Davydov-Yetter cohomology for Tensor Triangulated Categories

One way to understand the deformation theory of a tensor category $M$ is through its Davydov-Yetter cohomology $H_{DY}^{\ast }(M)$ which in degree 3 and 4 is known to control respectively first order deformations of the associativity coherence of $M$ and their obstructions. \ In this work we take the task of developing an analogous theory for the deformation theory of tensor triangulated categories with a focus on derived categories coming from algebraic geometry. We introduce the concept of perfect pseudo dg-tensor structure $\mathrm{\Gamma }$ on an appropriate dg-category $\mathcal{T}$ as a truncated dg-lift of a tensor triangulated category structure on $H^0(\mathcal{T})$ and we define a double complex $D{Y}^{\ast ,\ast }(\mathrm{\Gamma })$ and we see that the 4th cohomology group $HD{Y}^4(\mathrm{\Gamma })$ of the total complex of $D{Y}^{\ast ,\ast }(\mathrm{\Gamma })$ contains information about infinitesimal first order deformations of the tensor structure.