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For $M$ a simple $R$ module, and $N_i$ a family of $R$ modules, we have $$ \hom (M, \bigoplus_i N_i) \cong \bigoplus_i \hom (M, N_i) $$ as abelian groups. Since the direct sum is the coproduct in the category of $R$ modules, we could rewrite this as $$ \hom (M, \coprod_i N_i) \cong \bigoplus_i \hom (M, N_i). \tag{1} $$ This raises the question whether (1) is true for any abelian category, when $M$ is a simple object (defined I guess as an object such that any [regular?] monomorphism into it is an isomorphism). Does anyone know whether this is true?

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    $\begingroup$ We are restricting ourselves to cocomplete abelian categories. There is a natural injective map from RHS to LHS regardless of M being simple or not. Now is it surjective when M is simple? I wonder if you can use the fact general colimits are filtered colimits of finite colimits? $\endgroup$ – Rachmaninoff Feb 15 '15 at 6:46
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    $\begingroup$ A few points on the definition of "simple": 1. Every mono (and epi) is regular in an abelian category, so no need to worry about that. 2. The zero morphism $0 \to X$ is always mono, so you need to say that every mono is either an iso or zero. 3. The nlab gives the dual definition (using epis rather than monos) whereas wikipedia uses your definition. This certainly at least suggests that the notion probably isn't used much in the categorical literature! $\endgroup$ – tcamps Feb 15 '15 at 15:39
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It's not so much a question of simpleness as it is a question of finite generation. Indeed, in the case where $M$ is projective, the property in question is even equivalent to finite presentability.

Let $\mathcal{A}$ be an abelian category with filtered colimits. As was pointed out in the comments, infinite direct sums in $\mathcal{A}$ can be constructed using filtered colimits and finite direct sums. Thus, if $M$ is a finitely presentable object in $\mathcal{A}$, i.e. an object such that $\mathcal{A} (M, -) : \mathcal{A} \to \mathbf{Ab}$ preserves filtered colimits, then $\mathcal{A} (M, -) : \mathcal{A} \to \mathbf{Ab}$ also preserves infinite direct sums.

Now, suppose we have the following short exact sequence in $\mathcal{A}$: $$0 \longrightarrow K \longrightarrow P \longrightarrow M \longrightarrow 0$$ Then, for each object $N$ in $\mathcal{A}$, we have an induced exact sequence: $$0 \longrightarrow \mathcal{A} (M, N) \longrightarrow \mathcal{A} (P, N) \longrightarrow \mathcal{A} (K, N)$$ Suppose $P$ is finitely presentable. A little diagram chase then shows that the canonical comparison $$\bigoplus_{i \in I} \mathcal{A} (M, N_i) \to \mathcal{A} (M, \bigoplus_{i \in I} N_i)$$ is injective. In fact, it is bijective. For any morphism $M \to \bigoplus_{i \in I} N_i$ in $\mathcal{A}$, there is a finite subset $I' \subseteq I$ making the following diagram commute, $$\require{AMScd} \begin{CD} P @>>> \bigoplus_{i \in I'} N_i \\ @VVV @VVV \\ M @>>> \bigoplus_{i \in I} N_i \end{CD}$$ but $P \to M$ is an epimorphism and $\bigoplus_{i \in I'} N_i \to \bigoplus_{i \in I} N_i$ is a monomorphism, so there is a unique morphism $M \to \bigoplus_{i \in I'}$ making both triangles commute, i.e. $M \to \bigoplus_{i \in I} N_i$ factors through $\bigoplus_{i \in I'} N_i \to \bigoplus_{i \in I} N_i$.

If $\mathcal{A}$ is a locally finitely presentable abelian category, then any simple object in $\mathcal{A}$ is a quotient of a finitely presentable object. Indeed, by hypothesis, every object in $M$ is a filtered colimit of finitely presentable objects, so in particular there is an epimorphism $\bigoplus_{i \in I} N_i \to M$ where each $N_i$ is finitely presentable; but if $M$ is simple, that implies each $N_i \to M$ is either zero or an epimorphism, so there must be some $i \in I$ such that $N_i \to M$ is an epimorphism.

Thus, in view of Jeremy Rickard's counterexample, the conclusion must be that simple objects in abelian categories need not be finitely generated.

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It's not true in the opposite category of the category of vector spaces over a field. If you take $M$ to be a one-dimensional space then it is the (false) statement that $$\left(\prod_i V_i\right)^\ast\cong\bigoplus_iV_i^\ast.$$

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