Towards quantization of quasi-categories in modules : deformations and A∞-categories in the templicial context

In this thesis we initiate the algebraic deformation theory of templicial modules. After proving some technical results about the category of necklaces, we show invariance of quasi-categories in modules and of deg-projective templicial modules under flat infinitesimal deformation. We further define Hochschild cohomology for module-valued colax functors and show that it controles infinitesimal deformations and obstructions. Lastly we construct a faithful and conservative templicial nerve functor for A∞-categories and investigate its properties.