Cartesian simplicial sheaves model actual equivariant derived Hom

The ambient category and its Hom groups. By [F1] the category ${\mathrm{Sh}}_k(Z_{\bullet })$ is abelian with levelwise exactness, has enough injectives under AC via the coinduced objects $N_{p}I$, and its bounded derived category is actual with set-sized Hom groups computed by bounded-below injective replacements. This is the ambient category of the definition, with the ordinary shift and cone triangles of Derived category of an abelian category.

Closure under the triangulated operations. Shifts and isomorphisms preserve Cartesian cohomology and the constructibility of the degree-zero restriction because the structure maps of a shifted or isomorphic complex are the shifts or conjugates of the original structure maps. For a cone $C(f)$ of a morphism $f:A\to B$ of $D_{G,c}^b(X;k)$, the structure maps of $C(f)$ are the cones of the levelwise structure maps, and exactness of simplicial inverse image identifies the structural comparison for $C(f)$ with the map of cones of the structural comparisons for $A$ and $B$. Those two comparisons are quasi-isomorphisms by Cartesianity, so their cone comparison is a quasi-isomorphism by the long exact cohomology sequences. Thus $C(f)$ has Cartesian cohomology. The degree-zero restriction $C(f{)}_0$ is the cone of $f_0$ between constructible complexes, and by the common-stratification closure of [F2] it is again constructible. Hence $D_{G,c}^b(X;k)$ is a full triangulated subcategory of the ambient bounded derived category.

Conservative restriction. Let $A\in D_{G,c}^b(X;k)$ with $A_0\cong 0$ in $D^b({\mathrm{Sh}}_k(X))$. All cohomology sheaves of $A_0$ vanish, and by Cartesianity the structure morphism of $H^j(A)$ along the projection $d_0^p:Z_p\to Z_0=X$ identifies $H^j(A{)}_p$ with the pullback of $H^j(A_0)$ for every $p$, so every cohomology simplicial sheaf of $A$ vanishes. An object of the bounded derived category with vanishing cohomology objects is zero by [F2], so $A\cong 0$. Restriction to $X$ is thus conservative.

The finite resolution triples. Apply [F3] to the frame resolution of $G$: the simplicial map $p_N:G^{\bullet }\times P_N\to G^{\bullet }\times X$ is levelwise a locally trivial bundle with $r_N$-acyclic fiber, its induced pullback is compatible with the simplicial structure, and the locally split Čech descent of [F1] for the principal bundle $P_N\to X_N$ identifies the Cartesian-cohomology derived category of the nerve of $P_N$ with $D_c^b(X_N;k)$. Composing with $Q_N$ gives the identification of $D_{G,c}^b(X;k)$ with the full bounded resolution-triple subcategory of $D_c^{[b,a]}(X_N;k)$ in the stated range: an equivariant object $A$ corresponds to the triple $(Q_N(A),p_N^{\ast }A\cong q_N^{\ast }{Q}_N(A))$, and conversely a triple whose frame pullback comes from $X$ arises from an equivariant object. The bounded Hom comparison of [F3] is induced by pullback along $p_N$ and the descent equivalence, hence carries composition to composition; at $m=0$ it is a ring comparison ${\mathrm{End}}_G(K)\cong {\mathrm{End}}_{X_N}(Q_{N}K)$ for objects in the interval, and retract closure is inherited from [F3].

Idempotent completeness. Let $e:K\to K$ be an idempotent endomorphism of a bounded object $K\in D_{G,c}^b(X;k)$. Form the telescope in the bounded-below derived category of all simplicial sheaves: let $S={⨁}_{j\ge 0}K$ be the countable sum with the shift map $1-\mathrm{shift}(e):S\to S$ sending the $j$-th summand to the $j$-th summand by $\mathrm{i}\mathrm{d}$ and to the $(j+1)$-st by $-e$, and let $T=\mathrm{Cone}(1-\mathrm{shift}(e))$. Countable coproducts in ${\mathrm{Sh}}_k(Z_{\bullet })$ exist levelwise and are exact, and the cone is levelwise, so $H^j(T)={\mathrm{colim}}_j(H^j(K)\stackrel{H^j(e)}{\to }{H}^j(K)\stackrel{H^j(e)}{\to }\cdots )=\mathrm{im}{H}^j(e)$: the colimit of an idempotent is its image, and because $H^j(e)$ is an idempotent endomorphism of the simplicial sheaf $H^j(K)$ its image is a direct summand. The image of an idempotent of a simplicial sheaf is again a simplicial sheaf, is bounded, is Cartesian because it is a summand of a Cartesian object, and restricts on $X$ to the image of the idempotent $H^j(e{)}_0$, a summand of a constructible sheaf and hence constructible of finite rank. Thus $T$ is an object of the bounded-below ambient derived category with bounded Cartesian constructible cohomology, and by truncation it defines an object of $D_{G,c}^b(X;k)$. The first-summand inclusion $i:K\to T$ satisfies $ie=i$ by the telescope relations, and the map $S\to K$ with every component $e$ annihilates $1-\mathrm{shift}(e)$, so the triangle yields $p:T\to K$ with $pi=e$. Then $ip$ is an isomorphism on cohomology, hence by [F2] an isomorphism. Also $(ip{)}^2=i(pi)p=iep=ip$, since $ie=i$. An invertible idempotent is the identity, so $ip={\mathrm{i}\mathrm{d}}_T$ and $pi=e$, splitting the idempotent inside $D_{G,c}^b(X;k)$. Hence the category is idempotent complete.

Triangulated, locally small, conservative. Steps 1.1, 1.2 and 2.1 exhibit the ambient category and its Hom groups, the triangulated structure of $D_{G,c}^b(X;k)$, and conservativity of restriction; step 2.2 gives the finite resolution identification and the degree-zero ring and retract comparisons. No derived full faithfulness of restriction to $X$ is asserted or used.

Conclusion. Steps 1.1, 1.2, 2.1 and 2.2 give the triangulated, locally small model with conservative restriction and the finite resolution comparisons; step 2.3 gives idempotent completeness. AC is consumed exactly as recorded in [A1].