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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeAug 31st 2015
   • PermaLink
   Author: porton
   Format: MarkdownItexOnce I wrote [this nLab article](http://ncatlab.org/nlab/show/funcoid) as an extract from my new general topology theory. After this I was told that if I keep writing in nLab wiki, I could be considered as a person which writes in foreign lab book and may be banned (at least approximately so). Now I have released a good [draft of my book](http://www.mathematics21.org/binaries/volume-1.pdf) (I released it as Creative Commons and probably will never publish it "officially"). The draft represents the current research slice. It is dubbed as a draft because it most probably contains some minor errors. Please somebody review quickly my book and produce allowance for me to write extracts from my book in nLab wiki: I need a media to advertise my free ebook and nLab needs new research. Note that some concepts from my book apply to abstract category theory (notably my "cross-composition product" is defined for certain partially ordered (I mean with partially ordered Hom-sets) categories, probably for all interesting partially ordered categories in a sense). Also my "category with star morphisms" is an abstract category theory level stuff. These need to be presented in nLab.

   Once I wrote this nLab article as an extract from my new general topology theory.

   After this I was told that if I keep writing in nLab wiki, I could be considered as a person which writes in foreign lab book and may be banned (at least approximately so).

   Now I have released a good draft of my book (I released it as Creative Commons and probably will never publish it “officially”). The draft represents the current research slice. It is dubbed as a draft because it most probably contains some minor errors.

   Please somebody review quickly my book and produce allowance for me to write extracts from my book in nLab wiki: I need a media to advertise my free ebook and nLab needs new research.

   Note that some concepts from my book apply to abstract category theory (notably my “cross-composition product” is defined for certain partially ordered (I mean with partially ordered Hom-sets) categories, probably for all interesting partially ordered categories in a sense). Also my “category with star morphisms” is an abstract category theory level stuff. These need to be presented in nLab.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 1st 2015
   • (edited Sep 1st 2015)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>Please somebody review quickly my book do you realise it can take months to referee a paper of 20-30 pages? A book takes considerably longer, even for someone who is interested in the topic and familiar with the terminology and techniques. >nLab needs new research I think the nLab is doing quite well as it is. >These need to be presented in nLab Please note that the nLab is a notebook: people will write pages on things they are interested in or working on. It is not wikipedia, it does not aim for coverage of every conceivable topic. It does not "need" this or that material. If a regular here finds time out of their busy work and/or life to dip into your book *and finds the concepts they find there useful to their work* and with _interesting links to other parts of mathematics_, then they may write up some of this stuff. I searched for the word 'example' in your book and while I found many constructions and counterexamples to various properties, I did not find a single nontrivial example arising from outside your framework. If you'll take a few words from me, this is where the weakness of your work is. Mathematicians in general will learn things that help them with _their_ work, the problems _they_ are working on, not just learn about the latest thing because it subsumes in some abstract framework the general definitions of the things they work on. If you found, for instance, some important structure in set theory (just for example; I'm having trouble thinking of another realistic option) that was slightly mysterious, but when written out with your machinery, turned out to be a key and interesting construction, and your theorems proved something new _about the original problem_, then people might be interested. If your theorems merely prove things about properties of this structure when considered purely inside your theory, and tell us nothing new aside from that, then it doesn't get people interested. That's all from me for now.

   Please somebody review quickly my book

   do you realise it can take months to referee a paper of 20-30 pages? A book takes considerably longer, even for someone who is interested in the topic and familiar with the terminology and techniques.

   nLab needs new research

   I think the nLab is doing quite well as it is.

   These need to be presented in nLab

   Please note that the nLab is a notebook: people will write pages on things they are interested in or working on. It is not wikipedia, it does not aim for coverage of every conceivable topic. It does not “need” this or that material. If a regular here finds time out of their busy work and/or life to dip into your book and finds the concepts they find there useful to their work and with interesting links to other parts of mathematics, then they may write up some of this stuff.

   I searched for the word ’example’ in your book and while I found many constructions and counterexamples to various properties, I did not find a single nontrivial example arising from outside your framework. If you’ll take a few words from me, this is where the weakness of your work is. Mathematicians in general will learn things that help them with their work, the problems they are working on, not just learn about the latest thing because it subsumes in some abstract framework the general definitions of the things they work on. If you found, for instance, some important structure in set theory (just for example; I’m having trouble thinking of another realistic option) that was slightly mysterious, but when written out with your machinery, turned out to be a key and interesting construction, and your theorems proved something new about the original problem, then people might be interested. If your theorems merely prove things about properties of this structure when considered purely inside your theory, and tell us nothing new aside from that, then it doesn’t get people interested.

   That’s all from me for now.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 1st 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexTo add to what David said: people here are busy. There is no reason to expect they are going to drop what they are doing to review a long book with unusual idiosyncratic notation and terminology, and with no motivation other than your say-so that you think it's important, which in itself means nothing -- there are thousands of people who have pet theories, all clamoring for recognition. What is ironic is that you seem to feel entitled to being recognized here, and yet you yourself give no appearance of making an effort to engage with what other people are doing here. You say we "need new research" -- how on earth would you know? You don't try to follow anything we do here, do you? Actually, there is not time enough in the day for the research that goes on here already. The nLab is not a perch on which just anyone can sit and write about whatever they want. People have tried to do that, you know. It doesn't work. To make the project work, there needs to be an organic linking and cross-fertilization between new entries and what we already have, which covers thousands of concepts, the overwhelming majority of which have been tested in the peer-reviewed literature. This requires an awful lot of hard work and thinking through what others are saying and being careful and considerate about how things can mesh harmoniously. Stuff can't just be grafted on willy-nilly. So unless you yourself make a major effort to figure out how things like "cross-composition products" and "star morphisms" link up with and fit harmoniously with what we already have going on here -- which would require you to understand very well the contents of many nLab entries -- it's just not going to work and I'm afraid we'd have to say no to that.

   To add to what David said: people here are busy. There is no reason to expect they are going to drop what they are doing to review a long book with unusual idiosyncratic notation and terminology, and with no motivation other than your say-so that you think it’s important, which in itself means nothing – there are thousands of people who have pet theories, all clamoring for recognition.

   What is ironic is that you seem to feel entitled to being recognized here, and yet you yourself give no appearance of making an effort to engage with what other people are doing here. You say we “need new research” – how on earth would you know? You don’t try to follow anything we do here, do you? Actually, there is not time enough in the day for the research that goes on here already.

   The nLab is not a perch on which just anyone can sit and write about whatever they want. People have tried to do that, you know. It doesn’t work. To make the project work, there needs to be an organic linking and cross-fertilization between new entries and what we already have, which covers thousands of concepts, the overwhelming majority of which have been tested in the peer-reviewed literature. This requires an awful lot of hard work and thinking through what others are saying and being careful and considerate about how things can mesh harmoniously. Stuff can’t just be grafted on willy-nilly. So unless you yourself make a major effort to figure out how things like “cross-composition products” and “star morphisms” link up with and fit harmoniously with what we already have going on here – which would require you to understand very well the contents of many nLab entries – it’s just not going to work and I’m afraid we’d have to say no to that.

4. • CommentRowNumber4.
   • CommentAuthorporton
   • CommentTimeSep 1st 2015
   • PermaLink
   Author: porton
   Format: MarkdownItex@DavidRoberts Which kind of examples do you mean? Please provide an example of an example so that I could know what is needed. Note that every proximity space and every topological space is an example of a funcoid. I suspect you mean more elaborate examples. But what is "elaborate"?

   @DavidRoberts

   Which kind of examples do you mean? Please provide an example of an example so that I could know what is needed.

   Note that every proximity space and every topological space is an example of a funcoid.

   I suspect you mean more elaborate examples. But what is “elaborate”?

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 1st 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex> Please provide an example of an example so that I could know what is needed. That's for you to do: I can't do your research for you and I have my own work to do. > Note that every proximity space and every topological space is an example of a funcoid. Are there any examples that are not already known objects?

   Please provide an example of an example so that I could know what is needed.

   That’s for you to do: I can’t do your research for you and I have my own work to do.

   Note that every proximity space and every topological space is an example of a funcoid.

   Are there any examples that are not already known objects?

6. • CommentRowNumber6.
   • CommentAuthorporton
   • CommentTimeSep 1st 2015
   • PermaLink
   Author: porton
   Format: MarkdownItex> Are there any examples that are not already known objects? Not quite sure what you mean. Certainly, there are funcoids which are not corresponding to topological spaces, not corresponding to proximity spaces, and other things which you refer as "already known objects". An easy example to come up with (however probably having no significance) is a "hybrid" of a digraph and infinity: $D \sqcup (+\infty \times^{\mathsf{FCD}} +\infty)$ where $D$ is an arbitrary digraph and $+\infty$ is the well known filter "neighborhood of positive infinity" ($\sqcup$ means join on our lattice of funcoids). The above example is probably entirely useless. Another similar example: $D\sqcup T$ where $D$ is a digraph with vertices being real numbers and $T$ is the customary topology on the real line. (This example may probably be useful for something.) My purpose as I visioned it in deep past was not to "create" objects which are entirely new, but more of a more "fine" (algebraic) generalization of existing objects (such as topological spaces). The most interesting thing about my kinds of objects is probably that they generalize already known objects. For example, funcoids are a generalization of: (pre)topological spaces, proximity spaces, and also digraphs (I mean "binary relations"). An important aspect of the power of funcoids is that they generalize both discrete objects (graphs) and topological objects. For example, topological continuity and discrete continuity are described by the same formula. Also (this time with reloids) I generalize graphs, uniform spaces, Cauchy spaces; all of them are reloids. Well, after writing the above and some thinking, I can come up with a more practical example of a "not already known object": Cross-composition $X\times^{(C)} Y$ of any two typical topological spaces $X$ and $Y$. This object is what I call a pointfree funcoid whose both source and destination are what I call staroids (by the way, staroids themselves were not already known objects, as far as I know; and staroids deserve their own study especially because they appear in products). This product is not a topological space, but something new. Not sure, if what I've written above is the kind of answer which you wanted to receive.

   Are there any examples that are not already known objects?

   Not quite sure what you mean.

   Certainly, there are funcoids which are not corresponding to topological spaces, not corresponding to proximity spaces, and other things which you refer as “already known objects”.

   An easy example to come up with (however probably having no significance) is a “hybrid” of a digraph and infinity: $D \sqcup (+\infty \times^{\mathsf{FCD}} +\infty)$ where $D$ is an arbitrary digraph and $+\infty$ is the well known filter “neighborhood of positive infinity” ($\sqcup$ means join on our lattice of funcoids).

   The above example is probably entirely useless.

   Another similar example: $D\sqcup T$ where $D$ is a digraph with vertices being real numbers and $T$ is the customary topology on the real line. (This example may probably be useful for something.)

   My purpose as I visioned it in deep past was not to “create” objects which are entirely new, but more of a more “fine” (algebraic) generalization of existing objects (such as topological spaces). The most interesting thing about my kinds of objects is probably that they generalize already known objects. For example, funcoids are a generalization of: (pre)topological spaces, proximity spaces, and also digraphs (I mean “binary relations”). An important aspect of the power of funcoids is that they generalize both discrete objects (graphs) and topological objects. For example, topological continuity and discrete continuity are described by the same formula. Also (this time with reloids) I generalize graphs, uniform spaces, Cauchy spaces; all of them are reloids.

   Well, after writing the above and some thinking, I can come up with a more practical example of a “not already known object”:

   Cross-composition $X\times^{(C)} Y$ of any two typical topological spaces $X$ and $Y$. This object is what I call a pointfree funcoid whose both source and destination are what I call staroids (by the way, staroids themselves were not already known objects, as far as I know; and staroids deserve their own study especially because they appear in products). This product is not a topological space, but something new.

   Not sure, if what I’ve written above is the kind of answer which you wanted to receive.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 1st 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDavid, FYI the concept of 'funcoid' is equivalent to the concept of 'topogeny' as described at [[syntopogenous space]]. I spent some time last year investigating this stuff, as you can see from my web page '[topogeny](http://ncatlab.org/toddtrimble/published/topogeny)'. There are a number of equivalent notions here; my favorite one these days is that 'funcoids' from a set $X$ to a set $Y$ are in natural bijection with relations from $\beta X$ to $\beta Y$ in the pretopos of compact Hausdorff spaces.

   David, FYI the concept of ’funcoid’ is equivalent to the concept of ’topogeny’ as described at syntopogenous space. I spent some time last year investigating this stuff, as you can see from my web page ’topogeny’. There are a number of equivalent notions here; my favorite one these days is that ’funcoids’ from a set $X$ to a set $Y$ are in natural bijection with relations from $\beta X$ to $\beta Y$ in the pretopos of compact Hausdorff spaces.

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 2nd 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexTodd, yes I've looked at that, and I commend you for translating this to something people have a chance to understand. But where would such a thing arise in practice? I'm thinking of schemes: not just generalisation for its own sake, but providing constructions that just weren't possible for varieties, and which people wanted to use.

   Todd, yes I’ve looked at that, and I commend you for translating this to something people have a chance to understand. But where would such a thing arise in practice? I’m thinking of schemes: not just generalisation for its own sake, but providing constructions that just weren’t possible for varieties, and which people wanted to use.

9. • CommentRowNumber9.
   • CommentAuthorporton
   • CommentTimeSep 2nd 2015
   • PermaLink
   Author: porton
   Format: MarkdownItexAnother interesting example of a funcoid which is not a topological space, not proximity space, not digraph: $T\sqcap \{ (x,y)\in\mathbb{R}\times\mathbb{R} \mid y\ge x \}$. (Where $T$ is the funcoids corresponding to the customary topology on real line and I equate the set $\{ (x,y)\in\mathbb{R}\times\mathbb{R} \mid y\ge x \}$ with the corresponding funcoid) This is like "one-side-topology". We have "infinitely near points" for $x$ at its right side but not left side. We can get a $\pi$-length fragment of real line equipped with this funcoid and make a circle of it. Then we could travel continuously over the circle in the counter-clockwise, but not clockwise, accordingly this funcoid on the circle.

   Another interesting example of a funcoid which is not a topological space, not proximity space, not digraph:

   $T\sqcap \{ (x,y)\in\mathbb{R}\times\mathbb{R} \mid y\ge x \}$. (Where $T$ is the funcoids corresponding to the customary topology on real line and I equate the set $\{ (x,y)\in\mathbb{R}\times\mathbb{R} \mid y\ge x \}$ with the corresponding funcoid)

   This is like “one-side-topology”. We have “infinitely near points” for $x$ at its right side but not left side.

   We can get a $\pi$-length fragment of real line equipped with this funcoid and make a circle of it. Then we could travel continuously over the circle in the counter-clockwise, but not clockwise, accordingly this funcoid on the circle.

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 2nd 2015
    • (edited Sep 2nd 2015)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexAt the moment, much like syntopogenous spaces, topogenies (= funcoids) seem to me to be a mathematical backwater and presently I have no good answer to "what are they good for?", besides their subsuming a number of topology-ish notions such as proximity space, topological space, and so on, as Victor is wont to point out. Mike wrote on syntopogenous spaces [here](https://golem.ph.utexas.edu/category/2012/09/every_functor_is_a_coreflectio.html). At present I have no concrete plans to pursue these matters. I did spend a little time trying to determine whether the category of sets equipped with (reflexive) endofuncoids formed a cartesian closed (or better) category, without reaching a conclusion. Victor had been wondering about this question too. Since funcoids are a special case of relations in a monadic (and so a fortiori regular) category between free objects, I am mildly curious whether this general type of thing crops up elsewhere, and may try to think about that in an odd moment. Have you or anyone else seen that type of thing before?

    At the moment, much like syntopogenous spaces, topogenies (= funcoids) seem to me to be a mathematical backwater and presently I have no good answer to “what are they good for?”, besides their subsuming a number of topology-ish notions such as proximity space, topological space, and so on, as Victor is wont to point out. Mike wrote on syntopogenous spaces here.

    At present I have no concrete plans to pursue these matters. I did spend a little time trying to determine whether the category of sets equipped with (reflexive) endofuncoids formed a cartesian closed (or better) category, without reaching a conclusion. Victor had been wondering about this question too.

    Since funcoids are a special case of relations in a monadic (and so a fortiori regular) category between free objects, I am mildly curious whether this general type of thing crops up elsewhere, and may try to think about that in an odd moment. Have you or anyone else seen that type of thing before?

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 2nd 2015
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex@Todd I agree with your sentiments, I'm just pushing Victor to find something that makes his efforts more productive, rather than inventing new objects because he can. I've not seen anything like it before, no. @Victor you may like to read about directed topology. I'm also put in mind of continuous time Markov processes, and things like Brownian motion. But that's all I've got, and I don't have anything else to add to this conversation. If you could extract a five page paper that used non-baroque notation and gave some interesting applications of funcoids to other areas of mathematics while not claiming an amazing new theory that should be taken up by the maths community, then you might make more progress than so far. I stress _might_, and I can't promise to read such a paper or help you in any way, or even admit that this means you work "should" go on the nLab etc. Start small and work up.

    @Todd

    I agree with your sentiments, I’m just pushing Victor to find something that makes his efforts more productive, rather than inventing new objects because he can.

    I’ve not seen anything like it before, no.

    @Victor

    you may like to read about directed topology. I’m also put in mind of continuous time Markov processes, and things like Brownian motion. But that’s all I’ve got, and I don’t have anything else to add to this conversation. If you could extract a five page paper that used non-baroque notation and gave some interesting applications of funcoids to other areas of mathematics while not claiming an amazing new theory that should be taken up by the maths community, then you might make more progress than so far. I stress might, and I can’t promise to read such a paper or help you in any way, or even admit that this means you work “should” go on the nLab etc. Start small and work up.

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2015
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexVictor, what you\'ve been posting here in a search for examples is perhaps the most interesting stuff I\'ve read on funcoids. The material that I\'ve read before was such a rush of unmotivated definitions and complicated notation that I couldn\'t get a feel for them. (That may be what Bourbaki is like for people who don\'t know any abstract math already.) Here you\'re describing examples with visual metaphors that make them more interesting. Possibly this stuff is in your book already, but if so, then I never got that far before!

    Victor, what you've been posting here in a search for examples is perhaps the most interesting stuff I've read on funcoids. The material that I've read before was such a rush of unmotivated definitions and complicated notation that I couldn't get a feel for them. (That may be what Bourbaki is like for people who don't know any abstract math already.) Here you're describing examples with visual metaphors that make them more interesting. Possibly this stuff is in your book already, but if so, then I never got that far before!

13. • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 2nd 2015
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex@Toby As I said above, I checked every instance of the word 'example' in the document, and nothing like these recent examples are there that I remember. They are, apart from many instances of linguistic flourishes, all of the form 'Example: there is a [foo] with/without property [bar]', with little extrinsic motivation.

    @Toby

    As I said above, I checked every instance of the word ’example’ in the document, and nothing like these recent examples are there that I remember. They are, apart from many instances of linguistic flourishes, all of the form ’Example: there is a [foo] with/without property [bar]’, with little extrinsic motivation.

14. • CommentRowNumber14.
    • CommentAuthorporton
    • CommentTimeSep 2nd 2015
    • PermaLink
    Author: porton
    Format: MarkdownItexIt seems that today I've made up a new way to use funcoids in applied mathematics. Consider the following class of problems (I am almost sure such kind of problems happen in math applications.) Consider a plane (or $\mathbb{R}^n$ in general). Let to any point of the plane corresponds set of directions (set of unit vectors, I mean). Does there exists a smooth curve from point $A$ to point $B$ such that in every point of the curve its direction lies in the set of directions for this point? It seems that this problem can be reformulated in the language of funcoids: From the sets of directions we easily construct certain funcoid. Existence of such smooth curve is equivalent to existence of a continuous function from $[0;1]$ to our funcoid (it seems that we also need to add the condition that it smooth, I have not yet defined it quite formally). One advantage to use funcoids in this case, that when describing the curves we don't need to require that the curve is smooth. I think, it's a big advantage.

    It seems that today I’ve made up a new way to use funcoids in applied mathematics.

    Consider the following class of problems (I am almost sure such kind of problems happen in math applications.)

    Consider a plane (or $\mathbb{R}^n$ in general).

    Let to any point of the plane corresponds set of directions (set of unit vectors, I mean).

    Does there exists a smooth curve from point $A$ to point $B$ such that in every point of the curve its direction lies in the set of directions for this point?

    It seems that this problem can be reformulated in the language of funcoids: From the sets of directions we easily construct certain funcoid. Existence of such smooth curve is equivalent to existence of a continuous function from $[0;1]$ to our funcoid (it seems that we also need to add the condition that it smooth, I have not yet defined it quite formally).

    One advantage to use funcoids in this case, that when describing the curves we don’t need to require that the curve is smooth. I think, it’s a big advantage.

15. • CommentRowNumber15.
    • CommentAuthorporton
    • CommentTimeSep 2nd 2015
    • PermaLink
    Author: porton
    Format: MarkdownItexI've added my idea about traversing vector fields (see above) with funcoids to the draft file [addons.pdf](http://www.mathematics21.org/binaries/addons.pdf). This is a rather rough draft, don't expect to understand everything, there are errors and the proof is missing, just FYI. You may choose not to read it, because right now it's messy.

    I’ve added my idea about traversing vector fields (see above) with funcoids to the draft file addons.pdf.

    This is a rather rough draft, don’t expect to understand everything, there are errors and the proof is missing, just FYI. You may choose not to read it, because right now it’s messy.

16. • CommentRowNumber16.
    • CommentAuthorporton
    • CommentTimeSep 2nd 2015
    • PermaLink
    Author: porton
    Format: MarkdownItex@DavidRoberts Thanks for pointing me the definition of directed topological spaces. I propose a new way to construct a directed topological space. My way is more geometric/topological as it does not involve dealing with particular paths. Conjecture: Every directed topological space can be constructed in the below described way. Consider topological space $T$ and its subfuncoid $F$ (that is $F$ is a funcoid which is less that $T$ in the order of funcoids). Note that in our consideration $F$ is an "endofuncoid" (its source and destination are the same). Then a directed path from point $A$ to point $B$ is defined as a continuous function $f$ from $[0;1]$ to $F$ such that $f(0)=A$ and $f(1)=B$. Because $F$ is less that $T$, we have that every directed path is a path. ^^ I am going to add this to my draft.

    @DavidRoberts

    Thanks for pointing me the definition of directed topological spaces.

    I propose a new way to construct a directed topological space. My way is more geometric/topological as it does not involve dealing with particular paths.

    Conjecture: Every directed topological space can be constructed in the below described way.

    Consider topological space $T$ and its subfuncoid $F$ (that is $F$ is a funcoid which is less that $T$ in the order of funcoids). Note that in our consideration $F$ is an “endofuncoid” (its source and destination are the same).

    Then a directed path from point $A$ to point $B$ is defined as a continuous function $f$ from $[0;1]$ to $F$ such that $f(0)=A$ and $f(1)=B$.

    Because $F$ is less that $T$, we have that every directed path is a path.

    ^^ I am going to add this to my draft.