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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Re: source, sinks, and ?

[JeanBenabou]

In his mail, Mike Shulman wrote,

> Thanks to everyone who replied.  I did intend that the source and
> target be specified, i.e. to consider, for two given families of
> objects { x_i }_{i \in I} and { y_j }_{j \in J} (for which either I or
> J might be empty), a family of morphisms { x_i --> y_j }_{i \in I, j
> \in J}.  This reduces to the notion of sink (resp. cone) described by
> Reinhard when J (resp. I) is a singleton
>
> "Matrix" and "array" are both good words, although I agree that the
> non-composability in general makes "matrix" slightly misleading.

1 -  In the spirit of the word "array", which I proposed, I suggest
the following names for two special cases.
(i) When I = 1 , instead of "cone",  "column"
(ii) When J = 1 , instead of "sink",  "row"

This would have the following advantages:
(a) In the case of "matrices", i.e. when the product is defined, it
would fit with the usual matrix terminology.
(b) We wouldn't have to change our use of "cone" and "co-cone" over a
diagram D, rows and columns would be the special cases, when D is
discrete.
I have often used "rows" and "columns" in the context of general
"matrices", which I explained in my previous mail, without having met
any ambiguity or incompatibility


> One might also observe that such a family can be identified with a
> diagram indexed on the collage (or cograph) of a
> profunctor/distributor between discrete categories (specifically, the
> profunctor constant at 1).  But that doesn't immediately suggest a
> conciser name to my mind.

2- This "ad hoc" identification, apart from the fact that it "doesn't
immediately suggest a conciser name", needs complicated notions such
as distributors and collages. Moreover it  "doesn't immediately
suggest" generalizations. There is a very simple interpretation in
terms of the canonical fibration  Fam(C) --> Set which can be easily
generalized, and permits to define "arrays" for arbitrary fibrations
p: X --> S, provided S has finite products. With mild assumptions on
p and X, one can even define "matrices" and develop a "matrix calculus"

Best to all,

Jean

> One place such families occur is in what one might call "joint
> source/sink factorization systems".  For instance, in Ross Street's
> paper "The family approach to total cocompleteness and toposes," a
> "familially regular category" is defined to be one in which any such
> "array" with J finite factors into a strong-epic sink followed by a
> monic source, and strong-epic sinks are stable under pullback.
>
> Another is that just as the limit of a diagram is a cone over that
> diagram with a universal property, a *multilimit* of a diagram can be
> described as an "array" over that diagram (which we may regard as a
> family of cones with the same codomain) with a universal property.
>
> Mike
>

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Re: source, sinks, and ?

[Michael Shulman]

Thanks to everyone who replied.  I did intend that the source and
target be specified, i.e. to consider, for two given families of
objects { x_i }_{i \in I} and { y_j }_{j \in J} (for which either I or
J might be empty), a family of morphisms { x_i --> y_j }_{i \in I, j
\in J}.  This reduces to the notion of sink (resp. cone) described by
Reinhard when J (resp. I) is a singleton.

"Matrix" and "array" are both good words, although I agree that the
non-composability in general makes "matrix" slightly misleading.

One might also observe that such a family can be identified with a
diagram indexed on the collage (or cograph) of a
profunctor/distributor between discrete categories (specifically, the
profunctor constant at 1).  But that doesn't immediately suggest a
conciser name to my mind.

One place such families occur is in what one might call "joint
source/sink factorization systems".  For instance, in Ross Street's
paper "The family approach to total cocompleteness and toposes," a
"familially regular category" is defined to be one in which any such
"array" with J finite factors into a strong-epic sink followed by a
monic source, and strong-epic sinks are stable under pullback.

Another is that just as the limit of a diagram is a cone over that
diagram with a universal property, a *multilimit* of a diagram can be
described as an "array" over that diagram (which we may regard as a
family of cones with the same codomain) with a universal property.

Mike

On Tue, Jan 4, 2011 at 2:44 AM, Reinhard Boerger
<Reinhard.Boerger@fernuni-hagen.de> wrote:
> Hello,
>
> I am used to a slightly different terminology, which seems appropriate.
>
>> 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".
>
> For a sink, as I know it, the codomain should also be specified, i.e. a sink
> is given by an object y and a family of morphisms x_i --> y. If I is not
> empty, this does not matter, but for empty I at least y should be given. A
> cocone is given by an object y and a natural transformation from some
> functor to the constant functor with value y; her y is also specified. So  a
> sink is essentially a discrete cocone.
>
>   A family {
>> x --> y_j }_{j \in J} all with the same domain is called a "source" or
>> a "cone".
>
> These are the dual notions.
>
>> 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)?
>
> I do not know. Where does it occur? Probably the domain and codomain should
> also be specified, possibly even an arrow. If a non-empty collection of
> arrows behave similarly (e.g. is mapped to the same arrow by a given functor
> F), this means the same a saying that they all behave in the same way as a
> given arrow 8e.g are mapped to some special morphism by F). A collection of
> two objects x,y (prescribed domain and codomain) is something different; it
> does not give an arrow Fx -->Fy.
>
>
> Greetings
> Reinhard
>


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Re: source, sinks, and ?

[Reinhard Boerger]

Hello,

I am used to a slightly different terminology, which seems appropriate.

> 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".

For a sink, as I know it, the codomain should also be specified, i.e. a sink
is given by an object y and a family of morphisms x_i --> y. If I is not
empty, this does not matter, but for empty I at least y should be given. A
cocone is given by an object y and a natural transformation from some
functor to the constant functor with value y; her y is also specified. So a
sink is essentially a discrete cocone.

   A family {
> x --> y_j }_{j \in J} all with the same domain is called a "source" or
> a "cone". 

These are the dual notions.

> 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)?

I do not know. Where does it occur? Probably the domain and codomain should
also be specified, possibly even an arrow. If a non-empty collection of
arrows behave similarly (e.g. is mapped to the same arrow by a given functor
F), this means the same a saying that they all behave in the same way as a
given arrow 8e.g are mapped to some special morphism by F). A collection of
two objects x,y (prescribed domain and codomain) is something different; it
does not give an arrow Fx -->Fy.


Greetings
Reinhard



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Re: source, sinks, and ?

[Fred E.J. Linton]

On Sun, 02 Jan 2011 07:17:17 PM EST Michael Shulman <mshulman@ucsd.edu>
asked:

> ... 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)? ...

Looks more like a "mish-mash" to me :-) . Cheers, --  Fred




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