The Stacks project

7.37 Morphisms between points

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.

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

  1. Describe the automorphisms of the point of $\mathcal{T}_ G$ described in Example 7.33.7.

  2. Describe $\mathop{\mathrm{Mor}}\nolimits (p, p')$ in terms of $\mathop{\mathrm{Mor}}\nolimits (p_*, p'_*)$.

  3. 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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