The Zappa-Sz\'{e}p product of twisted groupoids

We define and study the external and the internal Zappa-Sz\'{e}p product of twists over groupoids. We determine when a pair (Σ₁,Σ₂) of twists over a matched pair (G₁,G₂) of groupoids gives rise to a Zappa-Sz\'{e}p twist Σ over the Zappa-Sz\'{e}p product G₁₂. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Sz\'{e}p twist Σ→ G₁₂ is a C*-blend of its subalgebras corresponding to the subtwists Σᵢ→ Gᵢ. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Sz\'{e}p twist Σ→ G₁₂ of two twists Σ₁→ G₁ and Σ₂→ G₂.