6.5 Presheaves of algebraic structures
Let us clarify the definition of presheaves of algebraic structures. Suppose that $\mathcal{C}$ is a category and that $F : \mathcal{C} \to \textit{Sets}$ is a faithful functor. Typically $F$ is a “forgetful” functor. For an object $M \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we often call $F(M)$ the underlying set of the object $M$. If $M \to M'$ is a morphism in $\mathcal{C}$ we call $F(M) \to F(M')$ the underlying map of sets. In fact, we will often not distinguish between an object and its underlying set, and similarly for morphisms. So we will say a map of sets $F(M) \to F(M')$ is a morphism of algebraic structures, if it is equal to $F(f)$ for some morphism $f : M \to M'$ in $\mathcal{C}$.
In analogy with Definition 6.4.4 above a “presheaf of objects of $\mathcal{C}$” could be defined by the following data:
a presheaf of sets $\mathcal{F}$, and
for every open $U \subset X$ a choice of an object $A(U) \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$
subject to the following conditions (using the phraseology above)
for every open $U \subset X$ the set $\mathcal{F}(U)$ is the underlying set of $A(U)$, and
for every $V \subset U \subset X$ open the map of sets $\rho _ V^ U: \mathcal{F}(U) \to \mathcal{F}(V)$ is a morphism of algebraic structures.
In other words, for every $V \subset U$ open in $X$ the restriction mappings $\rho ^ U_ V$ is the image $F(\alpha ^ U_ V)$ for some unique morphism $\alpha ^ U_ V : A(U) \to A(V)$ in the category $\mathcal{C}$. The uniqueness is forced by the condition that $F$ is faithful; it also implies that $\alpha ^ U_ W = \alpha ^ V_ W \circ \alpha ^ U_ V$ whenever $W \subset V \subset U$ are open in $X$. The system $(A(-), \alpha ^ U_ V)$ is what we will define as a presheaf with values in $\mathcal{C}$ on $X$, compare Sites, Definition 7.2.2. We recover our presheaf of sets $(\mathcal{F}, \rho _ V^ U)$ via the rules $\mathcal{F}(U) = F(A(U))$ and $\rho _ V^ U = F(\alpha _ V^ U)$.
Definition 6.5.1. Let $X$ be a topological space. Let $\mathcal{C}$ be a category.
A presheaf $\mathcal{F}$ on $X$ with values in $\mathcal{C}$ is given by a rule which assigns to every open $U \subset X$ an object $\mathcal{F}(U)$ of $\mathcal{C}$ and to each inclusion $V \subset U$ a morphism $\rho _ V^ U : \mathcal{F}(U) \to \mathcal{F}(V)$ in $\mathcal{C}$ such that whenever $W \subset V \subset U$ we have $\rho _ W^ U = \rho _ W^ V \circ \rho _ V^ U$.
A morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves with value in $\mathcal{C}$ is given by a morphism $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ in $\mathcal{C}$ compatible with restriction morphisms.
Definition 6.5.2. Let $X$ be a topological space. Let $\mathcal{C}$ be a category. Let $F : \mathcal{C} \to \textit{Sets}$ be a faithful functor. Let $\mathcal{F}$ be a presheaf on $X$ with values in $\mathcal{C}$. The presheaf of sets $U \mapsto F(\mathcal{F}(U))$ is called the underlying presheaf of sets of $\mathcal{F}$.
It is customary to use the same letter $\mathcal{F}$ to denote the underlying presheaf of sets, and this makes sense according to our discussion preceding Definition 6.5.1. In particular, the phrase “let $s \in \mathcal{F}(U)$” or “let $s$ be a section of $\mathcal{F}$ over $U$” signifies that $s \in F(\mathcal{F}(U))$.
This notation and these definitions apply in particular to: Presheaves of (not necessarily abelian) groups, rings, modules over a fixed ring, vector spaces over a fixed field, etc and morphisms between these.
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