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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