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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]