In his Tohoku article, Grothendieck says that the category $\mathbf{Ab}$ of abelian groups satisfies the axiom AB6, namely


"All small colimits exist in $\mathbf{Ab}$. Moreover for any index family $J$ and filtered categories $I\_j, j\in J$, with functors $I\_j\to \mathbf{Ab}, i\mapsto M\_i$, the natural map


$ 
\underset{\substack{\rightarrow \\ (i\_j\in I\_j)\_j}}{\lim} \prod\_{j\in J} M\_{i\_j} \rightarrow \prod\_{j\in J}\underset{\substack{\rightarrow \\ i\_j\in I\_j}}{\lim}M\_{i\_j} $


is an isomorphism."


It seems that this is saying that filtered limits and arbitrary products commute, but I don't think that this commutation happens.


Let us consider, for example (this is probably not the simplest example, but it is how I recognized the problem), a profinite space $S=\lim\_i S\_i$, projective limit of finite discrete sets $S\_i$, where $i\in I=\{\text{finite quotients of } S\}$. For all $n\in \mathbb{N}$, let us consider the family $I\_j=I$, with functors $I\_j\to \mathbf{Ab}, i\mapsto \mathrm{Map}(S\_i,\mathbb{Z})$ the abelian group of maps of sets from $S\_i$ to $\mathbb{Z}$.


The term on the left is, if I understood correctly, the abelian group


$\underset{\substack{\rightarrow\\(i\in I)\_n}}{\lim} \prod\_{n\in \mathbb{N}} \mathrm{Map}(S\_i,\mathbb{Z})=\underset{\substack{\rightarrow\\(i\in I)\_n}}{\lim} \mathrm{Map}(S\_i,\prod\_{n\in \mathbb{N}} \mathbb{Z})$


i.e. the abelian group of continuous function from $S$ to $\prod\_{n\in\mathbb{N}}\mathbb{Z}$, where the latter has the **discrete topology** (every map from the profinite $S$ to a discrete abelian group factors via some quotient $S\_i$).


On the other hand, the term on the right is


$\prod\_{n\in\mathbb{N}} \underset{\substack{\rightarrow\\ i\in I}}{\lim} \mathrm{Map}(S\_i,\mathbb{Z})=\prod\_{n\in\mathbb{N}}\mathrm{Cont}(S,\mathbb{Z})=\mathrm{Cont}(S,\prod\_{n\in\mathbb{N}}\mathbb{Z})$


which is the abelian group of continuous functions $S\to \prod\_{n\in\mathbb{N}}\mathbb{Z}$, where this time the latter has the **product topology**.


Since the continuous identity morphism from the discrete $\prod \mathbb{Z}$ to itself with the product topology is not a homeomorphism, the two abelian groups $\mathrm{Cont}(S,(\prod\mathbb{Z})^{\delta})$ and $\mathrm{Cont}(S,(\prod\mathbb{Z})$ cannot coincide for all profinite space $S$.


I am sure that in the formulation of Grothendieck there are no errors, so I guess I misunderstood the meaning of the left-hand side (the filtered colimit of the products): can someone help me?



Why does the category of abelian groups satisfy the axiom AB6?
The objects we are taking the limit over in the left side are $(i\_j \in I\_j)\_j$, i.e. tuples of, for each $j\in J$, an element $i\_j$ of $I\_j$. These are the same as elements of $\prod\_{j\in J}$.


Thus, the left side is a limit over $\prod\_{j\in J} I\_j$.


On the other hand, you have defined $I\_j = I$ for all $j$ and then taken the limit over $I$. Of course in this case $I$ embeds into $\prod\_{j\in J} I$ as a "diagonal" subset, but not every element is bounded by an element on the diagonal, unless $I$ is finite.


