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

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

is 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). $\square$

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