a dash of humor. There’s the Story Forum guy, who will dissect any tidbit of information as long as it keeps him from playing ever again. A relatively new group is the horde of unwashed meatserver players, drawn here primarily by the Marathon/Halo connection.

Today, I’m going to talk about something we don’t see as often.

His nickname is AP, and he’s just had a tough time of everything. If you take a look at his site, you will notice that he has more than a few “issues.” The first is a case of loch that far exceeds my own and approaches ahuxley’s. This man has thoughts we can’t even begin to decipher:

geen paper - backside (hand written)
problems
discused /`
did...	travel to computer lab  /\/\
going on bus  n/\/\/
intendance  /\/\nn
schoolaphobic  /`´\/\	math, science, exploritory/
depended on teachers  /\/\/\
industrial arts /\/\/\
1st  Quart...  /`´\/V

This example is actually very interesting to me; I am drawn now to anything loch. AP doesn’t seem too “stable” in that bit of writing, but he is actually quite brilliant. Take a look at the language he designed, called Bare:

Sample macro code for square root:

loop with (1<<(s::)/2) ratio 2 (s/result) until +delta≤1;

The “with” parameter (1<<(s::)/2) means the result starts with a bit magnitude of half the scale of the input. The “ratio 2” means that the result is only affected by the code by one half. And the “until +delta≤1” means that the positive (absolute value) difference calculated of each result must be at most one fraction.
This is the optimized equivalent of the Newton Method of finding the square root. It is not needed, since it is built in to the Bare languagel the root symbol with no second operand, “s*/;”.

I honestly don’t understand more than 10% of that. The rest of the language definition is nearly as cryptic. But I’m honestly intrigued by what he has written.

You might be wondering what this has to do with Marathon or the community. Well, the community part is tough, but I’ll get to that later. The Marathon part comes from his engine project, Old Durandal. The project is essentially a version of the Marathon 2 engine for old Motorola 68k computers, with enhancements like those in Aleph One. What at first seems like a hoax or a failure (given AP’s own comments about how terrible or useless the project is), I have actually seen his engine work. It’s far from perfect or wonderful, but he has actually accomplished something. It’s tough going for him because he’s not only programming for an old machine, he’s programming on it.

I started this article shortly after the foundation of JFO (on May 20, 2008), and I’ve waited this long to publish it because I sent AP a few questions. He took a while to reply, but I think he gives us some insight into his misunderstood persona. Here’s a slightly edited version of the correspondence.

(Try to read between the lines a little bit more)

K’ter l’oracne’ktr ESB’crkn rhl l’oac’rkthahrl tr’lac.
L’on t’hrl’ory, l’on l’oa’rhl’ktr tr’tract l’on t’rac r’ar r’arhl’rac.

(Try to read between the lines a little bit more)

Day 1

Dear Diary,

Today I checked Pfhorums as usual. There wasn’t much to see there. As usual lately. I mean, Moppypuppy got all huffy about his stupid topic (the one about the video, good God). There’s also a new scenario announcement: “lolnova.” It’s a joke scenario; whether the creators realize this or not is the only question in my mind.

ESB is getting some new adrenaline shot through its shriveled balls: some guy named “Godot” showed up. I swear something like this happened on Pfhorums two years ago… However, this Godot claims to be Hamish. I wouldn’t be surprised if it was just ukimalefu posting more of his redundant crap so ESB never dies. We were so close, too…

Three posts from glory.

Day 2

Dear Diary,

I thought about things a little yesterday, and I just realized I haven’t played at all in a month or two. The nights just after my router replacement were joyous, full of netgames. I think I hosted a hundred within a couple of weeks. It was a new leaf for me, after the many months of non-gaming. Things were looking up.

Then what? I got drawn back into the CLIQUE meta-game. And I like it. I’ve got some comments from outsiders on JFO now, which might be because of the recent Pfhorums outage. I think I like the people here better than I like the game itself, if you can believe that. Why play at all?

Day 3

Dear Diary,

#alephone lately has been mainly join/part messages and discussion about Smithy. Don’t get me wrong; Smithy is a great program that will add new life to Marathon. It’s just a few years too late for me. I talk a lot about this program and others, but I need to face the reality that I’m not planning on mapping ever again. Sure, I might make a level occasionally, but I think everyone agrees that my heyday was over by the time I started Pelikan. It’s another case of learned behavior: making a fuss out of Marathon when I have almost nothing to do with it any more. Why bother?

Day 4

Dear Diary,

I launched the 4GET MARARTHON campaign. Some might ask why, and I will answer them in the privacy of this journal: for me and for Wolfy. Both of us have things we want to accomplish that became twisted and unrecognizable at some point in the last four years. I want to write, and my prose lately has taken me to new and frightening places, causing me to draw unforeseen conclusions about myself. Treellama went to school for his music, a noble goal. We both need to forget Marathon. There’s a word other have used in the past for the act of forgetting the game, but I don’t dare utter it just yet.

Day 5

Dear Diary,

4GET has already produced fruit. I wrote more in Earth Mother. Wolfy finished a song he began months ago. There’s not much of a choice now.

Now

loch

~~~~~~~~~~

(Try to read between the lines a little bit more)

♫ Did you ever know that you’re my hero ♪

Marathon is the German Language of first-person shooters.

It is at times cumbersome, often full of subtleties, reviled or ignored by most unacquainted with its charm. To whom, then, do our own CLIQUE minds correspond?

Ryoko: Richard Wagner. Wagner is known for his dramatic, romantic, overblown musical style, the themes of which were eventually adopted by the Nazis. Ryoko’s maps are crazy big, grandiose, and so on. And he’s a Nazi.

He deserves nothing he has.

Treellama: Thomas Mann. Mann was an astounding writer, and is one of my personal favorites. He opposed the Nazis and went into exile during World War II. Treellama is pretty reasonable, and he privately hates CLIQUE. Like Treellama, Mann had better things to do than hang around those losers.

you guys are so negative :(

W’rkncacnter: This is where it gets tricky. I’m going with Karl Marx on the grounds that communism = JUICE.

HALF OF MY TEAM IS NOW OUT OF JAIL.

Thermoplyae: Thermo is kind of mysterious. He also likes numbers. Therefore, the most suitable choice is Gottfired Leibniz, one of two losers who invented calculus, and the man who brought us binary. I’m sure he would be just as content to program in ML as Thermo is.

*spills dixie cup beer*

Ray: If anyone had owned an imageboard based on Germany back in the day, it would have been Goethe. Let’s look at Goethe’s masterpiece, Faust, from a Ray’s perpective—that is to say, Faust v. Mephistopheles: King of Fighters Romanticism Edition!!

then you will be 1/infinity as cool as me

Irons: I’m half-genius, half-crazy. People try to understand me and usually fail (unless they are my CLIQUE contemporaries). I created an übermensch named Hotmodal. I am Nietzsche.

loch nits olmec taps you the eight-foot-tall Burton