Category-Theory and the modelling of $n$-ary functions (especially the $0$-ary functions)

Question

Hi have a questioning regarding the modelling of $n$-ary functions and constants.

First in the category of sets I know that the empty set $\emptyset$ is initial because for every other set $X$ there exists exactly one function $f$ such that $f: \emptyset \to X$ because there is no argument to choose from in the domain and set it's value, this function is unique.

I have some experience with functional programming languages and I heard that category theroy is applied in modelling them, but I am not so much into this stuff yet. But I asked myself how to model $n$-nary functions in category theory. My prior knowledge is

1) An $n$-nary function in mathematics could be modelled by cross-products, i.e a binary function on $\mathbb{N}$ is $f: \mathbb{N} \times \mathbb{N} \to X$.

2) In computer sciene and functional programming languages constants are sometimes modelled as $0$-ary functions (and in some algebraic approaches to model theory the constants are also interpreted as $0$-ary functions).

But here starts my conceptual confusion. First, the cross products of sets are also in the category of sets. So it would be now problem to represent a binary function by an arrow $f : \mathbb{N} \times \mathbb{N} \to X$. But what is the arity of the function $f : \emptyset \to X$. Because of $\emptyset \times X = \emptyset$, it should have all arities, i.e. it has $0$-arity, $1$-arity, binary and so on. So because it is $0$-arity too it should represent a constant, but what should this constant be?

But what really bothers me, how could the $0$-ary functions be represented in category theory (in the category of sets) as an arrows? I found some $0$-ary functions, the functions $f: \emptyset \to X$, but I am unsure how these could model constants?

Any hints or further suggestions for me, or is my attempt at modelling arity totally wrong?

Answer 1

Your mistake is in thinking that $\mathbb N^0$ is the empty set. But actually it isn't, it is a singleton set.

Set-theoretically, $\mathbb N^0$ is the set of all functions $\{\}\to\mathbb N$, and there is exactly one such function: the empty function. So $\mathbb N^0=\{\varnothing\}=1$, not $\varnothing$ itself.

In category theory, $\varnothing$ cannot be the unit of the product operation, because, as you have noticed $\varnothing\times A$ is not isomorphic to $A$ in general.

However $1\times A$ is isomorphic to $A$ in $\mathbf{Set}$, so $1$ is the correct domain to choose for a nullary function.

And clearly the possible functions $1\to B$ correspond exactly to the elements of $B$, which is what a "nullary function" intuitively ought to be.

Answer 2

This question has nothing to do with category theory.

A $n$-ary operation on $X$ is a map $\omega : X^n \to X$. For example, $2$-ary means binary, $1$-ary means unary. A $0$-ary operation is a map $X^0 \to X$. Since $X^0$ has exactly one element (it is the terminal object of $\mathsf{Set}$, but you don't need to know that here), this map corresponds to an element of $X$. So $0$-ary operations are constants.