  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 $\Gamma$ on an appropriate dg-category $\mathscr{T}$ as a truncated
dg-lift of a tensor triangulated category structure on $H^{0}(\mathscr{T})$ and
we define a double complex $DY^{\ast,\ast}(\Gamma)$ and we see that the 4th
cohomology group $HDY^{4}(\Gamma)$ of the total complex of
$DY^{\ast,\ast}(\Gamma)$ contains information about infinitesimal first order
deformations of the tensor structure.
