6.3 Presheaves
Definition 6.3.1. Let X be a topological space.
A presheaf \mathcal{F} of sets on X is a rule which assigns to each open U \subset X a set \mathcal{F}(U) and to each inclusion V \subset U a map \rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V) such that \rho ^ U_ U = \text{id}_{\mathcal{F}(U)} and whenever W \subset V \subset U we have \rho ^ U_ W = \rho ^ V_ W \circ \rho ^ U_ V.
A morphism \varphi : \mathcal{F} \to \mathcal{G} of presheaves of sets on X is a rule which assigns to each open U \subset X a map of sets \varphi : \mathcal{F}(U) \to \mathcal{G}(U) compatible with restriction maps, i.e., whenever V \subset U \subset X are open the diagram
\xymatrix{ \mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho ^ U_ V} & \mathcal{G}(U) \ar[d]^{\rho ^ U_ V} \\ \mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V) }
commutes.
The category of presheaves of sets on X will be denoted \textit{PSh}(X).
The elements of the set \mathcal{F}(U) are called the sections of \mathcal{F} over U. For every V \subset U the map \rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V) is called the restriction map. We will use the notation s|_ V := \rho ^ U_ V(s) if s\in \mathcal{F}(U). This notation is consistent with the notion of restriction of functions from topology because if W \subset V \subset U and s is a section of \mathcal{F} over U then s|_ W = (s|_ V)|_ W by the property of the restriction maps expressed in the definition above.
Another notation that is often used is to indicate sections over an open U by the symbol \Gamma (U, -) or by H^0(U, -). In other words, the following equalities are tautological
\Gamma (U, \mathcal{F}) = \mathcal{F}(U) = H^0(U, \mathcal{F}).
In this chapter we will not use this notation, but in others we will.
Definition 6.3.2. Let X be a topological space. Let A be a set. The constant presheaf with value A is the presheaf that assigns the set A to every open U \subset X, and such that all restriction mappings are \text{id}_ A.
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