| In this paper, a coseparable coring object in the tensor triangle category was inves-tigated, its comodule category which admits a pre-triangulation can be described by the separable comonad. This dissertation consists four sections.The first section is introduction. It introduces the background of monand and comonad and the main results of this thesis.The second section is preliminaries. The definitions of the trianglated category, pre-trianglated category, comonad and idempotent-complete category and also some lemmas are given.In the third section, the separated comonad was given. We prove that a left ad-joint functor is separable if and only if its corrseponding comonad is separable, then an important result of pre-trianglated category was obtained:If the functor L from an idempotent-complete suspended category D to a pre-triangulated category C is stably separable, and R is its right adjoint functor. The comonad LR of the adjunction is stable and exact. Then the category D is pre-trianglated category with distinguished trianglesΛbeing exactly the ones such that L(Λ) is distinguished in C. Moreover, both L and R are exact functors. IfΛis W - Comode, and W is stable separable and exact comonad, then conclusion still holds.In the fourth section, the n-triangle(n≥1) and triangulation of order N(N≥2) on the category was given. An important result of the triangulation of N order was obtained: If the functor L from an idempotent-complete suspended category D to a suspended category with a triangulation of N order C is stably separable, and R is its right adjoint functor. The comonad LR of the adjunction is stable and exact up to order N. Then the category D admits a triangulation of order N such that, for all n≥N, an n-triangleΛis distinguished in D exactly when the underlying n-triangle L(Λ) is distinguished in C. Moreover, both L and R are exact up to order N functors. |
