The Stacks project

Lemma 12.4.2. Let $\mathcal{C}$ be a preadditive category. The following are equivalent

  1. $\mathcal{C}$ is Karoubian,

  2. every idempotent endomorphism of an object of $\mathcal{C}$ has a cokernel, and

  3. given an idempotent endomorphism $p : z \to z$ of $\mathcal{C}$ there exists a direct sum decomposition $z = x \oplus y$ such that $p$ corresponds to the projection onto $y$.

Proof. Assume (1) and let $p : z \to z$ be as in (3). Let $x = \mathop{\mathrm{Ker}}(p)$ and $y = \mathop{\mathrm{Ker}}(1 - p)$. There are maps $x \to z$ and $y \to z$. Since $(1 - p)p = 0$ we see that $p : z \to z$ factors through $y$, hence we obtain a morphism $z \to y$. Similarly we obtain a morphism $z \to x$. We omit the verification that these four morphisms induce an isomorphism $x = y \oplus z$ as in Remark 12.3.6. Thus (1) $\Rightarrow $ (3). The implication (2) $\Rightarrow $ (3) is dual. Finally, condition (3) implies (1) and (2) by Lemma 12.3.10. $\square$

Comments (4)

Comment #539 by Nuno on

Minor typo: "induce an isomorphsm"

Comment #9223 by on

One could add another equivalent property:

(4) Every idempotent endomorphism is of the form , where and is a section of .

Proof. (3)(4). Clear.

(4)(3). Suppose that is a retraction of . Then is an idempotent endomorphism and thus so is . Hence, , where and is a section of . We already know that , , , so by Remark 12.3.6 it suffices to see , . Note , thus by monicity of . Analogously, .

There are also:

  • 1 comment(s) on Section 12.4: Karoubian categories

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09SH. Beware of the difference between the letter 'O' and the digit '0'.