Idempotent completion of linear categories and Yoneda

Question

Let $ \text{Vect} $ be the category of finite dimensional vector spaces over an algebraically closed field. The idempotent completion of a $\text{Vect}$-category $ \mathcal{C} $ may be though of in two different ways:

• As a category with objects $ (X,e_X) $ where $ X $ is in $ \mathcal{C} $ and $ e_X $ is an idempotent in $ \text{End}(X) $.
• As the the additive subcategory of $ \text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect}) $ (linear functors from $\mathcal{C}^{\text{op}} $ to $ \text{Vect}$) generated by direct summands of $ \mathbb{Y}(X) $ for $ X $ in $ \mathcal{C} $ (where $ \mathbb{Y} $ is the Yoneda embedding).

To an object-idempotent pair $ (X,e_X) $ one can associate the (contravariant) functor

$$ \begin{align} (X,e_X)^\sharp \colon \mathcal{C} &\to \text{Vect} \\ Y &\mapsto \{ f \in \text{Hom}(Y,X) \mid e_X \circ f = f \} \end{align} $$

which provides an equivalence between these two categories.

Question

From the above discussion we see that the following are equivalent

• $ \mathbb{Y}\colon \mathcal{C} \to \text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect}) $ is an idempotent completion.
• Every contravariant functor from $\mathcal{C}$ to $\text{Vect}$ is naturally isomorphic to $ (X,e_X)^\sharp $ for some $X$ in $\mathcal{C}$ and some idempotent $e_X$ in $\text{End}(\mathcal{C}) $.

Under what assumption on $ \mathcal{C} $ do these statements hold? For example, they hold if $ \mathcal{C} $ is semisimple with finitely many simple objects. I would like to know if they still hold without the assumption that $ \mathcal{C} $ has finitely many simple objects. I would also like to know if they hold when $\text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect})$ is semisimple (even if $ \mathcal{C} $ isn't).

Answer 1

Edit: This answer currently addresses a previous version of the question; here $\text{Vect}$ denotes the category of all vector spaces.

We can replace $C$ with its idempotent completion WLOG, so the question becomes: for $C$ an idempotent complete $k$-linear category, when is every ($k$-linear) presheaf $C^{op} \to \text{Vect}$ representable?

The answer is: iff $C$ is the zero category, by which I mean the $k$-linear category with only the zero object, or the empty category. (I'm a little confused as to whether the empty category should be regarded as being idempotent complete.)

Suppose $C$ is neither of these, so it has at least one nonzero object $c$. Then the representable presheaf $\text{Hom}(-, c) : C^{op} \to \text{Vect}$ takes at least one nonzero value. Now consider the presheaf $F(-) = \text{Hom}(-, c) \otimes W$ for an infinite-dimensional vector space $W$. We want to show that $F$ is not representable. It will suffice to show that $\text{Hom}(F, -)$ does not preserve colimits.

$F$ itself is the filtered colimit of the presheaves $\text{Hom}(-, c) \otimes V$ as $V$ runs over all finite-dimensional subspaces of $W$, and if $\text{Hom}(F, -)$ preserved this filtered colimit then every natural transformation $F \to F$ would factor through $\text{Hom}(-, c) \otimes V$ for some finite-dimensional $V$. But the identity natural transformation does not factor in this way, as we can see by plugging in $(-) = c$ (since $c$ is nonzero, $\text{End}(c)$ is also nonzero).