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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeMay 5th 2012
   • (edited May 5th 2012)
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   Author: porton
   Format: TextHow to define the cartesian product of some (say, of two) (not necessarily binary) relations in categorical terms? The direct product in the category Set would not work as it does not preserve the structure of relations. I did some googling and wikipediing on the subject and have found nothing. Is it really an unknown subject for the contemporary mathematics? Both if it is known or unknown we may create a page on nLab wiki to put there all we known about this subject and the proposal to study it further.

   How to define the cartesian product of some (say, of two) (not necessarily binary) relations in categorical terms?
   
   The direct product in the category Set would not work as it does not preserve the structure of relations.
   
   I did some googling and wikipediing on the subject and have found nothing. Is it really an unknown subject for the contemporary mathematics? Both if it is known or unknown we may create a page on nLab wiki to put there all we known about this subject and the proposal to study it further.
2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 5th 2012
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   Author: Mike Shulman
   Format: MarkdownItexFor any $n$, there is a category whose objects are sets equipped with an $n$-ary relation, and one could consider products in that category. Is that what you're after?

   For any $n$, there is a category whose objects are sets equipped with an $n$-ary relation, and one could consider products in that category. Is that what you’re after?

3. • CommentRowNumber3.
   • CommentAuthorporton
   • CommentTimeMay 5th 2012
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   Author: porton
   Format: MarkdownItexDear Mike, First, I'm not an expert in category theory. Second, I don't understand what are the morphisms in your category. Third, I'd prefer to consider objects (or morphisms) which may have different arity, that is one object may be an $n$-ary and other an $m$-ary relation where not necessarily $n=m$.

   Dear Mike,

   First, I’m not an expert in category theory.

   Second, I don’t understand what are the morphisms in your category.

   Third, I’d prefer to consider objects (or morphisms) which may have different arity, that is one object may be an $n$-ary and other an $m$-ary relation where not necessarily $n=m$.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2012
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   Author: Mike Shulman
   Format: MarkdownItexThe natural choice of morphism is a function which preserves the relation, i.e. for $f\colon (X,R) \to (Y,S)$, then $R(x_1,\dots,x_n)$ implies $S(f(x_1),\dots,f(x_n))$. One could also make the relationship an iff (so the morphisms are a sort of "elementary embedding") but the resulting category would be less well-behaved. What sort of morphism would you choose from an $n$-ary relation to an $m$-ary one if $n\neq m$?

   The natural choice of morphism is a function which preserves the relation, i.e. for $f\colon (X,R) \to (Y,S)$, then $R(x_1,\dots,x_n)$ implies $S(f(x_1),\dots,f(x_n))$. One could also make the relationship an iff (so the morphisms are a sort of “elementary embedding”) but the resulting category would be less well-behaved.

   What sort of morphism would you choose from an $n$-ary relation to an $m$-ary one if $n\neq m$?

5. • CommentRowNumber5.
   • CommentAuthorporton
   • CommentTimeMay 6th 2012
   • (edited May 6th 2012)
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   Author: porton
   Format: MarkdownItexMike, I don't know. It is the reason why I have asked in the forum.

   Mike, I don’t know. It is the reason why I have asked in the forum.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2012
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   Author: Mike Shulman
   Format: MarkdownItexWell, category theory isn't much good until you have a category to study. (-: What's your motivation for asking about $n$-ary relations?

   Well, category theory isn’t much good until you have a category to study. (-:

   What’s your motivation for asking about $n$-ary relations?

7. • CommentRowNumber7.
   • CommentAuthorTim_Porter
   • CommentTimeMay 7th 2012
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   Author: Tim_Porter
   Format: MarkdownItexThis reminds me of Goguen's Institutions. He wanted to include a signature morphisms in the data and a set with an n-ary relation is an example of a model of a particular relational signature. If you have a particular context in mind it may suggest a particular form of signature map. Have a look at the wikipedia article on [Institutions](http://en.wikipedia.org/wiki/Institution_%28computer_science%29)

   This reminds me of Goguen’s Institutions. He wanted to include a signature morphisms in the data and a set with an n-ary relation is an example of a model of a particular relational signature. If you have a particular context in mind it may suggest a particular form of signature map. Have a look at the wikipedia article on Institutions

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 7th 2012
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   Author: DavidRoberts
   Format: MarkdownItexAs a purely intellectual exercise, can we treat an m-ary relation as being an algebra of the free operad generated by the obvious tree, and consider a map from an m-ary operation to an n-ary operation as arising from a map of operads together with some other data?

   As a purely intellectual exercise, can we treat an m-ary relation as being an algebra of the free operad generated by the obvious tree, and consider a map from an m-ary operation to an n-ary operation as arising from a map of operads together with some other data?

9. • CommentRowNumber9.
   • CommentAuthorTim_Porter
   • CommentTimeMay 7th 2012
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   Author: Tim_Porter
   Format: MarkdownItexI should mention that I played around with Institutions when searching for functors between categories of models for various modal logics.

   I should mention that I played around with Institutions when searching for functors between categories of models for various modal logics.

10. • CommentRowNumber10.
    • CommentAuthorporton
    • CommentTimeMay 7th 2012
    • (edited May 7th 2012)
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    Author: porton
    Format: MarkdownItexDear Mike, My motivation is study of product of multifuncoids. The information about what I've called multifuncoids is in <a href="http://www.mathematics21.org/binaries/nary.pdf">this draft</a>. (If you want to understand it, you need first read <a href="http://www.mathematics21.org/binaries/filters.pdf">this article</a> and <a href="http://www.mathematics21.org/binaries/funcoids-reloids.pdf">this preprint</a> and also <a href="http://www.mathematics21.org/binaries/assoc.pdf">this short preprint</a>.) Multifuncoids are essentially some special $n$-ary relations. In this manuscript I define products of multifuncoids in several (non equivalent) ways. I'd like to have theorems which would show associativity of these products up to something like a natural transformation or a natural isomorphism. But I am unable to define this for multifuncoids being essentially $n$-ary relations.

    Dear Mike,

    My motivation is study of product of multifuncoids. The information about what I’ve called multifuncoids is in this draft. (If you want to understand it, you need first read this article and this preprint and also this short preprint.)

    Multifuncoids are essentially some special $n$-ary relations.

    In this manuscript I define products of multifuncoids in several (non equivalent) ways.

    I’d like to have theorems which would show associativity of these products up to something like a natural transformation or a natural isomorphism.

    But I am unable to define this for multifuncoids being essentially $n$-ary relations.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 7th 2012
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    Author: Mike Shulman
    Format: MarkdownItexUnfortunately, I don't have the time to read four papers to answer your question. (-: Do you know what a morphism of multifuncoids is? That seems like a natural place to start, by defining a category.

    Unfortunately, I don’t have the time to read four papers to answer your question. (-: Do you know what a morphism of multifuncoids is? That seems like a natural place to start, by defining a category.

12. • CommentRowNumber12.
    • CommentAuthorporton
    • CommentTimeMay 7th 2012
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    Author: porton
    Format: MarkdownItexI don't have any idea on how multifuncoids could be done into either morphisms or objects of a category. If I knew I would write this. The trouble here is that multifuncoids are essentially $n$-ary (where $n$ is an arbitrary index set) and two multifuncoids can have different arity, while morphisms in category theory are essentially binary. I don't have an idea how to combine these. Maybe someone may suggest me some higher category theory? The first trouble I see here is that I know no way to define direct product for these possible higher categories (I know no way of this first because I am not an expert in category theory).

    I don’t have any idea on how multifuncoids could be done into either morphisms or objects of a category. If I knew I would write this.

    The trouble here is that multifuncoids are essentially $n$-ary (where $n$ is an arbitrary index set) and two multifuncoids can have different arity, while morphisms in category theory are essentially binary. I don’t have an idea how to combine these.

    Maybe someone may suggest me some higher category theory? The first trouble I see here is that I know no way to define direct product for these possible higher categories (I know no way of this first because I am not an expert in category theory).

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMay 7th 2012
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    Author: Mike Shulman
    Format: MarkdownItexI don't think I can help you with the information you've provided. It could be that your multifuncoids don't arrange themselves naturally into a category; not all structures in mathematics do. Morphisms of higher arity can sometimes be dealt with using multicategories, but I can't tell whether that would be relevant for you.

    I don’t think I can help you with the information you’ve provided. It could be that your multifuncoids don’t arrange themselves naturally into a category; not all structures in mathematics do. Morphisms of higher arity can sometimes be dealt with using multicategories, but I can’t tell whether that would be relevant for you.

14. • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 8th 2012
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    Author: DavidRoberts
    Format: MarkdownItex@porton - you don't need to be an expert in category theory to understand the definition of a category and a product in a category. If what you are studying is as useful as you think it is, then it it is likely there will be some natural form of map between objects. I suggest you read and digest the early parts of MacLane's classic book, and see if you can come up with a definition of morphism and test it against your theory - that is the real test. Just knowing an abstract definition of morphism in some category of 'foo's isn't helpful at all if it isn't useful for foo-theory. As far as higher categories go, if there is a higher category definition, then there is a 1-category definition by truncation so I suggest you look at 1-categories first.

    @porton - you don’t need to be an expert in category theory to understand the definition of a category and a product in a category. If what you are studying is as useful as you think it is, then it it is likely there will be some natural form of map between objects. I suggest you read and digest the early parts of MacLane’s classic book, and see if you can come up with a definition of morphism and test it against your theory - that is the real test. Just knowing an abstract definition of morphism in some category of ’foo’s isn’t helpful at all if it isn’t useful for foo-theory. As far as higher categories go, if there is a higher category definition, then there is a 1-category definition by truncation so I suggest you look at 1-categories first.