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Atrium > Mathematics, Physics & Philosophy: Prove existence of exponential object - help request

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- CommentRowNumber1.
- CommentAuthorporton
- CommentTimeSep 10th 2013
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Author: porton
Format: MarkdownItexI recently have proved that my category $\mathbf{Fcd}$ (of continuous maps between endofuncoids) has both all small products and all small coproducts. See <a href="http://www.mathematics21.org/binaries/product.pdf">this draft article</a>. Now I try to prove that it has exponential objects (and so is cartesian closed). With this I encountered a difficulty. Let $G$ is a digraph that is an endomorphisms of the category $\mathbf{Rel}$. I denote $\operatorname{Ob} G$ the set of the vertexes and $\operatorname{GR} G$ the set of edges. It is not hard to prove that the definition 2.3 in <a href="http://arxiv.org/pdf/math/0605275.pdf">this arXiv article</a> of exponential object $\operatorname{MOR} ( G ; H)$ from a graph $G$ to a graph $H$ in the category of graphs can be written as: $( f ; g) \in \operatorname{GR} \operatorname{MOR} ( G ; H) \Leftrightarrow g^{- 1} \circ (\operatorname{GR} G) \circ f \subseteq \operatorname{GR} H$. (However, it may be different for digraph than for simple graphs, I haven't checked details here.) Now when I attempt to "transfer" this result from the theory of graphs to the theory of endofuncoids, I encounter this difficulty: If now $G$, $H$ are endofuncoids, then $\operatorname{MOR} ( G ; H)$ is an endofuncoid, not a pair of functions. I don't know how to deal with this. Could you help? Even a simple proof for the special case of graphs may be probably helpful. However the real research is about endofuncoids not graphs.

I recently have proved that my category $\mathbf{Fcd}$ (of continuous maps between endofuncoids) has both all small products and all small coproducts.

See this draft article.

Now I try to prove that it has exponential objects (and so is cartesian closed). With this I encountered a difficulty.

Let $G$ is a digraph that is an endomorphisms of the category $\mathbf{Rel}$ . I denote $\operatorname{Ob} G$ the set of the vertexes and $\operatorname{GR} G$ the set of edges.

It is not hard to prove that the definition 2.3 in this arXiv article of exponential object $\operatorname{MOR} ( G ; H)$ from a graph $G$ to a graph $H$ in the category of graphs can be written as:

$( f ; g) \in \operatorname{GR} \operatorname{MOR} ( G ; H) \Leftrightarrow g^{- 1} \circ (\operatorname{GR} G) \circ f \subseteq \operatorname{GR} H$ .

(However, it may be different for digraph than for simple graphs, I haven’t checked details here.)

Now when I attempt to “transfer” this result from the theory of graphs to the theory of endofuncoids, I encounter this difficulty:

If now $G$ , $H$ are endofuncoids, then $\operatorname{MOR} ( G ; H)$ is an endofuncoid, not a pair of functions. I don’t know how to deal with this.

Could you help?

Even a simple proof for the special case of graphs may be probably helpful. However the real research is about endofuncoids not graphs.
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeSep 10th 2013
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Author: Todd_Trimble
Format: MarkdownItexEven a simple proof of what for the special case of graphs? That the category of such is cartesian closed? I'll remark that categories of graphs (of whatever flavor) tend to be quasitoposes, which is even nicer than just being cartesian closed. Let me also remark that some of the the terminology can be confusing here; for a category theorist, "digraph" could mean "directed graph" (where the function $Edges \to Vertices \times Vertices$ can be any function, not necessarily an inclusion, which is what you seem to be referring to). It's just something to be aware of. I'm probably not understanding what the difficulty is (I've only just now looked at the definition of "funcoid", without much understanding), but: don't you _want_ $MOR(G; H)$ to be an endofuncoid? In other words, to get cartesian closure of the category of endofuncoids and endofuncoid maps, you'd want a universal endofuncoid map of the form $MOR(G; H) \times G \to H$, where $G$, $H$, and $MOR(G; H)$ are _endofuncoids_. Final comment: I don't know how it is for others, but the word "funcoid" just doesn't sound right to my ears, both because of the funky associations, and because it doesn't seem right from a linguistic standpoint. If it were me, I'd much prefer _functoid_. This is a small matter, perhaps, but little things like that can unfortunately get in the way.

Even a simple proof of what for the special case of graphs? That the category of such is cartesian closed?

I’ll remark that categories of graphs (of whatever flavor) tend to be quasitoposes, which is even nicer than just being cartesian closed.

Let me also remark that some of the the terminology can be confusing here; for a category theorist, “digraph” could mean “directed graph” (where the function $Edges \to Vertices \times Vertices$ can be any function, not necessarily an inclusion, which is what you seem to be referring to). It’s just something to be aware of.

I’m probably not understanding what the difficulty is (I’ve only just now looked at the definition of “funcoid”, without much understanding), but: don’t you want $MOR(G; H)$ to be an endofuncoid? In other words, to get cartesian closure of the category of endofuncoids and endofuncoid maps, you’d want a universal endofuncoid map of the form $MOR(G; H) \times G \to H$ , where $G$ , $H$ , and $MOR(G; H)$ are endofuncoids.

Final comment: I don’t know how it is for others, but the word “funcoid” just doesn’t sound right to my ears, both because of the funky associations, and because it doesn’t seem right from a linguistic standpoint. If it were me, I’d much prefer functoid. This is a small matter, perhaps, but little things like that can unfortunately get in the way.
- CommentRowNumber3.
- CommentAuthorporton
- CommentTimeSep 10th 2013
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Author: porton
Format: MarkdownItex> Even a simple proof of what for the special case of graphs? That the category of such is cartesian closed? Yes. But I need only to prove existence of exponential objects, because I know how to prove existence of all small products. > I’ll remark that categories of graphs (of whatever flavor) tend to be quasitoposes, which is even nicer than just being cartesian closed. I am going to investigate this for my category $\mathbf{Fcd}$, but the time to check whether is is a topos has not yet come. > but: don’t you want $\operatorname{MOR}(G;H)$ to be an endofuncoid? Yes, I want it to be an endofuncoid. > Final comment: I don’t know how it is for others, but the word “funcoid” just doesn’t sound right to my ears, both because of the funky associations, and because it doesn’t seem right from a linguistic standpoint. If it were me, I’d much prefer functoid. This is a small matter, perhaps, but little things like that can unfortunately get in the way. I'd better leave the term as is "funcoid": 1. It is in several draft articles on my site. 2. It is in my preprint book which is now in review, and also in an article which is in peer review. 3. It rhythms well with my other term "reloid": function -> funcoid, relation -> reloid. But if others will ask me to rename funcoid -> functoid, I may consider renaming. My writings about funcoids are not yet published, and in principle renaming is possible. P.S. It seems I have found a way to construct exponential objects in my category. Now I am going to start investigation.
Even a simple proof of what for the special case of graphs? That the category of such is cartesian closed?

Yes. But I need only to prove existence of exponential objects, because I know how to prove existence of all small products.

I’ll remark that categories of graphs (of whatever flavor) tend to be quasitoposes, which is even nicer than just being cartesian closed.

I am going to investigate this for my category $\mathbf{Fcd}$ , but the time to check whether is is a topos has not yet come.

but: don’t you want $\operatorname{MOR}(G;H)$ to be an endofuncoid?

Yes, I want it to be an endofuncoid.

Final comment: I don’t know how it is for others, but the word “funcoid” just doesn’t sound right to my ears, both because of the funky associations, and because it doesn’t seem right from a linguistic standpoint. If it were me, I’d much prefer functoid. This is a small matter, perhaps, but little things like that can unfortunately get in the way.

I’d better leave the term as is “funcoid”:
1. It is in several draft articles on my site.
2. It is in my preprint book which is now in review, and also in an article which is in peer review.
3. It rhythms well with my other term “reloid”: function -> funcoid, relation -> reloid.
But if others will ask me to rename funcoid -> functoid, I may consider renaming. My writings about funcoids are not yet published, and in principle renaming is possible.

P.S. It seems I have found a way to construct exponential objects in my category. Now I am going to start investigation.
- CommentRowNumber4.
- CommentAuthorTodd_Trimble
- CommentTimeSep 10th 2013
- (edited Sep 10th 2013)
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Author: Todd_Trimble
Format: MarkdownItex> but the time to check whether is is a topos has not yet come. Right; I understand. BTW: I very much doubt it's a topos. I have a vague hunch (or at least I wouldn't be very surprised if) it is a quasitopos. But anyway I think you are right to proceed as you are doing. > > but: don’t you want $\operatorname{MOR}(G;H)$ to be an endofuncoid? > Yes, I want it to be an endofuncoid. The reason I asked was because you said this in #1: > If now $G$, $H$ are endofuncoids, then $MOR(G;H)$ is an endofuncoid, not a pair of functions. I don’t know how to deal with this.

but the time to check whether is is a topos has not yet come.

Right; I understand. BTW: I very much doubt it’s a topos. I have a vague hunch (or at least I wouldn’t be very surprised if) it is a quasitopos. But anyway I think you are right to proceed as you are doing.

but: don’t you want $\operatorname{MOR}(G;H)$ to be an endofuncoid?

Yes, I want it to be an endofuncoid.

The reason I asked was because you said this in #1:

If now $G$ , $H$ are endofuncoids, then $MOR(G;H)$ is an endofuncoid, not a pair of functions. I don’t know how to deal with this.
- CommentRowNumber5.
- CommentAuthorTobyBartels
- CommentTimeSep 13th 2013
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Author: TobyBartels
Format: MarkdownItexMy response to the real mathematical question is the same as Todd\'s, so I\'ll make a linguistic comment. The infix ‘‑at‑’ is something of a null-op; think of words like ‘causation’, which is derived from ‘cause’ (rather than ‘causate’), when we could just as easily have used ‘causion’ (analogously to ‘fusion’). So ‘relation’ ~ ‘relion’ > ‘reloid’; analogously, ‘function’ > ‘functoid’. If funcoids haven\'t been formally published yet, then it\'s not too late to change it; and if they already under review, then it may be your last chance! You may have more important things to worry about, but all of the appearances of ‘funcoid’ are still under your control so far. You just have to edit your files and tell your reviewers!

My response to the real mathematical question is the same as Todd's, so I'll make a linguistic comment. The infix ‘‑at‑’ is something of a null-op; think of words like ‘causation’, which is derived from ‘cause’ (rather than ‘causate’), when we could just as easily have used ‘causion’ (analogously to ‘fusion’). So ‘relation’ ~ ‘relion’ > ‘reloid’; analogously, ‘function’ > ‘functoid’.

If funcoids haven't been formally published yet, then it's not too late to change it; and if they already under review, then it may be your last chance! You may have more important things to worry about, but all of the appearances of ‘funcoid’ are still under your control so far. You just have to edit your files and tell your reviewers!
- CommentRowNumber6.
- CommentAuthorporton
- CommentTimeSep 13th 2013
- PermaLink
Author: porton
Format: MarkdownItexI refer the word "funcoid" in a <a href="http://ijpam.eu/contents/2012-74-1/6/index.html">published article</a> (remark 9). The word "funcoid" is used as a forward reference (a reference to a draft of not yet published article). If I rename it now, that article could become an article with a grammar error. So I am yet unsure whether to rename. I'd like to hear more opinions, whether to rename or not to rename.

I refer the word “funcoid” in a published article (remark 9). The word “funcoid” is used as a forward reference (a reference to a draft of not yet published article).

If I rename it now, that article could become an article with a grammar error.

So I am yet unsure whether to rename.

I’d like to hear more opinions, whether to rename or not to rename.
- CommentRowNumber7.
- CommentAuthorporton
- CommentTimeSep 13th 2013
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Author: porton
Format: MarkdownItexAlso the word "funcoid" is used at a few wiki and wiki-like sites such as Open Problem Garden, and I would need some work to rename all of it.

Also the word “funcoid” is used at a few wiki and wiki-like sites such as Open Problem Garden, and I would need some work to rename all of it.
- CommentRowNumber8.
- CommentAuthorporton
- CommentTimeSep 13th 2013
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Author: porton
Format: MarkdownItexOne side comment: I've invented the term "foonkoid" in some deep past when I yet wrote in Russian language not English. "Functoid" (with t) would sound very bad in Russian. I realize we speak about English not Russian but may probably take this in the consideration, how it is transferred to other languages.

One side comment: I’ve invented the term “foonkoid” in some deep past when I yet wrote in Russian language not English.

“Functoid” (with t) would sound very bad in Russian. I realize we speak about English not Russian but may probably take this in the consideration, how it is transferred to other languages.
- CommentRowNumber9.
- CommentAuthorporton
- CommentTimeSep 13th 2013
- (edited Sep 14th 2013)
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Author: porton
Format: MarkdownItexOh well, also the word "funcoid" is used in my blog posts. Now I feel it's better to leave the term as is, but I'm open to hear more proposals to replace it.

Oh well, also the word “funcoid” is used in my blog posts.

Now I feel it’s better to leave the term as is, but I’m open to hear more proposals to replace it.
- CommentRowNumber10.
- CommentAuthorporton
- CommentTimeSep 14th 2013
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Author: porton
Format: MarkdownItexOh, well, now I have almost finally decided not to rename "funcoid" -> "functoid". This rename would not agree with denoting categories related with funcoids in three letter f.c.d. such as $FCD$ and $\mathbf{Fcd}$.

Oh, well, now I have almost finally decided not to rename “funcoid” -> “functoid”.

This rename would not agree with denoting categories related with funcoids in three letter f.c.d. such as $FCD$ and $\mathbf{Fcd}$ .

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Atrium > Mathematics, Physics & Philosophy: Prove existence of exponential object - help request