Idempotents in triangulated monoidal categories

In these notes we develop some basic theory of idempotents in monoidal categories. We introduce and study the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general idempotent decompositions of identity. If $\mathbf{E}$ is a categorical idempotent then $\operatorname{End}(\mathbf{E})$ is a graded commutative algebra. The same is true of $\operatorname{Hom}(\mathbf{E},\mathbf{E}^c[1])$ under certain circumstances, where $\mathbf{E}^c$ is the complement. These generalize the notions of cohomology and Tate cohomology of a finite dimensional Hopf algebra, respectively.