Lemma 7.37.1. Let \mathcal{C} be a site. Let u, u' : \mathcal{C} \to \textit{Sets} be two functors, and let t : u' \to u be a transformation of functors. Then we obtain a canonical transformation of stalk functors t_{stalk} : \mathcal{F}_{p'} \to \mathcal{F}_ p which agrees with t via the identifications of Lemma 7.32.3.
7.37 Morphisms between points
Proof. Omitted. \square
Definition 7.37.2. Let \mathcal{C} be a site. Let p, p' be points of \mathcal{C} given by functors u, u' : \mathcal{C} \to \textit{Sets}. A morphism f : p \to p' is given by a transformation of functors
f_ u : u' \to u.
Note how the transformation of functors goes the other way. This makes sense, as we will see later, by thinking of the morphism f as a kind of 2-arrow pictorially as follows:
\xymatrix{ \textit{Sets} = \mathop{\mathit{Sh}}\nolimits (pt) \rrtwocell ^ p_{p'}{f} & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) }
Namely, we will see later that f_ u induces a canonical transformation of functors p_* \to p'_* between the skyscraper sheaf constructions.
This is a fairly important notion, and deserves a more complete treatment here. List of desiderata
Describe the automorphisms of the point of \mathcal{T}_ G described in Example 7.33.7.
Describe \mathop{\mathrm{Mor}}\nolimits (p, p') in terms of \mathop{\mathrm{Mor}}\nolimits (p_*, p'_*).
Specialization of points in topological spaces. Show that if x' \in \overline{\{ x\} } in the topological space X, then there is a morphism p \to p', where p (resp. p') is the point of X_{Zar} associated to x (resp. x').
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