Weak action of a group on a category

Let $G$ be a group with unit $1$ (Left group actions, transitive actions, and faithful actions) and $Q$ a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). A weak action of $G$ on $Q$ is a choice of a functor $F_g\text{\hspace{0.17em}⁣}:Q\to Q(g\in G)$ for every element of $G$ (Covariant functor, identity functor, composite functor, and contravariant functor) such that

1. $F_1$ is the identity functor of $Q$, and
2. for all $f,g\in G$ the functors $F_{fg}$ and $F_f{F}_g$ are isomorphic, that is, there exists a natural isomorphism between them (Natural isomorphism).

No isomorphisms $F_{fg}\cong F_f{F}_g$ are chosen, they are not required to be compatible with the associativity of $G$, and no pentagon is imposed: the definition records only that such isomorphisms exist. This is Definition 2.6 of the source, stated there for a group and a category with the same comments.

Normalized coherent actions. In the identity-unit normalization, the action is coherent, or a genuine $2$-action, when isomorphisms ${\mu }_{f,g}\text{\hspace{0.17em}⁣}:F_f{F}_g⟶F_{fg}(f,g\in G)$ are chosen so that ${\mu }_{1,g}={\mathrm{i}\mathrm{d}}_{F_g}$, ${\mu }_{f,1}={\mathrm{i}\mathrm{d}}_{F_f}$ and the two composites $F_f{F}_g{F}_h\stackrel{{\mu }_{f,g}{F}_h}{\to }{F}_{fg}{F}_h\stackrel{{\mu }_{fg,h}}{\to }{F}_{fgh},F_f{F}_g{F}_h\stackrel{F_f{\mu }_{g,h}}{\to }{F}_f{F}_{gh}\stackrel{{\mu }_{f,gh}}{\to }{F}_{fgh}$ agree, the associativity (pentagon) condition. This is the normalized special case of a strong monoidal action. In the general definition, the unit constraint is a chosen natural isomorphism $u:F_1\Rightarrow {\mathrm{Id}}_Q$; when $F_1={\mathrm{Id}}_Q$, the unit triangles read ${\mu }_{f,1}=F_{f}u$ and ${\mu }_{1,g}=u{F}_g$, and need not be identity maps. No strictification to the normalized case is asserted. A coherent action with $F_1={\mathrm{Id}}_Q$ is in particular a weak action after forgetting its chosen compositors. For a general coherent action with only $u:F_1\Rightarrow {\mathrm{Id}}_Q$, first replace the identity component of the functor assignment by ${\mathrm{Id}}_Q$; the unit isomorphism and the compositors then supply the pairwise isomorphisms required by the weak definition. This replacement asserts no strictification of the coherence data.

Standing convention. The whole page uses "weak" in the sense of this definition and never silently substitutes a coherent action: whenever a compositor or pentagon argument would be needed, the weakness of the available data is stated.

Terminology. The functors $F_g$ are the components of the weak action, the assignment $g\mapsto F_g$ is its functor assignment, and a weak action is determined by the functor assignment with its prescribed identity component together with the existence of the pairwise isomorphisms in condition 2. We do not distinguish two weak actions that are naturally isomorphic componentwise.

Arbitrarily chosen pairwise isomorphisms of a weak action need not satisfy coherence: the companion-page counterexample Weak actions do not supply pentagon coherence data ↗ exhibits a weak action with chosen pairwise isomorphisms violating the pentagon. That example also admits identity compositors satisfying coherence, so it does not assert that no coherent choice exists.