Holmstrom Pseudo-abelian category

An Additive category $\mathcal{C}$ is called pseudo-abelian if of for every endomorphism $e \in \mathcal{C}(A,A)$ such that $e^2=e$, one can write $A$ as a direct sum $A_1 \oplus A_2$ such that $e$ is the composition of the projection $A \to A_1$ and the inclusion $A_1 \to A$. We call $A_1$ the image of $e$, and denote it by $e(A)$. In this situation, $A_2$ will be the image of $1-e$.

Let $\mathcal{C}$ be any additive category. Write $\mathcal{C}^{ps}$ for the category in which the objects are pairs $(A,e)$ where $A$ is and object of $\mathcal{C}$ and $e$ is an idempotent endomorphism of $A$, and where the morphisms from $(A,e)$ to $(A', e')$are given by the subgroup of $\mathcal{C}(A,A')$ of morphisms of the form $e' \circ f \circ e$. It is easy to verify that $\mathcal{C}^{ps}$ is pseudo-abelian, and we call it the pseudo-abelian envelope of $\mathcal{C}$. The obvious functor $\mathcal{C} \to \mathcal{C}^{ps}$ is fully faithful.

Balmer and Schlichting proved that a pseudo-abelian envelope of a triangulated category is also triangulated.

http://ncatlab.org/nlab/show/Karoubian+category

http://www.ncatlab.org/nlab/show/Karoubi+envelope

nLab page on Pseudo-abelian category