Universal algebra nullary operations

May 10, 2020

Recall universal algebra approach to definining algebraic structures. Instead of explicitly stating existence of certain elements – for exapmle, by designating a unit element in a group or associating an inverse – we replace them with a collection of functions.

With this in mind, if \(A\) is a group, a unit element can be replaced with 0-arity function that chooses an element of \(A\) . An inverse is a 1-arity function \(A \rightarrow A\) . A group operation (e.g. product) is a 2-arity function \(A \times A \rightarrow A\) . And so on.

Now on the other hand, a 0-arity function seems like a misnomer when thought of as number of arguments. If a function selects an element (e.g. a unit element), it can be associated with a function of a one element set \(\{*\} \rightarrow A\) that maps that one element set to an element of \(A\) . So, as a function, it actually takes 1 argument, not zero.

However, if we think of A as an object in the \(\mathbf{Sets}\) category of sets, we can associate each n-ary universal algebra function with a projection morphism from a finite product of n copies of \(A\) . Treating an empty product (a nullary operation) as a terminal object in a category (which coincides with the definition of a terminal object via a limit of an empty diagram) we obtain our “selector” function \(\{*\} \rightarrow A\) . Pretty neat.

What’s even neater is that by replacing the category of sets with a category of, say, topological spaces \(\mathbf{Top}\) , we can combine our universal algebra definition to obtain more sophisticated algebraic structures – in this case, since the morphisms are continuous, we obtain a definition of a topological group. And we only need to change the underlying category. Sweet.