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I'm reading Borceux - Handbook of categorical algebra 1, p.80-81

There is a proposition I had no problem understanding it: The forgetful functor $U:\textbf{Ab}\rightarrow \textbf{Set}$ preserves and reflects filtered colimits.

However, there appears a corollary right after this proposition that I do not understand why it follows from the above proposition. That is, "In $\textbf{Ab}$, finite limits commute with filtered colimits". How does this follow from the preceding proposition?

Let $\mathscr{C}$ be a small filtered category and $\mathscr{D}$ be a finite category and $F:\mathscr{C}\times \mathscr{D}\rightarrow \textbf{Ab}$ be a covariant functor. I know that $U$ preserves limits and filtered colimits, and filtered colimits commute with finite limits in $\textbf{Set}$.

Hence, we have the identifications as below:

$U(colim_C (lim_D F(C,D)))\cong colim_C(U(lim_D(F(C,D)))\cong colim_C(lim_D( U\circ F (C,D))) \cong lim_D(colim_C(U\circ F(C,D)))\cong lim_D(U(colim_C F(C,D)))\cong U(lim_D(colim_C F(C,D)))$.

However, this does not imply that $colim_C (lim_D F(C,D))\cong lim_D (colim_C F(C,D))$. How do I prove the corollary from the given proposition?

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1 Answer 1

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$U$ is conservative, and the isomorphism $[(a_i)]\mapsto ([a_i])$ from the colimit of limits to the limit of colimits in $\mathbf{Set}$ arises from a homomorphism of abelian groups, since in the end its components are compositions of projections from limits and inclusions to colimits which are themselves abelian groups morphisms.

A much more general argument is that $\mathbf{Ab}$, like any cocomplete category of models for an algebraic theory, is a reflective, filtered colimit-closed subcategory of a presheaf category (the keyword here is "locally presentable,") and filtered colimits commute with filtered limits in presheaves since limits and colimits are levelwise.

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