Faithful weak action

Let $(F_g{)}_{g\in G}$ be a weak action of a group $G$ on a category $Q$ in the sense of Weak action of a group on a category, with unit $1$ of $G$ (Left group actions, transitive actions, and faithful actions). The action is faithful if for every $g\mathrm{\ne }1$ the functor $F_g$ is not isomorphic to the identity functor of $Q$, that is, there is no natural isomorphism $F_g\cong {\mathrm{Id}}_Q$ of functors $Q\to Q$ (Natural isomorphism).

Equivalently, the action is faithful when the functor assignment $g\mapsto F_g$ is injective up to natural isomorphism: if $F_g\cong F_h$ then $g=h$, since $F_g\cong F_h$ is equivalent to $F_{h^{-1}g}\cong F_{h^{-1}}{F}_g\cong {\mathrm{Id}}_Q$ by the weak action property and the definition, so that $h^{-1}g=1$ for a faithful action.

Remarks on the notion.

• Faithfulness is a property of the functor assignment $g\mapsto F_g$ itself, not of an induced action on any invariant of $Q$. In particular a group element may act nontrivially on $Q$ while inducing the identity on a Grothendieck group or another functorial invariant; the pair of this definition with Equal actions on K_0 do not imply isomorphic derived autoequivalences ↗ records exactly that contrast.
• Because the components of a weak action are compared only through the existence of natural isomorphisms $F_{fg}\cong F_f{F}_g$, faithfulness is a property of the weak action and not of an underlying coherent $2$-action: nothing in the definition refers to the compositors ${\mu }_{f,g}$ of a coherent action, and the notion is well defined for a bare functor assignment.
• The source states this definition for the action of the braid group on $C_m$ and proves faithfulness in Corollary 1.2; the definition here is the general one used on this page, of which that statement is the instance The Khovanov-Seidel weak braid action is faithful.