Groups are in particular sets equipped with two operations: a binary operation (the group operation) and a unary operation (inverse) . Using these two operations, we can build up many other operations, such as the ternary operation , and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).

When we think of groups as objects of the category , where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?

It turns out that the answer is yes. The rest of this post will describe a general categorical definition of -ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.