Definition

More precisely, an operad $O$ in Set induces a monad $T$ on Set:

$T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$

Such a monad $T$ is equipped with a canonical weakly cartesian natural transformation to the monad $Sym$ arising from the commutative operad.

Finitary monads correspond to algebraic theories, and the analytic monads correspond to algebraic theories that are “linear regular”, that is to say, they can be presented using only equations where the same variables appear on both sides and exactly once on each side (see e.g. Szawiel-Zawadowski).

Properties

A theorem of Joyal Joyal states that there is a monoidal equivalence? between the monoidal category of endofunctors $Set\to Set$ that admits a weakly cartesian natural transformation to $Sym$ and the monoidal category of species, i.e., symmetric sequences in Set with the substitution product.

In particular, the category of analytic monads on Set is equivalent to the category of operads in Set.

The colored case

The correspondence carries over to colored operads (with a set of colors $C$) if we use the slice category $Set/C$ instead of Set.

The nonsymmetric case

A similar correspondence can be established for nonsymmetric operads?, except that we must include the data of a cartesian (not weakly cartesian) transformation to the monad of the associative operad, which is no longer unique.

See Example 4.2.14 in Leinster’s book Leinster.

The homotopical case

The correspondence generalizes to (∞,1)-categories, with some statements becoming more elegant. See Gepner–Haugseng–Kock GHK.

References

Last revised on June 24, 2021 at 12:34:55. See the history of this page for a list of all contributions to it.