Re: source, sinks, and ?

[Michael Shulman <mshulman@ucsd.edu>]

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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