Coherent action of a group on a category

Setting. Let $G$ be a group and let $\mathcal{C}$ be a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection); write $\mathrm{End}(\mathcal{C})$ for the strict monoidal category of endofunctors of $\mathcal{C}$ and natural transformations between them, with composition of functors as its product (Covariant functor, identity functor, composite functor, and contravariant functor, Natural isomorphism). Composition of endofunctors is strictly associative and its unit is ${\mathrm{i}\mathrm{d}}_{\mathcal{C}}$, so no associator constraint of $\mathcal{C}$ enters anything below.

Data. A coherent action of $G$ on $\mathcal{C}$ consists of:

1. a functor $F_g\text{\hspace{0.17em}⁣}:\mathcal{C}\to \mathcal{C}$ for every $g\in G$, with $F_1={\mathrm{i}\mathrm{d}}_{\mathcal{C}}$;
2. for all $f,g\in G$ a chosen natural isomorphism ${\mu }_{f,g}\text{\hspace{0.17em}⁣}:F_f{F}_g\Rightarrow F_{fg}$;
3. a chosen natural isomorphism $u\text{\hspace{0.17em}⁣}:F_1\Rightarrow {\mathrm{i}\mathrm{d}}_{\mathcal{C}}$, i.e. (since $F_1={\mathrm{i}\mathrm{d}}_{\mathcal{C}}$) a natural automorphism of the identity functor.

Pentagon. For all $f,g,h\in G$ the two composites $F_f{F}_g{F}_h\Rightarrow F_{fgh}$ agree: ${\mu }_{fg,h}\circ ({\mu }_{f,g}{F}_h)={\mu }_{f,gh}\circ (F_f{\mu }_{g,h}),$ where ${\mu }_{f,g}{F}_h$ is the whiskered natural transformation with components ${\mu }_{f,g,F_{h}X}$ and $F_f{\mu }_{g,h}$ the one with components $F_f({\mu }_{g,h,X})$.

Unit triangles. For all $f,g\in G$, ${\mu }_{f,1}=F_{f}u,{\mu }_{1,g}=u{F}_g\mathrm{.}$ Here $F_{f}u$ has components $F_f(u_X)\text{\hspace{0.17em}⁣}:F_f{F}_{1}X\to F_{f}X$, while $u{F}_g$ has components $u_{F_{g}X}\text{\hspace{0.17em}⁣}:F_1{F}_{g}X\to F_{g}X$. Since $F_1$ is the identity functor, these are natural transformations $F_f\Rightarrow F_f$ and $F_g\Rightarrow F_g$, respectively. They need not agree: for a general natural automorphism $u$ of ${\mathrm{i}\mathrm{d}}_{\mathcal{C}}$, naturality does not imply $u{F}_g=F_{g}u$, whose components are $u_{F_{g}X}$ and $F_g(u_X)$. The two equalities are the left and right unit axioms for a monoidal functor $G\to \mathrm{End}(\mathcal{C})$ with unit constraint $u^{-1}\text{\hspace{0.17em}⁣}:{\mathrm{i}\mathrm{d}}_{\mathcal{C}}\Rightarrow F_1$ and the discrete monoidal structure on $G$.

Relation to a weak action. The underlying functors form a weak action of $G$ on $\mathcal{C}$ in the sense of Weak action of a group on a category: the chosen ${\mu }_{f,g}$ in particular exhibit isomorphisms $F_{fg}\cong F_f{F}_g$ for all $f,g$. The converse does not hold: a weak action records no chosen compositors and imposes no pentagon, and a coherent action is precisely a weak action equipped with chosen compositors and a chosen unit satisfying the pentagon and the two unit triangles above. The pentagon is part of the data; it is not a formal consequence of the existence of isomorphisms $F_{fg}\cong F_f{F}_g$.

Rouquier's instance. In the application of this page, $\mathcal{C}=K^b(R\text{-grmod})$. For $g\mathrm{\ne }1$, the functor $F_g$ is left tensoring by the invertible object $G_g$ of Rouquier's rigidification; set $F_1={\mathrm{i}\mathrm{d}}_{\mathcal{C}}$ and use the canonical unit identification $R{\otimes }_R-\cong {\mathrm{i}\mathrm{d}}_{\mathcal{C}}$ for the empty-word object $G_1=R$. The isomorphisms ${\mu }_{f,g}$ are induced by the unique maps $m_{f,g}:G_f{\otimes }_R{G}_g\to G_{fg}$ compatible with the canonical comparisons $c_{t,u}$ of Canonical comparisons between standard graph tensor products in the derived category, with the canonical tensor unit identifications when an index or product is $1$. The map $m_1:G_1\to R$ corresponds, under the unit identification, to $u={\mathrm{i}\mathrm{d}}_{\mathcal{C}}$. Rouquier's construction produces the pentagon and unit triangles by lifting associative graph multiplication together with the additive internal shifts of signed words: in this normalization the derived models $R_{\pi (v)}(-e(v))$ are the pullback of the strict $W\times \mathbb{Z}$ action along $v\mapsto (\pi (v),e(v))$, where $\pi$ is the permutation projection and $e$ the signed word exponent; the lifts are fixed by localization isomorphisms and normalized uniqueness. The definitions make sense for an arbitrary group and category, and no Hecke algebra enters.

Degenerate cases. If $G$ is the trivial group then $F_1=\mathrm{i}\mathrm{d}$ and the data reduce to natural automorphisms ${\mu }_{1,1}$ and $u$ of ${\mathrm{i}\mathrm{d}}_{\mathcal{C}}$. The unit axioms give ${\mu }_{1,1}=u$; with this value the pentagon is automatic. Thus a coherent action of the trivial group may still have any invertible natural automorphism $u$ as its unit; taking $u=\mathrm{i}\mathrm{d}$ gives the identity example. If $\mathcal{C}$ is the one-object category attached to a monoid then the definition reduces to the usual coherence data on that monoid. For a $k$-linear category, an invertible scalar multiple of the identity is a natural automorphism, so $u$ need not be the identity. Once the compositors are fixed, however, the unit triangle at $f=g=1$ forces $u={\mu }_{1,1}$.