source, sinks, and ?

source, sinks, and ?

[Michael Shulman]

A family of morphisms { x_i --> y }_{i \in I} in some category, all
with the same codomain, is called a "sink" or a "cocone".  A family {
x --> y_j }_{j \in J} all with the same domain is called a "source" or
a "cone".  Is there a name for a family of the form { x_i --> y_j }_{i
\in I, j \in J} ?  A cylinder?  Or a frustrum (since I \neq J)?

Mike


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Re: source, sinks, and ?

[burroni]

Dear Mike,

In fact, we can call it a matrix (of (I x J)- type)  with coefficients  
in the category. When the category is a category of modules, we have  
that way a natural generalization of classical matrices.

Cheers,
Albert


Michael Shulman <mshulman@ucsd.edu> a écrit :

> A family of morphisms { x_i --> y }_{i \in I} in some category, all
> with the same codomain, is called a "sink" or a "cocone".  A family {
> x --> y_j }_{j \in J} all with the same domain is called a "source" or
> a "cone".  Is there a name for a family of the form { x_i --> y_j }_{i
> \in I, j \in J} ?  A cylinder?  Or a frustrum (since I \neq J)?
>
> Mike

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Re: Re: source, sinks, and ?

[JeanBenabou]

Dear Albert,

I "half agree" with you. The term "matrix" has been used by by Freyd- 
Scedrov in their allegories, by myself in my paper on distributors,  
and probably by many other persons.
However in both cases the situation was such that one could define  
the "product" of two matrixes and get an allegory or more generally a  
bi-category, and I suggest it should be restricted to such cases. In  
the general situation where the product is not defined, I suggest the  
word "array" (with entries in C, if we want to specify C)

Bonnne Année,
Jean


Le 3 janv. 11 à 23:40, burroni@math.jussieu.fr a écrit :

> Dear Mike,
>
> In fact, we can call it a matrix (of (I x J)- type)  with  
> coefficients in the category. When the category is a category of  
> modules, we have that way a natural generalization of classical  
> matrices.
>
> Cheers,
> Albert
>

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Re: source, sinks, and ?

[Tom Prince]

On 2011-01-02, Michael Shulman wrote:
> A family of morphisms { x_i --> y }_{i \in I} in some category, all
> with the same codomain, is called a "sink" or a "cocone".  A family {
> x --> y_j }_{j \in J} all with the same domain is called a "source" or
> a "cone".  Is there a name for a family of the form { x_i --> y_j }_{i
> \in I, j \in J} ?  A cylinder?  Or a frustrum (since I \neq J)?
>
> Mike

A join?

   Tom


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