# What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?

This is an embarrassingly simple question, but I was not able to find a definitive answer from literature search.

Suppose one has some collection of functions $f_1: X \to Y_1, \dots, f_n: X \to Y_n$ on a common domain $X$. Then one can form the function $(f_1,\dots,f_n): X \to Y_1 \times \dots \times Y_n$ in the usual fashion:

$$(f_1,\dots,f_n)(x) := (f_1(x),\dots,f_n(x)).$$

My question is: what does one call the function $(f_1,\dots,f_n)$? I had (without thinking much about it) used to call it the "direct sum" of $f_1,\dots,f_n$, before realising that this actually had no relation with the usual meaning of direct sum. In category theory, one might call $(f_1,\dots,f_n)$ the "product" of $f_1,\dots,f_n$, but this could get confusing if $f_1,\dots,f_n$ already take values in some ring, so that there is also a pointwise product. Is there some other commonly accepted term for describing the function $(f_1,\dots,f_n)$? It doesn't sound quite right grammatically to refer to it as the "tuple" of $f_1,\dots,f_n$, and "concatenation" or "join" don't quite seem to fit either.

• In the case $n = 2$, I would call it the pairing. Similarly, one has "tripling", "quadrupling", and so in general I would call it the tupling of the list $f_1, \ldots, f_n$. – Todd Trimble Oct 5 '15 at 21:54
• One option might be to call this the "diagonal" of the Cartesian product? I don't know though if such a name is standard outside of optimization where this "diagonal" arises as a part of the so-called "product-space trick" – Suvrit Oct 5 '15 at 22:38
• Suvrit's comment is to some extent supported by Engelking's General topology, which is very precise about terminology. It uses the term "diagonal of mappings" and the notation $\bigtriangleup_{i=1}^n f_i$. (the definition is at the bottom of page 79 in the 1989 edition) – Niels J. Diepeveen Oct 6 '15 at 0:16
• it IS the product (object) of the (objects) $(f_i : X \to Y_i)$ in the comma category $X \backslash YourCat$. So you could perfectly well call your map the (comma) product of those maps, and leave out "comma" when it gets tiresome. – Jesse C. McKeown Oct 6 '15 at 1:22
• As $X_1\times\cdots \times X_n$ is called cartesian product I use the same name for $f=(f_1,\ldots,f_n)$. – Jochen Wengenroth Oct 6 '15 at 6:27

I was encouraged to make my comment an answer:

In the case $n = 2$, I would call it the pairing. Similarly, one has "tripling", "quadrupling", and so in general one might call it the ($n$-)tupling of the list $f_1, \ldots, f_n$. And indeed that is what the nLab calls it: see here.

Whatever this should be called, I would not call it the cartesian product of $f_1, \ldots, f_n$. The product is a functor $\mathcal{C}^n \to \mathcal{C}$ whose value at a morphism $(f_1: X \to Y_1, \ldots, f_n: X \to Y_n)$ of $\mathcal{C}^n$ is rather the morphism $f_1 \times \ldots \times f_n: X \times \ldots \times X \to Y_1 \times \ldots \times Y_n$ of $\mathcal{C}$, and it's the latter that I would call the product of $f_1, \ldots, f_n$.

I will see if I can track down further citations for $n$-tupling.

• If the maps happen to go into a field, "pairing" has the minor misfortune of being a term one might also use to describe an inner product of $f_1$ and $f_2$. To make matters worse, come to think of it, an inner product is often denoted $(f_1, f_2)$. Fortunately, there's an easy fix: just refer to $(f_1, f_2) \colon X \to Y_1 \times Y_2$ as the tupling even when you're only tupling two things (and hope nobody adopts this terminology for multilinear functionals, I guess). – Vectornaut Oct 6 '15 at 18:47
• @Vectornaut I'd think if you referred to the pairing $(f_1, f_2): X \to Y_1 \times Y_2$ or $\langle f_1, f_2 \rangle: X \to Y_1 \times Y_2$, then context should usually be sufficient to disambiguate this sense of pairing from bilinear pairing, since a product $Y_1 \times Y_2$ of two rings is never a field. :-) – Todd Trimble Oct 6 '15 at 19:27
• tupling is nice; though one could perhaps also use 'stringing' but that may strike the wrong chord.... It is also interesting to note that syntactically, 'tupling a list' as noted above, literally means 'parenthesizing' it --- the $f_i$ and the commas are already there! – Suvrit Oct 6 '15 at 22:37
• I'm accepting this answer as being (a) one with a little bit of precedent (from the n-lab), (b) a term which does not require specialised knowledge in order to understand, and (c) the most popular of the alternatives suggested here. I also like the fine distinction between the tuple $(x \mapsto f_1(x),\dots,x \mapsto f_n(x))$ of $f_1,\dots,f_n$ and the tupling $x \mapsto (f_1(x),\dots,f_n(x))$ of $f_1,\dots,f_n$; these two concepts are canonically and naturally equivalent and it is a fairly safe abuse of notation to call them both $(f_1,\dots,f_n)$, and the similar names support this. – Terry Tao Oct 8 '15 at 16:45
• As a synonym you could also use bunching. – Aditya Guha Roy Nov 1 '19 at 17:19

You could call $$\mathbf r(t)=\langle x(t), y(t), z(t)\rangle$$ the vector function (or vector?) of $(x,y,z)$.

• I thought of mentioned this name but then I saw this mathworld.wolfram.com/VectorFunction.html and decided to pass...but I like this name nevertheless – Suvrit Oct 6 '15 at 18:42
• I wanted to say something like "vector-valued function with components $f_i$" but this only really works for appropriate $Y_i$. – JP McCarthy Oct 6 '15 at 18:56

Perhaps you can take inspiration from computer science.

In "APL: an interactive approach", Gilman and Rose talk about the catenation and lamination operators for joining together appropriately shaped arguments which (in my view) are multidimensional arrays. I think 'catenating' functions is appropriate for this usage. (Gilman, Leonard, and Allen J. Rose. APL: an interactive approach. Krieger Publishing Co., Inc., 1992. I have the second edition from an earlier year at home, with catenation on page 138, I think.)

If you decide that you prefer something else, be bold and define a term. Although I like Brendan McKay's suggestion of 'assemblage', for some reason 'amalgam' appeals to me. However, it is important to know how you end up using the name and the construct: there may be other things that better deserve those names. Further, be sure to point out similar but different uses (e.g. 'amalgamated product'). (You know that already, but a reminder shouldn't hurt.) You might even give a nod to history and note how such things were named and used in early vector calculus and linear algebra texts.

Gerhard "Can Say You Were Ill-advised" Paseman, 2015.10.06

A reasonable case could be made for calling your tuple-function a multi-span in whichever category $C$ your $\{f_i\}_1^n$ inhabit. When $n=2$, this reduces to the usual span given by roof-diagrams which look like this:

$$Y_1 \stackrel{f_1}{\gets} X \stackrel{f_2}{\to} Y_2$$

In this happy $n=2$ case you can actually compose these guys provided $C$ admits pullbacks; in this case, you have a bicategory of spans in $C$. For larger $n$, things are much less structured (as far as I know).

The obvious dual gadgets (unfortunately, but not unexpectedly, called multi-cospans) have actually appeared in work by Grandis, pdf here.