# A mathematical intuition for empty variants and tuples

The goal of this post is to introduce sum types and product types from a very high level perspective, and to try and derive an intuition for what the meaning of tuple<> and variant<> should be. The mathematics in this post are purposefully kept a bit vague, because being more formal would make the post heavier.

This post was lifted from an email I was going to send in answer to this message on the Boost.Devel mailing list. The email was getting a bit long and I thought it was a good candidate for a short blog post, so here we are. Check out the initial thread if you’re interested in the std::variant proposal.

Without getting into category theoretical stuff, we know that a tuple is what’s called a product type, i.e. the type tuple<A, B> is essentially equivalent to the cartesian product of the types A and B. Similarly, a variant is what’s called a sum type, i.e. the type variant<A, B> is essentially equivalent to a disjoint union of A and B, which is a set containing all elements of A and all elements of B, with duplicates.

Tuples and variants with arities other than 2 are simply syntactic sugar. Indeed, we have that

where $\simeq$ means “is morally equivalent to”. This relation of being “morally equivalent to” can be formalised with the notion of isomorphism in category theory. We won’t define isomorphisms in this post to keep things light, but for the purpose of this post you can think of two types as being “morally equivalent” if they can be converted to one another without any loss of information.

Returning to the above equivalences, we can see that arities higher than 2 are really just provided for convenience. Indeed, you could express a n-ary tuple or variant by applying the above equations recursively, and you would always end up with nested binary tuples or variants. Hence, for the remainder of this post, we will only consider tuples and variants with arities lower than or equal to 2. Continuing on this road, we can see that some types have a special interaction with these product types and sum types. Indeed, consider the following type, often called the “unit type”:

This is a singleton type, which contains only one possible value. Not very interesting, except that when you use it in a tuple, it makes the tuple “morally equivalent” to the other type in the tuple:

Indeed, that is because the Unit type does not bring any new information to the tuple, so having a tuple<Unit, T> and having a T carries the same amount of information. If you think about a tuple as a product, you can kinda imagine the Unit type as being 1:

Now, suppose you had the following type, often called the “bottom type”:

This unfriendly type is such that you can never create an object of that type. Whereas the Unit type was like a set containing a single element, this type is like an empty set, because there is no object of that type. Now, that object interacts with variants in a similar way as Unit did with tuples:

Indeed, having a variant<Bottom, T> is like having “either a Bottom or a T”. However, you know you can never have a Bottom, because an object of this type can’t exist. The only possibility left is that you have a T, which makes variant<Bottom, T> “morally equivalent” to T. If you think of a variant as a sum, you can kinda imagine the Bottom type as being 0:

Back to the initial question about lower arity tuples and variants. First, I think no one will be surprised if I say that unary tuples and variants should satisfy

This is indeed not very surprising if you think about the meaning of a tuple and a variant; clearly, a tuple or a variant with a single type in it should be “morally equivalent” to the type itself. Now, for the empty tuples and variants, I think the most mathematically correct definition would be

Hence, you could create an object of type tuple<>, but you can’t create an object of type variant<>; trying to do so should be a compile-time error. This intuition is backed by the usual behavior of products and sums on empty collections as defined in mathematics. Indeed, we normally define

This is convenient in mathematics, and I also find it quite intuitive. Naively, I think the design of variant<> and tuple<> should follow this.

### Edits

15 Jul 2015: Add more information about isomorphisms and reformulate the intro