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I don’t see why you bother with the pre-category [category without identities, what I’d call a semicategory] version. After defining categories with star-morphisms, it seems obvious enough to me what a semicategory with star-morphisms would be, but people don’t usually need semicategories anyway.
Actually, I don’t see what the purpose of the concept is at all. Your research on funcoids etc is clearly not meaningless, but I don’t see the connections to the greater body of mathematics. (I’ve tried to read Funcoids and Reloids, but I find it wordy and unmotivated.) If you want to integrate your new research into the nLab, I can see the potential value of that, but it would probably work better on a personal web. From there, you could link seamlessly to established nLab articles, which (when relevant) could link back to your personal web’s pages.
But even before that, you should establish yourself as a helpful contributor to what is already on the nLab. (Actually, that is a prerequisite for having a personal web; see also discussion on the policy.) Many of the regulars here are complaining (privately and publicly) that your work is inappropriate for the nLab, because it is obscure work that doesn’t connect to everything else that we do. Of course, much of the stuff on the nLab is obscure to the general mathematician, but at least we know how it connects, which we don’t know about your stuff.
So if you first contribute to articles that are not about subjects directly related to your original research, then we can see that you’re a good scholar, then you can get a personal web, put your material on it, develop connections to the main nLab, then we can integrate it into the main nLab. Otherwise, we see you as a stranger writing in our lab book, so to speak, and your stuff will probably just be erased in the end.
Saying that something “may be useful” does not make a connection, nor does “thinking” that there are many connections. Even if one can rephrase subject X in terms of language Y, that does not make Y connected to mathematics through X. (E.g. one could rephrase all of computer science in terms of a Turing tarpit, but that does not make the tarpit interesting or connected to mathematics.) A further requirement is that Y provide some insight or tools or results that are of interest to mainstream mathematicians, particularly those studying subject X.
If you could use your theory to prove something that was independently conjectured by mathematicians studying proximity spaces, then that might be a valid connection. Or if it gave some insight that they appreciated. But unless and until such a connection exists, the nLab is not a place for potentially blind-alley speculations. In order for a wiki-structured site to remain useful, it has to maintain a high signal-to-noise ratio. You understandably believe the subjects you have invented to be signal rather than noise, but without some evidence of that, they look like noise to the average mathematician (and thereby reflect negatively on the nLab).
(As a piece of friendly advice, my experience suggests that valuable mathematics generally arises by building outwards from subjects that are understood and valued, rather than making up something totally new and then looking for connections to existing things. You are of course free to pursue whatever direction you please, but the nLab needs to preserve a certain baseline of value for its readers.)
In the literature, “precategory” is mostly yet another name for quiver. I actually prefer this name because many notions from “category” (e.g. hom, functor, enrichment, slice, etc.) make sense in a precategory though their behavior is not as rich as in real categories.
Whether you should include precategories/semicategories depends entirely on whether including them serves any purpose.
I don’t actually say that you should do with out them. I say that I don’t see what their purpose is. And I don’t see what the purpose of categories with star-morphisms is. And I mostly don’t see what the purpose of funcoids is (although at least I know that they’re supposed to be an approach to general topology that includes both proximity spaces and pretopological spaces). And I mostly don’t see what the purpose of your entire research is either. That’s the problem.
For general topology that includes proximity spaces and (not all but most in practice) pretopological spaces, I vaguely know Gerhard Preuß’s Convenient Topology. See this introduction with motivating properties of the system that make it sound attractive. Avoiding mention of s and s is old hat, but cartesian closed categories of spaces are not.
OK, that’s good: Preuß’s semiuniform convergence spaces are a special case of your funcoids, and funcoids also include all pseudotopological spaces (and not just the symmetric ones as Preuß’s spaces do). So if people like Preuß’s spaces (which certainly some mathematicians do) but don’t like that they leave out non-symmetric topological spaces, then this is a point in your favour.
I wonder if you’d care to go through the 12 points that Preuß puts in favour of the category of semiuniform convergence spaces and say how many you know or suspect hold for funcoids (or reloids, or pointless funcoids, etc). Cartesian closedness is the first of these.
Can you say anything about the other 11? Whether for reloids, pointfree funcoids, or whatever?
(By the way, is it correct that reloids and pointfree funcoids are both generalisations of funcoids? And are pointfree reloids a further generalisation?)
My advice to you, Victor, is to write first for yourself, and then for your blog, an essay which explains why you are pursuing this line. You should use no technical definitions, but be precise. If you have mathematical reasons why you are generalising spaces to funcoids then I haven’t seen them (not that I’ve looked at your work in depth, but every then I look just in case). By mathematical reasons I mean that you can prove theorems (see Mike’s second paragraph in #2) that otherwise are not easy, or awkward, or very difficult with the usual techniques/machinery.
But from what little I have read you look like you have philosophical reasons why you want to work with your constructions (the ’algebraic’ in algebraic general topology). You should be aware that there is lots of research on generalising topological spaces along these lines for various philosophical reasons or foundational choices. Talking about how your work relates to these programmes will clarify your own thinking and may also help others see why you are pursuing this line.
Until you can get to the point where you could motivate an average pure mathematician to want to think about funcoids, I don’t know think you will see any take-up of your work.
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