Re: source, sinks, and ?

["Reinhard Boerger" <Reinhard.Boerger@FernUni-Hagen.de>]

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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