$C_{pq}$-Injective Diagrams and a Combination Theorem for Minimal Models

We study diagrams of commutative differential graded algebras (DGAs) over the orbit category ${\text{sO}}_G$ in the context of equivariant rational homotopy theory. For $G=C_{pq}$ with $p,q$ distinct primes, we give necessary conditions for injectivity. We prove a combination-type result: the equivariant wedge of injective diagrams over ${\mathcal{O}}_{C_p}$ and ${\mathcal{O}}_{C_q}$ with retract structure maps yields an injective diagram over ${\mathcal{O}}_{C_{pq}}$ with a level-wise minimal model. As an application, we construct examples of $C_{pq}$-formal spaces.