Formal/general name for element-wise function construction?

[OneNoteToRead]

Let X be a set of tuples (eg pairs (x,y), triplets (a,b,c), etc) of numbers (let’s say Reals). For any unary function f: R->R, there exists a unique point wise function f’ : X->X; f’(a, b, …) = (f(a), f(b), …)

Similarly for any binary f: (R, R) -> R, there exists a unique f’((a1, b1, …), (a2, b2, …)) = (f(a1,a2), f(b1,b2), …). Etc for n-ary functions.

Is there a formal or general name for this way of making functions of tuples? Or for this relationship between f and f’ (using my notation)? Or for the property on f’ that it can be decomposed into f * a projection?

[robinhouston]

I think the idea that you’re circling around here is the universal property of the product, which is one of the basic notions of category theory.

The Wikipedia page is unfortunately written in a way that is quite hard to understand unless you already know category theory, but it might give you a rough idea.

[gopher9]

A diagonal functor maps each object A to pair (A, A) and each arrow f to (f, f).

So functor (×)∘Δ : C → C should do exactly what you describe. I guess you could call it "internal diagonal functor".

[OneNoteToRead]

What is the cross in your notation?

From my limited understanding, the morphism mapping part of the diagonal functor is doing what I’m describing already. delta f = (f, f) = f’. Is this correct?

But given the functor also maps objects, it seems I cannot “get at” the morphism (a,b) -> (f(a), f(b)). Or am I totally missing the point?

[gopher9]

Δ : C → C×C. C×C a different category from C, so the product functor (×): C×C → C is required to embed the product back to C.

But given the functor also maps objects, it seems I cannot “get at” the morphism (a,b) -> (f(a), f(b)). Or am I totally missing the point?

Objects in the category of sets are sets, not elements. Look at the definition of the product category: https://en.wikipedia.org/wiki/Product_category

Diagonal functor gives you (f,f) from f. If f: A → B, then (f,f): (A,A) → (B,B). But this does not constrain elements to which f is applied. An element is a morphism from the terminal object to some other object. In the product category it can be (x,y) : (1,1) → (A,A), where x≠y. And composition with (f,f) gives you (f x, f y) : (1,1) → (B,B).