Idempotents in triangulated monoidal categories

Abstract

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 $\mathrm{End}(\mathbf{E})$ is a graded commutative algebra. The same is true of $\mathrm{Hom}(\mathbf{E},{\mathbf{E}}^c[1])$ 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.

References in corpus (2)

• A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in S2 x S1
• A polynomial action on colored sl(2) link homology

Cited by in corpus (1)

• Constructing categorical idempotents