# Talk:Refinement (category theory)

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 This article was accepted from this draft on 22 March 2018 by reviewer GeoffreyT2000 (talk · contribs).
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The following comments were in the draft before it was accepted. GeoffreyT2000 (talk) 23:03, 22 March 2018 (UTC)

Sorry, but I have too difficult a time distinguishing the maze of definitions of category theory from their parody at https://cemulate.github.io/the-mlab/ to have much in the way of useful opinions. I can't see anything here that is recognizable as matching https://ncatlab.org/nlab/show/refinement though, and I would think that the nlab definition of what a "refinement" means in this context would be the more canonical one. The book reference without any specific page numbers is completely useless, most of this is completely lacking in inline references, the other two references listed are WP:PRIMARY, and the writing here is far too WP:TECHNICAL. I'd want to see this rewritten based on secondary sources, in a style that could be read by non-specialists, with more thorough referencing, before accepting it. —David Eppstein (talk) 04:35, 8 February 2018 (UTC)

I wrote this draft because this is the dual notion to the notion of envelope, so that was just for symmetry. The source is my monograph [1], where this notion is discussed in detail (and it is completely different to the notion of refinement mentioned in nLab, https://ncatlab.org/nlab/show/refinement). I think there are no secondary sources. If this contradicts the rules of Wikipedia, I think we should delete this page, excuse me for my mistake in this case. Eozhik (talk) 05:47, 8 February 2018 (UTC)

I have just added pages in the notes. Of course, this does not change the situation too much. Eozhik (talk) 06:00, 8 February 2018 (UTC)

It looks plausible, given that it seems to be a cut-n-paste of envelope, with all the arrows reversed. .. Oh, wait, I see that you wrote that, too. I don't have the mental energy to think about what this is actually saying, so I will nit-pick instead. (1) You should mention that there is this unrelated concept of refinement, which is by the way a staple of topology textbooks, so that the reader is not left thinking that maybe this is some categorification of that particular concept. (2) Please do not use "bornologification" in the first sentence. Or the second, or the third. (3) "saturation of an LCS" -- is there a WP article for that? Because it should be linked, and if it doesn't exist, it should be written. Viz. something that provides a simpler, low-brow example. (4) from what I can tell, ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$ are not just any classes of morphisms, but are morphisms that have X as codomain. Right? You should say so. Is there a secret comma category in here? Or rather, a slice category, so that ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$ are both slice categories??? I know what comma categories are, so if that's what ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$ are part of please say so. (5) When you say "there exists a unique morphism", this is normally called a universal morphism Is that what you've got here? (6) which then makes the whole thing smell of cone (category theory) or maybe limit (category theory) but I can't quite tell. Some kind of weird or special case limit or cone? Like a cone with respect to ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$??? Be kind to the befuddled reader- weave in topics and concepts that the reader might already be familiar with, wikilinking to those WP articles to place it in context. (7) Before giving the hard-core definition, it would be good to give a general overview, in "plain English" -- it could say something like this: "whenever one has a Banach space, or a convex topological vector space, one typically wants to refine the blah (??) so as to get a finer blah. This allows blah". Motivate the reader for why they should even care to burn the brain cells to understand the rest of the article. So maybe take example 1. from the envelope article, and spin it out into 4-5-6 sentences, and place that before the formal defnition. I mean, I know basic stuff about TVS'es and Banach algebras, basic stuff about category theory; make this article easy to read. (Not every WP math article has this "informal introduction" but many or most should) (8) Use more wikilinks. Like stereotype algebra, C*-algebra Banach algebra - we've got articles on these already. Link them. (9) I see that you wrote stereotype algebra and wrote stereotype space six years ago. That makes you an old-timer. Why are you asking for reviews of your drafts?? 67.198.37.16 (talk) 08:32, 26 February 2018 (UTC)

67.198.37.16, thank you for the review. It will take me some time to understand whether it is possible to correct the text in the way you suggest, or, perhaps, it will be better just to ask people to delete this page. I'll answer your questions here anyway:
1. I think, this is indeed another concept, it is not related to the refinements used in topology. I do not know how the situations like this are resolved in Wikipedia.
2. I agree about the "bornologification". I do not think that this is a popular term, and this is one of the reasons why I have doubts about the usefulness of this article in Wikipedia.
3. The same with the "saturation of an LCS" (and the corresponding page does not exist in WP).
4. For the formal definition of refinement it is not necessary to require that ${\displaystyle X}$ is a codomain of ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$. If there are no morphisms in ${\displaystyle \Gamma }$ which have ${\displaystyle X}$ as a codomain, then the further conditions are not fulfilled (and the refinement does not exist). On the other hand, if there are no morphisms in ${\displaystyle \Phi }$ which have ${\displaystyle X}$ as a codomain (but the morphisms in ${\displaystyle \Gamma }$ with ${\displaystyle X}$ as a codomain exist), then the further conditions are fulfilled automatically, and the existence of refinement depends on the other properties of the category (for example, if this is a category with zero object then the refinement is always zero in this situation)[2].
5. I did not encounter the term universal morphism in the context you decribe. If this is popular, I have no objections.
6. I don't think that it will be easier to read this text if we explain this notion in terms of cones or limits. Maybe for specialists this will be useful, I don't know.
7. I also don't know if this notion can be explained for non-specialists in "plain English". This also inspires me some doubts in the usefulness of this article.
9. I don't know Wikipedia's customs well, excuse me.
In addition, I have just found that the first picture is wrong. I'll correct it. Eozhik (talk) 11:20, 26 February 2018 (UTC)
Well, don't delete the article; its always good to have content. Regarding codomains: So I actually tried to understand this article, just now, (unlike last time) ... and I still don't understand it. The diagram shows ${\displaystyle \forall \varphi \in \Phi }$ which suggests that all ${\displaystyle \varphi }$ have X as the codomain. Because that is what ${\displaystyle \forall }$ means. Also, since you are using the symbol ${\displaystyle \in }$ this implies that ${\displaystyle \Phi }$ is a set, not a class. (So that ${\displaystyle \Phi }$ is a Hom-set.) So can K be any category, or must it be a small category, or a locally small category?
The article starts with the words "Suppose K {\displaystyle K} K is a category..." and so the reader naturally assumes that the article describes something that can hold for ... almost any category, say, the category of sets or the category of groups, and so as I read it, I try to imagine that K is Sets, and then I try to imagine an enrichment/refinement for sets. I can't (I'm pretty stupid), so I jump ahead to the examples: which suggests that K cannot be "any category", but can only be the category LCS. Is that correct? Are there examples for other categories, e.g. Sets, groups, group actions, sheaves?
Also, please note: the word "enriched" is already used in category theory to mean enriched category. I believe that this would be a completely unrelated concept. However, enriched categories do something funny to Hom-sets, while here, you have some class ${\displaystyle \Phi }$ ... (which is maybe a Hom-set?), and you are adding some kind of funny additional structure to it. I am very much a novice in these areas, so you do not have to take my comments seriously. 67.198.37.16 (talk) 18:21, 28 February 2018 (UTC)
67.198.37.16, the writing ${\displaystyle \forall \varphi \in \Phi ...\exists !\varphi '}$ in the first diagram is supposed to mean the condition

${\displaystyle \forall \varphi \ {\Big (}\varphi \in Hom(B,X)\cap \Phi \Rightarrow \ \exists !\varphi '\ (\varphi '\in Hom(B,X')\ \&\ \varphi =\sigma \circ \varphi '){\Big )}}$

So there is no need for ${\displaystyle \Phi }$ to be a subset in ${\displaystyle Hom(B,X)}$. At the same time there is no need for ${\displaystyle \Phi }$ to be a small set, it can be a class (containing ${\displaystyle \varphi }$ as elements).
I indeed assume that the category is locally small (i.e. all ${\displaystyle Hom(B,X)}$ are sets), but this is just because I do not understand how this can be otherwise: I was brought up in the Morse-Kelley theory, I translate everything into this language, and when I see a mapping ${\displaystyle (B,X)\mapsto Hom(B,X)}$, for me this means that ${\displaystyle Hom(B,X)}$ must be small sets.
In the category Sets the enrichements and the refinements also can be constructed, for example if we choose ${\displaystyle \Gamma =\Phi }$ as the class of all embeddings (not just mappings, but embeddings) with ordinal numbers as domains then the refinement of a set ${\displaystyle X}$ is the maximal ordinal number containing in ${\displaystyle X}$ (it can happen that this is ${\displaystyle \varnothing }$).
Of course, apart from Set and LCS there are other categories where (actually, everywhere) one can construct other examples, but since this notion was suggested not long ago, nobody did this up to now. In the source that I cite the simply connected covering is an example of a refinement (but under a more complicated definition of refinements, that is why I do not give it in this text).
The enrichments in this text indeed have nothing in common with the enriched categories.
The more I think about this, the more I tend to think that this text needs to be deleted. I actually do not see a necessity to make efforts for preserving this draft, I would prefer if somebody would delete it. Or, I don't know, perhaps, one could conservate it somehow for the future. Eozhik (talk) 19:42, 28 February 2018 (UTC)
By the way: perhaps it is possible to merge this text with the article "envelope"? One could make a little section there like "Dual notion: refinement". Eozhik (talk) 05:39, 1 March 2018 (UTC)

I see nothing wrong with merging; of course, it is up to you as a manual, labor-intensive process. I apologize for the confusing questions that I ask (and you are doing a good job of answering them) - I guess my central point here is that it is wise to assume that the typical reader will be "just like me" -- will know just enough math to be interested in reading this, but not enough math to really guess what it's about. I stumbled across this by accident, while looking for something else. I read the first sentence, decided its interesting, and probably something I should know and understand.. and then got stuck. The concept is interesting; perhaps you could assign the task of creating examples to some of your students? Besides Set, the other interesting categories (to me) would be the monoidal categories that are not cartesian closed. Most of the math world is mostly interested in cartesian closed ones, so an example for some category of group actions or continuous group actions (principle fiber bundles) or some other "quotidian" sheaf or topos would be nice. A less polite way of saying this is that I am both stupid, and lazy, and expect insight to be served to me on a silver platter with champagne. 67.198.37.16 (talk) 19:17, 1 March 2018 (UTC)

Publish/Keep: Anyway, to be clear: I have no objections to keeping and publishing this article as it is. It does need more work, but then, so do 98% of the other math articles on WP, almost all of which are deficient or woefully lacking in some way. So it would be in good company, at least :-) 67.198.37.16 (talk) 19:26, 1 March 2018 (UTC)
Dear 67.198.37.16, there is no need to apologize. First, there is nothing confusing in your questions. This is normal that when somebody is asking questions about the field where he is not a specialist, the specialists perceive his questions as too simple. At Mathoveflow I am asking questions on topics beyond my field of expertise, and all the way I have a feeling that most of what I am asking is strange for specialists (and they show me this from time to time). But when doing this (and this is the second thing in what I wanted to point out) I follow the idea that it is specialist's duty to explain the details for newcomers. Since otherwise the system stops working. I am againts the idea that to explain something it's enough to drop a hint. It's clear for me that it's not enough. A specialist must make efforts on explaining what he is doing. This is not optional, he must spend his time on this.
This is actually why I am writing articles here in Wikipedia. For me this is not the main tool, but one of them. Wikipedia's advantage is that it is a good place for references. I can give a link to a WP article, for example, to envelope, and all my listeners immediately understand what I am talking about.
I do not know what to do with this article. If there is a possibility to publish it as it is, I would prefer this, because inside the article devoted to envelopes this text will look a bit like a foreign body. I am sure the new examples will appear in future because we are actively working in this field.
On the other hand if there will appear a person who could delete this text (I do not know how to do this), this will save me from the necessity to watch it. That would be a relief for me. Eozhik (talk) 08:43, 2 March 2018 (UTC)
1. ^ Akbarov 2016, p. 52.
2. ^ Akbarov 2016, p. 53.