The Stacks project

6.13 Stalks of presheaves of algebraic structures

The proof of Lemma 6.12.1 will work for any type of algebraic structure such that directed colimits commute with the forgetful functor.

Lemma 6.13.1. Let $\mathcal{C}$ be a category. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that

  1. $F$ is faithful, and

  2. directed colimits exist in $\mathcal{C}$ and $F$ commutes with them.

Let $X$ be a topological space. Let $x \in X$. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$. Then

\[ \mathcal{F}_ x = \mathop{\mathrm{colim}}\nolimits _{x\in U} \mathcal{F}(U) \]

exists in $\mathcal{C}$. Its underlying set is equal to the stalk of the underlying presheaf of sets of $\mathcal{F}$. Furthermore, the construction $\mathcal{F} \mapsto \mathcal{F}_ x$ is a functor from the category of presheaves with values in $\mathcal{C}$ to $\mathcal{C}$.

Proof. Omitted. $\square$

By the very definition, all the morphisms $\mathcal{F}(U) \to \mathcal{F}_ x$ are morphisms in the category $\mathcal{C}$ which (after applying the forgetful functor $F$) turn into the corresponding maps for the underlying sheaf of sets. As usual we will not distinguish between the morphism in $\mathcal{C}$ and the underlying map of sets, which is permitted since $F$ is faithful.

This lemma applies in particular to: Presheaves of (not necessarily abelian) groups, rings, modules over a fixed ring, vector spaces over a fixed field.

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