Re: source, sinks, and ?

[JeanBenabou <jean.benabou@wanadoo.fr>]

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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