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

Abstract: We study diagrams of commutative differential graded algebras (DGAs) over the orbit category $\sO_G$ 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.