ScalingStacks

Construction 4.6. For any category $C$ enriched in a symmetric monoidal category $V$, one may of course form the opposite enriched category $X^{\mathrm{op}}$. This is an involution on the category of $V$-enriched categories.

Inducting this yields a free action $\rho$ of $(\mathbb{Z}/2)^{n}$ on $\cat_{n}$. We will show in a moment that in fact this action produces an equivalence between $(\mathbb{Z}/2)^{n}$ and the monoidal full subacategory of $\Fun(\gaunt_{n},\gaunt_{n})$ spanned by the autoequivalences.

For now, let us restrict this action: we note that $\rho$ restricts to an action on the globular category $\mathbb{G}_{n}$.