A structure on a set consists of a collection
for each natural number such that
is a Boolean subalgebra of ,
If , then and belong to ,
The set belongs to ,
If denotes the projection onto the first coordinates, then the image belongs to whenever .
Morally, a structure on a set is the collection of subsets of a finite power of which are definable with respect to a first-order language that has been interpreted in (such an interpretation being called a structure of the language). Indeed, the definable sets of such a language do form a structure in the present sense. Conversely, to any structure in this sense, we may introduce a relational language whose -ary relation symbols are named by the elements of , and then the tautological structure of this language on , where each relation is interpreted as the set that names it, is a structure of this language.
Each structure on induces a bicategory of relations: the objects are natural numbers, and 1-cells are triples where , ordered by inclusion. (It is indeed not difficult to show that 1-cells are closed under set-theoretic relational composition.)