Talk:Inverse limit

Terminology source?

The intro paragraph introduces the term "injective limit" for the direct limit. Is this terminology standard? I have not heard of it. My guess is that the author mixed up injective and *inductive*. Link to the edit which added the claim. I will replace injective by inductive unless somebody comes up with a source for 'injective.' https://en.wikipedia.org/w/index.php?title=Inverse_limit&diff=prev&oldid=1049708539

 — Preceding unsigned comment added by 100.34.251.31 (talk) 19:05, 29 August 2022 (UTC) 

Article has become too long-winded and unpractical

The old version of this page was extremely useful as written. The current version (3/7/2021) seems to be better suited for a project like nLab rather than Wikipedia. Less is more here. WP:BECONCISE Furthermore, the edit history is plagued with single line edits (~700 edits since October 2020, often only minutes apart), making it impossible to have any version control. I tried rolling back the page to its September 14, 2020 version, but I don't have the credentials to do it.

Derived functor

There should be some mention of the derived functor, ${\varprojlim }^{1}$. Likewise for the direct limit, if somebody happens to know how that works.

Done long ago.

Decreased clarity

I think that this article has changed for the worse. The version of 14 September 2020 was, for example, much clearer and more to the point.

The definition now given is overtly complicated. The point seems to be lost by introducing many different notations. I think the article should be more closely written to how limits are actually used by mathematicians in practice.

I advice to write the definition short and succinct, using the full language of category theory, and then offering an unpacking paragraph for the interested reader. — Preceding unsigned comment added by 85.230.178.159 (talk) 15:57, 10 June 2021 (UTC)

Lost

I'm a little lost on the Formal Definition. Theres 7 different 'i's. Which 'i's are bounded to which expressions? Could someone perhaps use a different letter for different expressions? —Preceding unsigned comment added by 68.81.242.10 (talk) 18:56, 7 October 2010 (UTC)

I've had a go at clarifying this. Is it better now? maybe some editors think it is now too fussy. ComputScientist (talk) 19:07, 7 October 2010 (UTC)

"Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category."

The article says "Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category." This can give the impression that inverse limits always exist in a category of algebraic objects. I don't know to which extent this is true, but here is a counterexample if the poset is not required to be directed : let I = {a, b} be a set with two distinct elements, ordered by equality (thus not directed); let K_a a field with 2 elements, let K_b a field with 3 elements. The (only) corresponding projective system has no inverse limit in the category of fields, because there would then exist field homomorphisms from the limit field to K_a and to K_b, which is impossible since field homomorphims are injective. Additionnally, if the poset I is empty, the inverse limit doesn't exist in the category of fields, since there are no final objects in this category. Marvoir (talk) 07:58, 7 October 2012 (UTC)

"Short exact sequence of inverse systems"

Morphisms in C^I are not defined in the article. Is it a full subcategory? Or are the morphisms homomorphisms that commute with the transition maps? Surely the short exact sequence must have something to do with the structure of the transition maps?! 129.215.104.100 (talk) 13:54, 26 April 2013 (UTC)

Gluing or ungluing?

The article at the present states that the inverse limit amounts to a glue together operation. Perhaps some people can feel this way. I worked (did research) with inverse limits quite a bit, and my intuition is just the opposite: the inverse limit, when projections are surjections or epimorphisms (a typical case), is a result of gradual ungluing consecutive spaces--the approximating spaces are glued in themselves, but less and less so in the limit. When projections are not onto then a different intuition applies; in particular the intersection is a special case of an inverse limit. Once again, there is no gluing here, not at all. Wlod (talk) —Preceding undated comment added 08:40, 16 May 2013 (UTC)

I agree. Gluing is usually an instance of colimits, i.e. inductive limits. Even for sets, pullbacks are intersection while pushouts are unions, and the latter feels much more like gluing than the first,even more so in the category of topological spaces. It feels weird to begin the page with such a statement! 37.161.220.0 (talk) 09:38, 20 January 2020 (UTC) Qwertj

Agreed also! Gluing is for colimits. Jesuslop (talk) 20:55, 19 May 2022 (UTC)

Two redundant pages, merging needed ?

The following page Inverse system also deals with Inverse System, in the language of categories and functors. Maybe it could be integrated as a part of this page ? At least, link between them are needed. --Tilwen (talk) 13:52, 9 December 2015 (UTC)

• I will include links, but I would not recommend merging; in fact, I oppose it. For example, vector and vector space are two distinct articles. One might recommend to clear up some redundancy which might be present; I currently don't have the time for that. --Mathmensch (talk) 17:23, 6 January 2016 (UTC)

• Hmmm... the article Inverse limit already defines Inverse system, and in fact, it defines it in a simpler and more readable fashion than inverse system does -- even if you know basic category theory, the current inverse system article is obtuse and pointlessly arcane, missing an intuitive overview, missing examples. So, OK, maybe a merge is not appropriate, but clearly inverse system needs a cleanup. 67.198.37.16 (talk) 19:11, 5 July 2016 (UTC)

Difficult to use and incommensurate with parallel articles

The article has become very difficult to use. Already the third paragraph of the introduction, on threads, is sure to alienate non-experts. This topic should be much more accessible; it is a fundamental notion in many areas of mathematics, and concise, useful descriptions are easy to find.

Additionally, the article is now incommensurate with the article on direct limits, which is the dual notion in category theory, and should read roughly in parallel. See also the articles on products and coproducts, which are far more simply and clearly presented, and whose contents are just as relevant. Ebrussel (talk) 19:25, 9 July 2021 (UTC)

@David Eppstein: Could you have a look at this article? Seems to be a one man show, and I don't know the first thing of category theory. Hope I'm not a bother here. Horsesizedduck (talk) 01:31, 12 July 2021 (UTC)

Oh, hell, it's one of Mkgrupa's messes again. Horribly WP:TECHNICAL to the point of unreadability. Mkgrupa seems more interested in Mathematics Made Difficult than in making any attempt at being informative to people who are not already intimately familiar with this material. I think the version by User:Joel Brennan from early February, before Mkgrupa made hundreds of edits, is far more usable, and have reverted to that. I don't think there's any point in trying to salvage value from the later edits. But in general, this sort of issue is better brought to the attention of WT:WPM than to me personally, because you can find a larger number of editors there one of whom is more likely to take a personal interest in the specific topic of the article. —David Eppstein (talk) 01:59, 12 July 2021 (UTC)

Assuming that both versions are technically ok (which I cannot judge), why not publish both of them, under different titles (similar to group (mathematics) and group theory)? - Jochen Burghardt (talk) 08:55, 12 July 2021 (UTC)

Evaluating the article, it seems that the verbose, Mkgrupa version is not technically okay for the following reasons:

• The vast majority of it is uncited and quite a bit of that is original research, especially proofs or explanations. If it can be found in several textbooks or journals, we should cite them and summarize them. If not, it's not wiki-worthy.
• Most of the article is written like a textbook or mathematical journal article, both of which it oughtn't be (WP:NOT). The entire purpose of Wikipedia scientific articles is to give a broad sketch and then link to reliable sources for readers that need more. By trying to provide a comprehensive overview of the subject and not providing any references for individual claims, the Mkgrupa version of this article is simultaneously too confusing for new readers and not helpful enough to advanced readers.
• It reproduces material that is better placed in other articles. In fact, some sections of this article (such as Pullbacks and Products) are comparable in size to the entire main articles on those subjects! It's almost like someone took a normal article and expanded each blue link by including the whole article. It's like writing an article about blood and saying "Blood is a type of human tissue in liquid form and contains hemoglobin, which contains iron, an element of the periodic table with atomic number (a number referencing the number of protons and electrons in an atom) 26.

If the material can be rewritten so that each major claim is referenced with a textbook or journal article, I would recommend shoving each big chunk into its own article and summarizing it here. However, unsourced original research is not good for those articles either. So, for now, I recommend not using the reverted material. However, I'm just one editor and respect whatever consensus happens.Brirush (talk) 15:15, 12 July 2021 (UTC)

Don't use the information that I added. Also, the current version of this article has 1 inline citation "This same construction may be carried out if the $A_{i}$'s are sets,[1] semigroups,[1] topological spaces,[1]" Inline citations are missing and need to be added. Mgkrupa 15:28, 12 July 2021 (UTC)

May I suggest the following? Suppose a new article was created, say, in draft space, that used the February version of the article as an introduction, followed by Mgkrupa's extensions and elaborations. The result could then be submitted to (for example) the Wiki Journal of Science. Once reviewed, revised and accepted, it could then be moved to article space, either under the same name, or under a new title: Review of Inverse Limits. This path has already been taken by others: see Category:Wikipedia articles published in peer-reviewed literature. I am suggesting this path, because the current path, of asking existing WP editors to jump in and review, correct, revise and improve Mgkrupa's version is an overwhelming demand on time and resources. The participation in WikiProject Mathematics is too thin to do this justice. 67.198.37.16 (talk) 19:27, 13 July 2021 (UTC)

The only content that is original is the interpretations, the notation $\operatorname {Sys} _{M}$ which I chose to try to minimize the amount of unnecessary indices displayed on the page (with limits, it is easy to get lost in a sea of indices), and some of the concrete examples (such as limits of $\left(\left(\mathbb {R} ^{i}\right)_{i\in \mathbb {N} },\left(\Pr {}_{ij}\right),\mathbb {N} \right)$ and limits of cones into the space $\mathbb {R} ^{\mathbb {N} }$) are original but they are so trivial, it would be like condemning some particular example of a metric space $(M,d)$ and point $p\in M$ where the closure of an open unit ball $\operatorname {Cl} _{M}\left(B_{1}(p)\right)\neq \{m\in M:d(m,p)\leq 1\}$; technically examples like this are original research but it is so trivial that the example is essentially interchangeable with any number of other trivial examples that get the same point across, which is why no one cares about such examples and why they appear throughout mathematics article. In fact, there is nothing the least bit "hard" about 95% of the statements that former article; its length is due to it being more detailed than it needed to be. But I wrongly wrote that article as a textbook, which I should not have done per WP:TEXTBOOK. I looked at the original state of the article and thought to myself "this article is useless as an aid to students learning about inverse limits" so I wrote the article that I thought would help people learning about inverse limits, emphasizing some of the finer easily-missed technical details that I wish I had initially known about (hence why some statements have proofs while others don't). I should not have done this. Many people do not like the article that I wrote (had I been aware of this earlier, I would have stopped editing it) and I also don't want to deal with rude remarks from users like David Eppstein. I do not care about the content that I added. I made a mistake editing this article and I've learned from it. I do not want to deal with this article anymore. Mgkrupa 01:58, 16 July 2021 (UTC)

Thank you David Eppstein for having rolled back the article to a more readable version. I am sure Mgkrupa had good intentions, but the old version is a lot more usable, even if it does not contain all the fine points that could be discussed about this topic. PatrickR2 (talk) 05:28, 8 August 2021 (UTC)

Prufer group example

The third item in the list of examples, about the groups $\mathbb {Z} /p^{n}\mathbb {Z}$ and the homomorphisms $\mathbb {Z} /p^{n}\mathbb {Z} \rightarrow \mathbb {Z} /p^{n+1}\mathbb {Z}$ induced by multiplication by $p$ is a direct system and not an inverse system, with a resulting direct limit instead of inverse limit. The same example is already present in the Prüfer group and Direct limit articles and should be removed from the current article. If people agree, I can remove it. (I know every direct limit is an inverse limit in the opposite category, but that's not useful here.)

Furthermore, for my own benefit, I think the direct limit would be the Prüfer group, and not the circle group $\mathbb {R} /\mathbb {Z}$ as stated. Would you agree? PatrickR2 (talk) 05:59, 8 August 2021 (UTC)

This has now been removed. PatrickR2 (talk) 22:49, 11 August 2021 (UTC)