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nLab > Latest Changes: Pages on cubical sets

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  1.  
    • CommentRowNumber1.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 14th 2016
    • (edited Apr 14th 2016)

    I have begun making quite extensive changes to the pages concerning cubical sets.

    Perhaps most significantly, I have begun trying out a somewhat different style to the usual one at the nLab. I would like the entries concerning cubical sets to be concise, ideally short, and just contain mathematics. However, I of course do not wish to remove previous work. What I have done therefore is to shunt what was before at cube category and cubical set to new pages: cube category - exposition and cubical set - exposition. The name for the latter two is maybe not the best, but I’m not sure what would be better. Any suggestions? I have then re-written cube category, now called category of cubes with a redirect, and cubical set, in the style I have in mind.

    I have also created cubical truncation, skeleton, and co-skeleton and cubical Kan complex. I have also edited homotopy hypothesis for 1-types to remove material which is now present at one of the afore-mentioned entries.

    Many things could be added. Here is a TODO list for the moment (excluding homotopy hypothesis for 1-types).

    1) Expand upon the monoidal structure on cubical sets at cubical set. Explicit description and construction. On own page. [High priority, but will take some time, so may not get done for a while.]

    2) Draw what a horn looks like in dimensions 1 and 2 at cubical set, and indicate the same for a boundary. [High priority, and quick to do.]

    3) Give an explicit generators and relations description of □\square, but on a separate page. [Lower priority. Does not take much time. I do not think we should give a proof; the only way to convince oneself that the relations are correct is to write out a proof oneself.]

    I have defined a cubical horn at cubical set in a way which is slightly novel, I suppose. Because I like everything to be constructively valid, I prefer to avoid a ’removing a face from the boundary’-like definition. But this could be added as a remark to give intuition, once the pictures of a horn are added.

    I am working towards making further progress on homotopy hypothesis for 1-types, but needed the notion of a cubical Kan complex.

    To keep a consistent style, it is likely that I will keep a close eye on the pages concerning cubical sets, and heavily edit any deviations from the style I am beginning to put in place. If there are objections to this, let me know. I guess this is maybe the first significant example of ’re-factoring’ on the nLab, so it will be interesting to see what people think of it!

    I will eventually add one of those panels with links on the right hand side which includes all pages in the new style on cubical sets, called something like ’cubical sets’.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2016

    it is likely that I will keep a close eye on the pages concerning cubical sets, and heavily edit any deviations from the style I am beginning to put in place. If there are objections to this, let me know.

    On general grounds, I’m uncomfortable with this. That doesn’t mean I will have objections to what you plan specifically in this case; maybe I’ll really like it and really prefer it. But on general grounds, I’m uncomfortable with authors staking claims to nLab pages.

    For what it’s worth, I have a personal web which is not open to other people’s edits (actually I have two, but one is invisible to the public); this I find very useful for cases where I want “ownership”. Sometimes stuff I’ve written on my personal web gets ported over to the main page, but that entails the possibility of others’ edits (I admit that I would feel annoyed if someone rewrote it substantially, but that hardly ever happens). If you don’t already have a personal web, that might be something good to have (and I personally would be intersted to see what you put on there!).

    I guess this is maybe the first significant example of ’re-factoring’ on the nLab

    I’m not sure I understand what is meant by ’refactoring’. Is it consonant with this definition: “Code refactoring is the process of restructuring existing computer code – changing the factoring – without changing its external behavior. Refactoring improves nonfunctional attributes of the software.” ?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 14th 2016

    I don’t really think it’s up to one person to decide that the entries on any one subject ought to be styled in one way or another, especially if that style is significantly different from the rest of the nLab. In general, we try to just add or improve, rather than delete. Why not merge your new content with what was already on the cubical set pages?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2016
    • (edited Apr 15th 2016)

    It’s good to add technical detail to pages where it is missing. But many readers may want to see the informal idea of a concept first, before getting presented “Definition 1”. It is good style, I think, to first have a few informal paragraphs as lead-in, and then the technical detail. We once laid out the Template page to suggest this pattern. What #1 calls the “usual style” of the nLab seems to refer to the style of just the Idea-sections of entries. On fully developed pages, these Idea sections are just the first sections before the technical sections.

    • CommentRowNumber5.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 15th 2016
    • (edited Apr 15th 2016)

    Just some very quick clarifications before I must fly.

    Regarding #2 and #3: I did not mean at all to suggest any sense of ownership, or staking of claims. My position is as follows. I would like to add a substantial amount of stuff on cubical sets (and other things using them). I consider what is now at cubical set - exposition and cube category - exposition to be a mess from an organisational point of view, and also rather verbose. Whereas I would like a precise place to point to something, without having to point to some section about five-eighths of the way down a long page, surrounded by a lot of stuff that is irrelevant for the purpose at hand. I do not have the energy to try to merge my material into a wider reworking of the entirety of those pages, as Mike suggests, but I have the energy to do what I have done: get the page in the style I would like it, without losing anybody else’s work.

    So rather than being anything to do ownership, it is a pure question of style. I am suggesting a new style for certain pages, which are more like conventional mathematial documents, and are also deliberately intended to be short and easy to scan. Rambling discussion, which can of course be very valuable too, is moved to other pages. Once the style is in place, anybody could edit to keep that style: I am just saying that I think a good, tight style can be a good thing, and I am willing to enforce it on these pages.

    Regarding #4: I am not against informal discussion, and intend to add some to the pages I created yesterday. However, on such ’mathematical’ pages, I would prefer that this be very focused, and only be added if it gives real insight, just as in a conventional mathematical document. It could be in an introductory section, or through remarks, etc. But I would separate this from lengthier expositional material. Take for example the first section at cubical set - exposition. This kind of thing is great, and I have written similar stuff myself elsewhere. But as soon as one has understand anything about cubical sets, one does not need to read it anymore. Whereas one can always have a need for quickly recalling a precise definition/proof, etc. This is why I propose a separation.

    But of course I will follow what people wish, I am just suggesting something new. Feel free to make alternative suggestions. I am open to a personal web if people would prefer that, and then other people could merge whatever they like from that into the nLab.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 15th 2016

    I would definitely support Richard having a personal web should he want one, for his material is unique and certainly worth a home here.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 15th 2016

    Those two exposition pages have query boxes appearing prominently. This is a sign that they haven’t been put into the standard recent nLab style since we realised fairly soon that such boxes were a distraction and any such discussion could happen at the nForum. So the pages being a “mess” may be more of a sign of their age than that standard nLab style should be avoided.

    But then it does sound as though you want some pages to be more like a Bourbaki textbook. If this goes through in the two cases here, I can’t see why it wouldn’t apply to hundreds of other pages, so it’s quite a serious decision.

    Do you find more recently worked pages problematic? Take, e.g., derivator, is there anything difficult in someone wanting to skip the motivation and go straight to the definition to click on that link. (Now I see that has an old query box too, so perhaps this isn’s so up-to-date.)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2016
    • (edited Apr 15th 2016)

    My suggestion: let’s not further discuss this. Let Richard go ahead do his edits. And when he is done, then anyone who cares about the entries on cubical sets may go and try to clean up the earlier messy entry (which is indeed necessary) and then merge the result with Richard’s piece to a new beautiful entry that properly follows the structure of the Template page: first a nice informal introduction, then a nice technical section.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 15th 2016

    I support Urs’s suggestion for this page.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 15th 2016

    I second Todd’s support. B-)

    • CommentRowNumber11.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 16th 2016
    • (edited Apr 16th 2016)

    Thanks all!

    I forgot to say in #5, in reply to Todd, that, yes, by ’refactoring’ I had in mind something akin to the usage in programming. For in the end, as Urs hints, probably the existing content will be more or less the same as before, and the style not so very different from the remainder of the nLab. Indeed, with regard to the existing content, it is more of a re-structuring/re-working than removing anything, though I am also adding content that was not there before.

    Regarding #7, I’ll not reply in detail, in deference to #8 - #10. I’ll just say that I wouldn’t really see much of an analogy to Bourbaki. It is more that my suggestion is for the technical part of a page to be more like a fragment of a mathematical paper. But of course there are different ways of writing the latter; I myself am not overly fond of mixing in a narrative with formal statements (definitions, theorems, etc), which is the prevailing style on the nLab, and, in my experience, in the vast majority of papers. Thus the contrast may be starker than it would otherwise be.

    Regarding the page derivator, this is maybe an unlucky choice, as I happen to have worked with these rather a lot, so may have strong opinions! I’ll refrain from any specific comments, but one thing I would certainly do is to divide up the page into many smaller ones.

  12. I have, by the way, begun working on homotopy groups of a cubical Kan complex, with a few tweaks at cubical set. Quite a lot left to do, still.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 16th 2016
    • (edited Apr 16th 2016)

    Looking at homotopy groups of a cubical Kan complex, I see the following diagram:

    □ n−1 →p □ 0 y(I i−1⊗i ε⊗I n−i)↓ ↓x □ n →σ X \array{ \square^{n-1} & \overset{p}{\rightarrow} & \square^{0} \\ y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n-i}) \downarrow & & \downarrow x \\ \square^{n} & \underset{\sigma}{\rightarrow} & X }

    If you don’t like the placement of the left-hand vertical arrow, you can use the \mathllap command (or \mathrlap for notations to the right of arrows); click Source for the code. It comes out looking more like this:

    □ n−1 →p □ 0 y(I i−1⊗i ε⊗I n−i)↓ ↓x □ n →σ X \array{ \square^{n-1} & \overset{p}{\rightarrow} & \square^{0} \\ \mathllap{y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n-i})} \downarrow & & \downarrow x \\ \square^{n} & \underset{\sigma}{\rightarrow} & X }

    Here it is with the ’xx’ in \mathrlap:

    □ n−1 →p □ 0 y(I i−1⊗i ε⊗I n−i)↓ ↓x □ n →σ X \array{ \square^{n-1} & \overset{p}{\rightarrow} & \square^{0} \\ \mathllap{y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n-i})} \downarrow & & \downarrow \mathrlap{x} \\ \square^{n} & \underset{\sigma}{\rightarrow} & X }

    Other tricks may be found at the HowTo page; see for instance here.

  14. Thanks very much, Todd, I had meant to ask about this!

    At some point, I plan to have a look at svg/mathml as it stands now, to see whether I can add some nice functionality to instiki for creating commutative diagrams. But I’m not sure exactly when I’ll do this, so it’s good to have something that looks OK for the moment!

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2016
    • (edited Apr 19th 2016)

    I myself am not overly fond of mixing in a narrative with formal statements (definitions, theorems, etc),

    Nobody is. Nobody is promoting to mix narrative with technical discussion. But most mathematical articles start with an introduction before becoming technical, and that’s good style

    which is the prevailing style on the nLab

    It used to be prevailing in the beginning, years back. But if you just cursorily check what is going on in other threads on the nForum, you’ll see plenty of other material.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 19th 2016

    I rather suspect that Richard prefers something yet more formal. To continue what you quoted:

    I myself am not overly fond of mixing in a narrative with formal statements (definitions, theorems, etc), which is the prevailing style on the nLab, and, in my experience, in the vast majority of papers.

    So even most journal articles are not as he would wish.

    I think there is something rather significant at stake here. We have been motivated all along, from the days of the n-Category Café, by the thought that there is something like a “what’s really going on here” that needs some amount of narrative to be conveyed, perhaps also with the idea that “things properly understood have a deep simplicity”.

    This ideology is pervasive around these parts. Mike said somewhere that his writing for the blog informed his writing for the HoTT book. Would I be right, Richard, in thinking that you are not overly fond of its style?

    • CommentRowNumber17.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 19th 2016
    • (edited Apr 19th 2016)

    Regarding #15: I do not think that we disagree on anything, Urs. I am not at all against exposition, both through an introduction and elsewhere in the text. The way that the entries that I have begun on cubical sets look at the moment should not be taken as definitive with regard to how I would like them to look in the end.

    Regarding #16: The only point at which, it seems to me, that I differ from convention is that I like the technical parts to be formally arranged into environments, and I like to avoid a ’chatty” style. Whereas most papers in my experience will provide some kind of connecting text between the formal environments. In my opinion, such text is often vacuous (“This brings us to the first of several lemmas”); sometimes leads to confusion over where the formal mathematics begins and ends; often goes hand in hand with a chatty style of proof that can be hard to follow, and hops over details; and, if it has some content, is difficult to cite.

    However, I tried to make clear in #11 that this is only a matter of presentational style that I happen to prefer, not a matter of right or wrong, and that, in the end, the pages I am working on will probably not be so different from many others on the nLab.

    In summary, I don’t consider that anything significant is stake; in fact, I consider it rather a trivial matter. I am myself entirely in agreement that all deep ideas are rooted in simplicity, when properly understood and explained. I don’t see anything in what I am doing that contradicts this. To re-iterate, I am all for exposition; it is only a particular style of incorporating the exposition that I am not fond of; and I am not suggesting that anybody else should change their way of writing, I am only explaining that I prefer to write slightly differently.

    I am not very well acquainted with HoTT book, so do not really have an opinion on its style.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 19th 2016

    Thanks for explaining your position, Richard, and reassuring me. The kind of chattiness you mention doesn’t seem terribly helpful.

    • CommentRowNumber19.
    • CommentAuthorRodMcGuire
    • CommentTimeApr 20th 2016

    One reason to have a brief overview at the top is that when someone Googles a math phrase Google might choose to extract that summary as an overview and present it before search results, or as a right side bar. Google often does this for things that can be found in Wikipedia such as google.com/search?q=sea+slug

    I know nothing about Google’s algorithms to choose a top or right summary or nothing, nor of how much of the selection process is hand crafted. For example, why does google.com/search?q=natural+number extract a top summary from WhatIs.com rather that Wikipedia (the top search result) and why does google.com/search?q=Lawvere%27s+fixed+point+theorem extract a top from MathOverflow rather than nLab (the top search result)?

    Anyway, this is a long-term issue and Google’s rules may be changed by the time that anybody reads this. Having nLab results promoted may depend on having a nLab fan on the Google search team. (I wonder how much Google has worked with Wikipedia in order to structure the entries into an extractible form?).

    Anyway I am mostly pleased with Google’s highlighting and prioritizing of search results (except when the results might be considered consumer items or things like food recipes, or fashion, or celebrities when I am not looking for them - such may cause clogs).

    • CommentRowNumber20.
    • CommentAuthorfastlane69
    • CommentTimeApr 20th 2016
    • (edited Apr 20th 2016)

    and also rather verbose.

    .

    In summary, I don’t consider that anything significant is stake; in fact, I consider it rather a trivial matter.

    I am a no-nothing un-affiliated with the n-Lab but I strongly disagree with these viewpoints.

    Some of us newcomers consider that verbosity non-trival and, in several cases, that verbosity is the only starting point or “hook” that I/we have to understanding the technical aspects.

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