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• CommentRowNumber1.
• CommentTimeAug 24th 2014

I have renamed the entry pretriangulated dg-category to stable dg-category. I think this is a logical move that emphasizes the close relationship with the notions of stable (infinity,1)-category and stable model category. If anyone disagrees I would be interested to hear why. I have also edited the body of the page to give an exposition more in line with modern references like Cisinski-Tabuada. However I have adopted the term dg-presheaf for what is usually called (right) dg-module. This seems much more natural to me, but again I am open to hearing any arguments against it. When I get a minute I will also update the entry dg-category to match the conventions of this page.

Also, I never liked the term “quasi-equivalence”. I think that this should just be called equivalence of dg-categories, or maybe Dwyer-Kan equivalence if this is too ambiguous. Any thoughts?

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

I have created version 1 of pretriangulated category. I am glad that people work on it and the page looks better now, I am not happy with the new title except as a redirect.

Who is the title page for ? The titles are not a battlefield for name-changing agenda but a key word for finding. And most people whom I know have a main complaint about the $n$Lab that it is too hi-brow, unfriendly to conventionally educated mathematician and use much of its own lingo. Pretriangulated dg-category is a standard term. If you want a nonstandard something it can be a redirect. Once you convince over 50% of main researchers in the field of dg-categories to change the terminology it will be a moment to change it here. Before that we loose users and loose the feeling of unity with the tradition from which we taken most of the material. also heard complaints from people that much material is borrowed and than reshaped n nonrecognizable form so authors of original work feel offended.

Besides there is a mathematical reason not to do that. Namely

1. there is more than one model category structure on dg-categories and in some of them all dg-categories in characteristic zero can be considered as being avatars of stable infinity categories, I think.

2. pretriangulated dg-categories are also interesting in characteristic zero, where the stable interpretation is not quite to the point!

3. the terminology in $A_\infty$-world.

• CommentRowNumber3.
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

I thought that the whole point of the nLab is to expose the n-point of view, not to cater to the “conventionally educated” mathematician or to become another Wikipedia. And this is not the only instance of the nLab adopting different terminology than found in the literature. In fact the nLab is probably the one who should take a lead in using new terminology when it is necessary. In this case, I’m pretty sure I could convince the experts on dg-categories (e.g. Töen or Keller) that this terminology is better (in fact I bet they probably already believe this themselves).

1. Yes, there are many model structures on dg-categories. The ones usually considered are the usual Dwyer-Kan model structure, which can then be localized to the quasi-equiconic model structure presenting stable dg-categories, and then further to the Morita model structure presenting karoubian stable g-categories. I am planning to write some details on this at model structures on dg-categories. However I don’t understand what this has to do with the terminology. Or is your point that stable dg-category should mean “karoubian pretriangulated dg-category” (due to the comparison result of Cohn)? That seems reasonable maybe, but I should note that Cisinski has an alternative version of the comparison where stable linear infinity-categories are equivalent to (not necessarily karoubian) pretriangulated dg-categories.

I don’t understand what you mean with 2.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

I thought that the whole point of the nLab is to expose the n-point of view, not to cater to the “conventionally educated” mathematician or to become another Wikipedia.

nLab is a practical tool for our work and depends on wider community of supporters, not a blog which would be primarily for propaganda or discussion list. We do have discussion list and we do have a couple of blog sections for these purposes (nForum Notices and nCafe). It happens to be that the community which has started it has several central common biases in interest like the interest in higher categories, in mathematical physics, in foundations (including logics, philosophy, model theory, topos theory, type theory); the “n” is the slogan for this whole circle. This circle is quite a good approximation to Urs’s own interests in last decade (constantly expanding!). People who wrote papers on pretriangulated categories should not be scorned with attachments like “conventional” nor be approached with such an attitude – these are among the top people who made our work and that beautiful n-subject possible. Most importantly, $n$Lab depends heavily on accessibility and support from wider community. One can always expand the extra-hi-brow points of view in properties and additional definitions, there is no need to replace the standard terminology in the basics like titles and so making unrecognizable (and who will know how to find the new terminology unless one already read the entry?). It is sometimes good to choose new terminology at places where there are hesitations and where there are multiple formalisms; though it should normally be left to the creators of the formalisms and theories. Classical 1-categorical community has largely allienated themselves from the wider mathematical community by insisting on their hi-brow specialized words. When a practical mathematician says strictly full subcategory they say full and replete knowing that an outsider won’t know what replete means. As a consequence their work is less known and most category journals were not listed in main databases which you need to get a position. Only recently TAC was included in Current Contents and other databases; even MathSciNet is not counting citations in most category journals like Cahiers as many of us discussed before (ask Julia Cheng who done some analysis of the MathReviews case). Some of us suffered in career (including with job positions) because category theory has historically lower status among scientists. Founders of this club, like Baez, spent much effort in trying to popularize the subject and cross the barriers. For young supertalents like you it may be attractive to be a extremist with uncompromising manners, elitistic style and terminology, and refrain from classics. I heard that some mathematicians dislike main contributors of $n$Lab because they redo their work and present it as new. It is not true that we do present it with such a meaning, but the attitude happens partly on the fault that we do not have time, patience or attitude to avoid such impressions. I thought that doing such a service to the community will help well known people from $n$Lab in their career (beyond their own usage of $n$Lab). But I heard that some people oppose some of the best people in $n$Lab in their career (even Urs) for doing $n$Lab kind of things (by wrong rationale which I just explained about seeing us as redoing somebody’s else’s work and making it unrecognizable!).

$n$Lab is used much beyond the higher category community and also supported by larger community of people who are more common to share standard practices in mathematics. We apply our category knowledge to concrete problems and would like to have those $n$Labified to larger extent. We do care that the contributors are sensitive mathematicians and writers and responsibly contribute to $n$Lab more than they have an $n$-lineage and blood.

model structures on dg-categories. However I don’t understand what this has to do with the terminology

The meaning of “equivalent” must be stated in some formalism for the collection of all dg-categories, e.g. model structure.

By 2 I mean that dg-categories over a field in prime characteristic are not a model for (k linear) stable infinity categories. So the two terminologies will never be fully compatible. This kind of result was known before new results of Cohn, which I was not aware, so thanks for the reference.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 24th 2014

Zoran,

you keep reiterating allusions to complaints you hear as now in #2 above, where you say

also heard complaints from people that much material is borrowed and than reshaped n nonrecognizable form so authors of original work feel offended.

I urge you please to complement such statements with a minimum of details that would give “us” a chance to react. Keep the person who complained anonymous if necessary, but let us know which entry comes across as offensive and why.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 24th 2014
• (edited Aug 24th 2014)

Regarding the question of how to name the entry:

I can see both points. Adeel’s suggestion is clearly what the community ought to have chosen as the standard term, while it hasn’t. (This is a mild case, at times I am being amazed at just how careless people choose terminology.)

I am agnostic about how best to resolve it here, but I make this remark from experience in similar cases:

for Google searches, what the entry is actually titled by is not as important as which keywords prominently link to it. If lots of $n$Lab entries refer to the redirect “stable dg-categories”, then Google will draw its conclusions.

Therefore as a compromise we could keep the entry name “pretriangulated dg-category” and then make sure that the entry text right away explains that that this is the traditional terminology but that conceptually the actual meaning is “stable dg-category in the sense of ’stable $\infty$-category’ “.

(And not to forget that also the term “stable ’$\infty$-category” could be criticized (and has been here before) since what is stable here is not so much the category itself but rather its objects. Math terminology will never be optimal or even just good, we’ll have to accept it. )

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeAug 25th 2014
• (edited Aug 25th 2014)

Re: #5 hear hear!

• CommentRowNumber8.
• CommentTimeAug 25th 2014

Ok, I’ve renamed the page back to pretriangulated dg-category.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeAug 25th 2014
• (edited Aug 25th 2014)

5 Urs. I did before tell about milder complaints about abstractness and so many times before as they are common among most external users whom I meet, but I am sure this is the very first time I speak of the other problem of the “authorship” complaints, as I became aware of the seriousness of the issue only within last several months. Please do not mix totally different issues. I am trying to ammortize this by concrete work (did anybody include more extra references in $n$Lab ?), I can not press the people who I talk to to detail their problems beyond trying to explain them how the things happen so that they have more understanding (but most serious complaints I became aware through 3rd, intermediate, parties who are not responsible for details and in those cases I know just of generalities, sometimes not even names).

chance to react

We are editing hundreds of entries. We react when we can, but when a major mathematician (read somebody with extensive contributions to the fields we cover) has a complaint it is usually against very many issues. One of my collaborators got a complaint in a review of his work that his work s a weak work from “orbit of John Baez” or alike arbitrary statement probably based on some history this referee had with appreciation of his/her own work. It is largely fault of the referee, not of my collaborator. However, being generally careful in this or that manner can diminish such issues and neglect can aggravate and sometimes really cause new histories to start. The attitude of calling the predecessors outdated, classically educated, “wikipedia” and similar will not help. The $n$Lab should be the meeting point for the points of view of various schools of thought, and as far as this succeeds it will be its strength. You know that I have spent massive efforts trying to achieve that with my contributions.

About pretriangulated again: the question of nonzero characteristic is I think decisive to me. There are similar technical notions in algebra where a geometric notion behind is morally better but the technical realization may depend on subtle conditions. For example, quasi-free algebras in the sense of Cuntz and Quillen are noncommutative case of formal smoothness. Now it is tricky to rename it so, as the commutative formally smooth algebras are not a case of this but rather the case of formally smooth in that commutative (sub)category. That is to say, the subcategory of commutative algebra does not inherit the geometric characterization. So, the technical notion coming from algebra is still sometimes useful, though in specialized work one could go with the appropriate formal smoothness terminology and theory as far as one pleases.