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• CommentRowNumber1.
• CommentAuthorPieter
• CommentTimeJan 31st 2019
Hi,

being new to the forum and the nLab, I would like to make sure my intentions on using this align with yours.

I'm a researcher in theoretical computer science, interested in formalizing the semantics of models of execution of real-time and cyber-physical systems.
While working on a set of lecture notes on the topic, I noticed I like to use category theory "as a compass".
I am developing my own behavioral models, and by considering them as objects and morphisms in a category, I have found many small `tweaks' that make my definitions work better.
I go by the rule: "if the standard category theoretic definitions give me what I hoped to get in the first place, I'm on the right track", and that has helped me a lot (even though it also slows me down terribly).

Having done this for a while, I'm now coming to a point where notions like product, limit, and monomorphism indeed reflect the intuitions I would like them to reflect.
Furthermore, recently I have found out that I should actually be looking at my categories as concrete categories, and then embeddings turn out to be useful.
Together with a colleague we even found 'upgrades' of known categorical definitions, which seem helpful.

And that's the point when it all of a sudden stopped... or at least changed...

Now, whenever I meet a new topic I would like to investigate, I try to think of the 'natural' way to represent it in terms of objects and morphisms,
rather than thinking about the best way to do it "within" my own category (categories) of choice first. And while this is fun, it leaves me with the
problem that I invariably come up with definitions that sound very plausible, but that I cannot find (quickly) in the basic literature on category theory.

For example, yesterday I decided I need a notion of "remainder". The practical problem at hand requires me to study a morphism from the behavior
of a complex system to the behavior of a component, and then "divide out" the behavior of the component to obtain the behavior of "the rest of the system".
I consider the complex system as something that defines a "relation" between components, by synchronizing them in some unknown way.
I think that jointly monic families (the dual of jointly epimorphic family) are a nice way to capture such relations categorically (something that is not really mentioned on that page, by the way...),
so my way of "dividing" comes down to (I don't know how to make diagrams here yet...):

* given a map f : X --> Y, find (the smallest) object Z (which I'll call the remainder of f) for which there exists a map g : X --> Z that makes the family (f,g) jointly monic.

where "smallest" means that any other object Z' with map g' : X --> Z' that makes (f,g') jointly monic has a unique map k : Z --> Z' such that kg = g'.

I think it is a rather natural notion, but I can't find it anywhere (at least not quickly, and my colleagues are getting on with the problem, so this is a 'side thing' and should not cost hours of library research).

Now, what I would like to ask (finally) is:

1) Is this the right forum to get suggestions on references to (anything like) the definition above? (If so, please comment!!)
2) Is nLab the right place to start a page and write down my notes on researching the above definition, and others that may come up?
3) How do you/we deal with "original work" on topics (even before it is published elsewhere)?
Can I just start a page on a topic, name it whatever I see fit, and "see what happens"?
How can I indicate that pages are "standard", "pretty standard", or "have not passed the test of peer-review anywhere yet"?
A page may be "needing review" because it is "standard stuff that may not have been explained correctly", or because it is "new stuff, and the definitions may not make sense really yet".
4) Should I ask permission first here, before making changes to other peoples pages (for example, adding the typical use of jointly monic families as a generalization of relations to the page jointly epimorphic family)?
Or do you just do it first, and hope for forgiveness (which is much quicker, but it is also a bit rude I suppose).

Kind regards,

Pieter
• CommentRowNumber2.
• CommentAuthorAlec Rhea
• CommentTimeJan 31st 2019
Greetings Pieter, I'm new here too so I can't really address questions 1-3. For question 4, whenever you make an edit (or create a new page I'd imagine) it automatically creates a thread here for other people to see/comment on, and if someone has an issue with the edits they can bring it up here. Accordingly it seems that the community is okay with you making edits that are well thought out, with the caveat that they might get debated here/rolled back once made if something is amiss.
• CommentRowNumber3.
• CommentAuthorPieter
• CommentTimeFeb 1st 2019
Thanks, Alec

Perhaps I should just give it a go then.
Which would make question 3 my most pressing next question.

The pages I read so far are of a rather 'enceclopedic' style.
Does anyone have an example for me of a page that really contains "notes of ongoing research"? That would be helpful.
• CommentRowNumber4.
• CommentAuthorRichard Williamson
• CommentTimeFeb 1st 2019
• (edited Feb 1st 2019)

Hi Pieter, welcome! The following are my personal opinions, not official ones, though hopefully they do not differ too much from the official ones. Thanks for checking here!

1) Is this the right forum to get suggestions on references to (anything like) the definition above? (If so, please comment!!)

Definitely! Maybe just post your question with a more specific title so people are more likely to see it.

2) Is nLab the right place to start a page and write down my notes on researching the above definition, and others that may come up?

Definitely it sounds like the nLab place is an appropriate place to record information about what you’re looking for. Naturally, try to fit it into the existing style and find existing places where it might be incorporated.

3) How do you/we deal with “original work” on topics (even before it is published elsewhere)? Can I just start a page on a topic, name it whatever I see fit, and “see what happens”? How can I indicate that pages are “standard”, “pretty standard”, or “have not passed the test of peer-review anywhere yet”? A page may be “needing review” because it is “standard stuff that may not have been explained correctly”, or because it is “new stuff, and the definitions may not make sense really yet”.

This is a tricky one. In principle, we encourage original work. I personally would very much ilke to encourage research being done in the open. Unfortunately, though, our past experience has made us a little wary. I think maybe best would be to try first to make some edits/additions to existing nLab pages, of course creating new ones if there is something missing. But for entirely new research, maybe hold off a little. If all goes well, in the end we could for example give you a personal web where you could develop your research, which you later could move to the main nLab. But if you have some research that you think it is already appropriate to create a new page about, you can always post a summary here in the nForum, and others can offer some advice as to whether a new page would be appropriate.

4) Should I ask permission first here, before making changes to other peoples pages (for example, adding the typical use of jointly monic families as a generalization of relations to the page jointly epimorphic family)? Or do you just do it first, and hope for forgiveness (which is much quicker, but it is also a bit rude I suppose).

You can just go ahead! Maybe take a few smaller steps first rather than massive edits just to get feedback.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 1st 2019

Hi Pieter. To add to what Richard said:

The nLab is made up of a combination of standard material that any maths or physics wiki might have, exposition from an nPOV perspective (again standard from within some part of the category theory using community), and then also some research notes which may include research in progress. Properly personal research tends to be confined to personal webs.

What we tend to be nervous about is someone arriving at the nLab with their own idiosyncratic agenda and starting out at the end of this list, presenting their idiosyncratic ideas as standard. There’s so much that could be written in terms of standard material, and elaborations of existing entries, and this generates the good will that might allow for a personal web.

As to

making changes to other peoples pages

on the nLab itself (not the personal webs) there are no individual’s pages.

So if you’d like to help, try out working on some pages. This generates a comment box to report on changes. You’ll soon get the idea of what’s wanted or unwanted.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeFeb 1st 2019

Welcome! As has been noted, although the nLab is definitely open to original and even unpublished research, we have to be somewhat wary of newcomers arriving and immediately starting to write only about their own research, as we have had some experience with cranks trying to use us as a platform. I think you are off to a pretty good start, but just be aware that until we know you better, it’ll help to be especially careful to also contribute to existing pages, connect the new things you write about to existing related concepts, make sure you’ve asked first to find out whether something already has a name before inventing a new one, give as many citations as possible (to other people in addition to yourself) to make the point that the concept is important and widespread, etc.

One way to make meta-notes about a page such as “work in progress”, “original research”, “this definition may not be quite right yet”, etc. is to use query and standout boxes. Unfortunately I’m not sure I can point to any pages that are currently “in progress” in this way.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeFeb 1st 2019

given a map f : X –> Y, find (the smallest) object Z (which I’ll call the remainder of f) for which there exists a map g : X –> Z that makes the family (f,g) jointly monic.

where “smallest” means that any other object Z’ with map g’ : X –> Z’ that makes (f,g’) jointly monic has a unique map k : Z –> Z’ such that kg = g’.

I don’t think I’ve seen this notion before. Can you give some examples of categories where such a thing exists? I don’t think it does in $Set$.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeFeb 1st 2019

Thinking on the fly here…

Doesn’t $g=id\colon X\to X$ give you a remainder? I mean, surely $X \leftarrow X \stackrel{f}{\to} Y$ is jointly monic (it gives the graph of $f$ as a relation, assuming binary products), and, given any other $g'\colon X\to Z'$ whatsoever, take $k=g'\colon Z=X\to Z'$… ??

But Mike’s comment makes me pause.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeFeb 1st 2019

Hmm, maybe I had the universal properties the wrong way around in my head?

• CommentRowNumber10.
• CommentAuthorPieter
• CommentTimeFeb 2nd 2019
Okay, this definitely sounds like the place to be :-)

Now, I got myself into trouble by starting a 'multiple topic post'... I still have to learn how to behave on forums like this, clearly...

@Urs: yes, I could use some help with the references. Did not come around to 'figuring out what the right syntax would be'. So if it is a quick thing for anyone, please just edit my attempt at a reference on the Prefix Order page. Otherwise, I'll figure out how in due time (i.e. when the start of the semester is over, and I have fewer education tasks...)

@Richard and David: what I get from your comments is that original research is "tricky" because some people tend to present things immediately as if it has been around forever, even though it is original and unestablished work. That is precisely what I wanted to avoid. I posted a page on Prefix Orders on "the" wikipedia, and I felt that I had to write in this nPOV style because that is what wikipedia 'feels' like. But I found it annoying. I wanted the notion to become 'visible' to a larger audience, in order to get feedback and reactions to it. It had been published, so I felt it was okay to use the term. But it doesn't feel right to treat it as "encyclopedia material" yet. That is why I got excited about the idea of nLab being an "open lab-logbook".

In general, I'd love to add "standard stuff" as well, but I have to pay attention that the work I do also "pays off" for myself. So, for me personally, I would like to create pages about things that I recently published and make links to existing work. And whenever I encounter something "standard" in the basis of my own work that is not discussed yet, I'll do my best to write a decent standard page for it.

The most important reason for me to have an "open logbook" about my current research, is that the category theoretic musings are really a 'side job' for me. I have hardly any colleagues close to me that can give me input. So working online and immediately getting reactions like the ones above, pointing out possible flaws in my definition, really really helps me. Question is whether I should create a page, or just post a message on the forum for such a thing. But I feel posting a page is "better" for me, because it shows the "current status" of my work.

I get the point that I should "earn" a place here first, so that I can have personal pages for this kind of stuff. That keeps the "original work" focused in a relatively controlled area of the wiki.
So for the time being, I'll try to stick to making edits to pages and adding some "original but published" work.
• CommentRowNumber11.
• CommentAuthorPieter
• CommentTimeFeb 2nd 2019
Now, I guess the best way to discuss the "remainder" problem, is by starting a new tread for it...