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In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of showing this?

I am mainly interested in this because you use this fact for the computation of stalks of a sheaf, and usually this is proven by saying: "check it if you don't believe it"

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  • $\begingroup$ I don't know about "slick", but there is certainly a general way of showing this for categories of finitary algebraic structures. $\endgroup$ – Zhen Lin Nov 11 '14 at 11:02
  • $\begingroup$ I would be interested in this way. Do you have any resources? The nLab wasn't very helpful (or I overlooked something) $\endgroup$ – Andre Knispel Nov 11 '14 at 18:05
  • $\begingroup$ Well, if you know how to do it for a few examples it is easy to extrapolate (provided you understand what finitary algebraic structures are). $\endgroup$ – Zhen Lin Nov 11 '14 at 18:35
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1) Let $T$ be a monad on a cocomplete category $C$. It is a general and trivial fact that the forgetful functor $\mathsf{Mod}(T) \to C$ preserves all colimits which $T$ presveres. In particular, if $T$ preserves filtered colimits (one then says that $T$ is finitary), then the forgetful functor does so. If $T$ is given by a theory of finite operations, $C$ has products which preserve filtered colimits in each variable, then $T$ is finitary.

2) Stalks of sheaves on $X$ commute with colimits because the stalk functor is left adjoint; in fact it is given by $i^*$ where $i : \{x\} \to X$, with right adjoint $i_*$.

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