Re: source, sinks, and ?

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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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 ?

[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 ?

[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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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: 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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