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Let $C$ be a category with filtered projective limits and coproducts (denoted by $+$). Then for all filtered projective systems $\{X_i\}$, $\{Y_i\}$ in $C$ there is a canonical morphism

$$\alpha ~:~ \lim_i X_i + \lim_i Y_i \longrightarrow \lim_i (X_i + Y_i).$$

Question. For which categories $C$ is this an isomorphism? In other words, when does the coproduct functor $+ : C \times C \to C$ preserve filtered projective limits? More specifically, let's say $C$ is algebraic, i.e. the category of algebras of some type $(\Sigma,E)$. For which $(\Sigma,E)$ is $\alpha$ an isomorphism?

Examples. $\bullet$ For the categories $(\mathrm{Set})$, $(\mathrm{Set}_*)$, $(\mathrm{Mod}(R))$, $(\mathrm{CMon})$ it is an isomorphism (direct computation). $\bullet$ For $(\mathrm{CRing})$ it is no isomorphism (take $X_n = \mathbb{Z}/p^n$ and $Y_n = \mathbb{Z}/q^n$ for primes $p \neq q$, then $\alpha$ is $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_q \to 0$). $\bullet$ It also fails for $(\mathrm{Grp})$ (take $X_n = Y_n$ the free groups with generators $x_1,\dotsc,x_n$ resp. $y_1,\dotsc,y_n$ with $X_{n+1} \twoheadrightarrow X_n$ mapping $x_{n+1} \to 1$. Then $(x_1 \cdot y_1 \cdot \dotsc x_n \cdot y_n)_n \in \lim_n (X_n * Y_n)$ does not lie in the image of $\alpha$). A similar counterexample works in $(\mathrm{Mon})$.

The examples mentioned above indicate that the property is related to disjoint coproducts. If there is no general necessary/sufficient criterion available, can you give some more examples?

Added. The dual question is also interesting: Let $C$ be a category with filtered colimits and products. When is the canonical morphism $\beta : \mathrm{colim}_i (X_i \times Y_i) \to (\mathrm{colim}_i X_i) \times (\mathrm{colim}_i Y_i)$ an isomorphism? This is true for $C=(\mathrm{Set})$, and then for every algebraic category since filtered colimits and products are reflected. This property guarantees that for every topological space $X$ the associated-sheaf functor $\mathrm{PSh}(X;C) \to \mathrm{Sh}(X;C)$ preserves products and that the stalk maps $\mathrm{Sh}(X;C) \to C$, $F \mapsto F_x$ preserve products.

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It seems like any Grothendieck abelian category satisfies this, almost by definition... So (1) Ind(C) for any small abelian category, (2) Qcoh(X), (3) Ch(A) for one of those A's. –  Dylan Wilson Jan 31 '12 at 10:23
Thanks! Can you add this as an answer? I don't see why it should be true for Grothendieck categories (the axiom don't tell us anything about infinite projective limits), so it would be nice to include a proof :). –  Martin Brandenburg Jan 31 '12 at 10:27
Any additive category surely satisfies this, as in an additive category finite coproducts are the same thing as finite products (and products, being a particular case of projective limits, always commute with projective limits). –  Leonid Positselski Jan 31 '12 at 11:24
@Leonid: Of course! Thanks. –  Martin Brandenburg Jan 31 '12 at 12:36

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