The Stacks project

Lemma 12.4.3. Let $\mathcal{D}$ be a preadditive category.

  1. If $\mathcal{D}$ has countable products and kernels of maps which have a right inverse, then $\mathcal{D}$ is Karoubian.

  2. If $\mathcal{D}$ has countable coproducts and cokernels of maps which have a left inverse, then $\mathcal{D}$ is Karoubian.

Proof. Let $X$ be an object of $\mathcal{D}$ and let $e : X \to X$ be an idempotent. The functor

\[ W \longmapsto \mathop{\mathrm{Ker}}( \mathop{Mor}\nolimits _\mathcal {D}(W, X) \xrightarrow {e} \mathop{Mor}\nolimits _\mathcal {D}(W, X) ) \]

if representable if and only if $e$ has a kernel. Note that for any abelian group $A$ and idempotent endomorphism $e : A \to A$ we have

\[ \mathop{\mathrm{Ker}}(e : A \to A) = \mathop{\mathrm{Ker}}(\Phi : \prod \nolimits _{n \in \mathbf{N}} A \to \prod \nolimits _{n \in \mathbf{N}} A ) \]

where

\[ \Phi (a_1, a_2, a_3, \ldots ) = (ea_1 + (1 - e)a_2, ea_2 + (1 - e)a_3, \ldots ) \]

Moreover, $\Phi $ has the right inverse

\[ \Psi (a_1, a_2, a_3, \ldots ) = (a_1, (1 - e)a_1 + ea_2, (1 - e)a_2 + ea_3, \ldots ). \]

Hence (1) holds. The proof of (2) is dual (using the dual definition of a Karoubian category, namely condition (2) of Lemma 12.4.2). $\square$


Comments (0)


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Your email address will not be published.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 05QV. Beware of the difference between the letter 'O' and the digit '0'.