Fix a countably infinite set of names $𝔸$. A nominal set [1] is a set $𝑋$ equipped with an action by the group of finite permutations $\text{𝖯𝖾𝗋𝗆}(𝔸)$, such that every element $𝑥\in 𝑋$ has a finite support.

A finite set of names $𝑆\subseteq 𝔸$ supports $𝑥$ if any permutation fixing $𝑆$ pointwise also fixes $𝑥$.

The category of nominal sets is equivalent to the Schanuel topos. Under this equivalence, a nominal set $𝑋$ corresponds to a functor $𝕀\to \text{𝐒𝐞𝐭}$, where $𝕀$ is the category of finite sets and injections, given by mapping a finite set of names $𝑡$ to the set of elements supported by $𝑡$: