Sure, the CLIQUE likes to ruin Mararthon, but have any of us managed to do this?

Stats collected so far this month:

$ grep -i survival misc.log | wc
229 4496 25219
$ grep -i emfh misc.log | wc
228 4198 21697

I know we have an OSH award for maps, but to me, this seems like an even greater accomplishment. Congratulations to Megabyte for ruining the metaserver!

A. Huxley
c/o Goulburn Correctional Centre
PO Box 264
Maud St
Goulburn NSW 2580

Job wanted: Unemployable

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

Dugit and friends have been the subject of much discussion on #alephone, especially in light of some recent Pfhorums threads. To confirm the public’s suspicions that CLIQUE is more concerned with posturing than anything else, here’s what half of us really think of shitposting:

[thermoplyae] jesus christ
[thermoplyae] i was just thinking “man the pfhorums sure are active, this is totally contrary to that graph that irons had”
[thermoplyae] but 2/3 of the last posts were made by DUGIT, and now i understand what you guys are so uppity about
[thermoplyae] i’m gonna do steve’s job and see if i can flag any of these for shitposting
[Stevedollars] if you report posts I’m more likely to actually do something
[Stevedollars] I’ve only deleted two or three posts today
[ray] it’s a good thing shitposting kingpin patrick is suspended and quality poster dugit is able to go about his posting^†
[thermoplyae] well i lied, i don’t really know what shitposting even means, so i’m not going to flag anything
[Wrkncacnter] i took a screenshot from last night
[Wrkncacnter] it’s pretty great
[ray] nobody knows what shitposting means
[Wrkncacnter] ^
[treellama] ^^
[Wrkncacnter] that’s why it’s so fun to talk about
[ray] the only thing we can agree on is that dugit is a loser

^†: Whatever CLIQUE feels about shitposting has had little bearing on Steve, who’s taken the opportunity to seize some Pfhorums political power^‡ by suspending patrick and leaving the Dugit/WJ/Envy cell undisturbed.
^‡: hahaha

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

In honor of the Fat Sam video being mirrored by RAYLABORATORIES, here is the CLIQUE guide to preventing uncomfortable Maintenance Closet Incidents in YOUR building. Follow the six easy steps of BE LORD.

1. Ban fat kids. This is a simple precaution that can save millions of dollars. Fat kids are practically born to be picked on, and when one fat kid establishes dominance over another, it is only a matter of time before he goes looking for an unlocked maintenance closet. Cut straight to the root of the problem by removing all fat kids from the premises.

2. Enforce beackpeack protocol. Beackpeacks are valuable tools, but they cause blind spots in wearers and are more often than not the starting point of a given Maintenance Closet Incident. If and only if beackpeacks are absolutely necessary, they should be worn on the beack at all times, and removed only when the wearer is alone and ready to place the beackpeack in storage. Otherwise, beackpeacks must be prohibited.

3. Lock all maintenance closets. It might seem like an obvious step, and it is. Most readers will move on to the next point before they finish this sentence. Even so, it is imperative that the custodial staff perform daily closet-sweeps. They must check all maintenance closet interiors, lock all closets, and make sure that no existing closets have disappeared or no new closets have appeared.

4. Outlaw three-syllable laughter. Many experts recognize that Maintenance Closet Incidents are triggered on both ends by small vocal ticks that come from one, or both, of the participants. The most common trigger by far is the reflexive three-syllable aspirate laugh. Don’t let dormant Maintenance Closet Offenders awaken; stem the tide with silence.

5. Require assistance paperwork. Should a person wish to give or receive help, he or she must place the request in writing using an approved Assistance Form. In the (hopefully) unlikely event that a Maintenance Closet Incident does occur, it must be possible to determine who the culprits were, and all liability must be traced to the involved assistance giver and receiver. Any assistance in progress, if observed, must be challenged through a request to see both  participants’ forms.

6. Don’t ever releacks. A single moment of releacksation can cost your facility a billion dollars under present-day socialist law. If you ever cut corners on the above steps; if you ever let customers or employees believe they can get away with the violation of your policies; if you ever turn a blind eye to any suspicious behavior–those billion dollars will only be the beginning of your worries.

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.

I HAVE JUST FOUND OUTE

ESB is Empty

ESB’s tent is broken: the last echos of *NM*
Clutch and sink off the top page. The sound of uki
Crosses the blackness, unread. The readers are departed.
JFO, run softly, till I end my post.
ESB bears no new messages, announcements of carnage fests,
Discussions of Eternal, comparisons of weapons, trolls, spammers,
Or other testimony of activity. The readers are departed.
And the CLIQUE, the loitering heirs of the old Bungie crew;
Loched, have left no addresses.
By the ruins of TGI I sat down and wept…
JFO, run softly, till I end my post,
JFO, run softly, for I speak not loud nor boast.

But at my back from an unlocked maintenance closet I hear
The slam of a door, and chuckle spread from ear to ear.

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)