symmetric monoidal (∞,1)-category of spectra
Given a natural number $n$, an n-ary operation on a set $S$ is a function
from the function set $\mathrm{Fin}(n) \to S$ to $S$ itself, where $\mathrm{Fin}(n)$ is the finite set with $n$ elements.
The arity of the operation is $n$.
In general, if the natural number $n$ is not specified, these are called finitary operations.
Sets equipped with finitary operations are also called finitary magmas (or “finitary groupoids” in older terminology which now clashes with another meaning of groupoid, see at historical notes on quasigroups).
More generally, a finitary operation in a multicategory is just a multimorphism.
More generally, one could use an arbitrary set instead of a finite set. However, the generalizations are only definable in closed multicategories, rather than any multicategory.
Every set $S$ with an $n$-ary operation $\phi$ comes with an endomorphism called the $n$-th power operation
where $S \overset{diag_n}{\longrightarrow} S^n$ is the diagonal morphism.
Wikipedia, n-ary group
Steven Duplij, Polyadic Algebraic Structures, IOP (2022) [ISBN:978-0-7503-2646-9]
Last revised on May 12, 2024 at 20:53:44. See the history of this page for a list of all contributions to it.