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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.
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.
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.
Regarding your mathematical question:
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 .
Thinking on the fly here…
Doesn’t give you a remainder? I mean, surely is jointly monic (it gives the graph of as a relation, assuming binary products), and, given any other whatsoever, take … ??
But Mike’s comment makes me pause.
Hmm, maybe I had the universal properties the wrong way around in my head?
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