nLab symmetric endofunctor

Idea

A symmetric endofunctor is an endofunctor together with actions of the symmetric groups on its automorphism groups.

Definition

Let $\Phi$ be a graded monoid in the category of groups, e.g. the graded monoid $\Sigma = (\Sigma_n)_{n \ge 0}$ of symmetric groups.

A symmetric endofunctor $F : C \to C$ is an endofunctor together with the data of, for each $n \in \mathbf{N}$, a group homomorphism

$$\Phi_n \to \Aut(F^{\circ n})$$

where $F^{\circ n}$ is the $n$-fold composite of $F$.