gluing and sheaves

Author: grthomson

.obsidian/graph.json

@@ -67,6 +67,6 @@
   "repelStrength": 13.6934931506849,
   "linkStrength": 0.76138698630137,
   "linkDistance": 110,
-  "scale": 0.3807590342068506,
-  "close": false
+  "scale": 0.4032705228026572,
+  "close": true
 }

maths/algebraic-geometry/knutson_notes.md

@@ -97,7 +97,6 @@ T \to X \times_Z Y
 $$
 making the entire diagram commute.
 
-<br>
 <br>
 
 ```tikz
@@ -133,3 +132,190 @@ making the entire diagram commute.
 
 \end{document}
 ```
+
+<br>
+
+### Definition 1.2
+
+Let $\tau$ be a topology and $\mathcal{D}$ a category with products.
+
+A **presheaf** on $\tau$ with values in $\mathcal{D}$ is a functor
+
+$$F : \mathcal{C}^{\mathrm{op}} \to \mathcal{D}.$$
+
+
+A **sheaf** is a presheaf satisfying the following condition:
+
+If $\{ U_i \to U \} \in \mathrm{Cov}_\tau$, then the diagram
+
+<br> 
+
+```tikz
+\usepackage{tikz-cd}
+\usepackage{xcolor}
+
+\definecolor{mypink}{HTML}{FF69B4}
+
+\begin{document}
+
+% scaled product symbol (no extra packages)
+\newcommand{\Bigprod}{\mathop{\scalebox{1.35}{$\prod$}}}
+
+\begin{tikzcd}[
+  scale=1.2,
+  transform shape,
+  row sep=3em,
+  column sep=3.5em,
+  text=mypink,
+  draw=mypink,
+  every to/.style={draw=mypink, line width=0.9pt},
+  arrows={-Stealth}
+]
+|[font=\huge]| F(U)
+  \arrow[r, "{\raisebox{3pt}{\small\textit{\(\pi\)}}}"] &
+|[font=\huge]| \Bigprod_i F(U_i)
+  \arrow[r, shift left=0.7ex, "{\raisebox{3pt}{\small\textit{\(\pi_1\)}}}"]
+  \arrow[r, shift right=0.7ex, "{\raisebox{3pt}{\small\textit{\(\pi_2\)}}}"'] &
+|[font=\huge]| \Bigprod_{i,j} F(U_i \times_U U_j)
+\end{tikzcd}
+
+\end{document}
+
+```
+
+<br>
+
+is **exact**.
+
+(Exactness here means that $\pi$ is the equalizer of $\pi_1$ and $\pi_2$.)
+
+
+### Note: Equalizer
+
+### Definition (Equalizer)
+
+Let $\mathcal{C}$ be a category and let
+
+$$X \;\substack{\xrightarrow{\ f\ }\\[-0.6em]\xrightarrow[\ g\ ]{}}\; Y$$
+
+be two parallel morphisms in $\mathcal{C}$.
+<br>
+
+An **equalizer** of $f$ and $g$ is an object $E$ together with a morphism
+
+$$e : E \to X$$
+
+such that
+
+$$f \circ e = g \circ e,$$
+
+and which is **universal** with this property:  
+<br>
+for any object $T$ with a morphism $t : T \to X$ satisfying
+
+$$
+f \circ t = g \circ t,$$
+there exists a unique morphism
+
+$$u : T \to E$$
+
+such that
+
+$$e \circ u = t.$$
+<br>
+
+```tikz
+\usepackage{tikz-cd}
+\usepackage{xcolor}
+
+\definecolor{mypink}{HTML}{FF69B4}
+
+\begin{document}
+
+\begin{tikzcd}[
+  scale=1.3,
+  transform shape,
+  row sep=3.2em,
+  column sep=3.5em,
+  text=mypink,
+  draw=mypink,
+  every to/.style={draw=mypink, line width=0.9pt},
+  arrows={-Stealth}
+]
+|[font=\huge]| T \arrow[d, dotted, "u"'] \arrow[r, "t"] & |[font=\huge]| X \arrow[r, shift left=0.7ex, "f"] \arrow[r, shift right=0.7ex, "g"'] & |[font=\huge]| Y \\
+|[font=\huge]| E \arrow[ru, "e"'] & &
+\end{tikzcd}
+
+\end{document}
+```
+
+### Back to Sheaves
+
+## The Sheaf Condition — Intuitive Explanation
+
+Let $U \in \mathcal{C}$ be an object covered by a family of morphisms $\{U_i \to U\}$, and  
+$F : \mathcal C^{op} \to \mathcal D$ be a presheaf (assigning data to objects).
+
+Because $F$ is contravariant, each $U_i \to U$ induces a restriction map
+
+$$F(U) \to F(U_i).$$ 
+
+Collecting these gives a single map
+
+$$\pi : F(U) \to \prod_i F(U_i).$$
+
+The pieces $U_i$ overlap pairwise and each overlap is represented in the fiber product
+
+$$U_i \times_U U_j.$$
+
+From the two projection maps
+
+$$U_i \times_U U_j \to U_i, \qquad U_i \times_U U_j \to U_j,$$
+
+applying $F$ produces two restriction maps. Bundling these for all $(i,j)$ gives
+
+$$\pi_1, \pi_2 : \prod_i F(U_i) \rightrightarrows \prod_{i,j} F(U_i \times_U U_j).$$
+
+The sheaf condition says that
+
+$$F(U) \xrightarrow{\pi} \prod_i F(U_i)$$
+
+is the **equalizer** of $\pi_1$ and $\pi_2$:
+it picks out exactly those families of local data that agree on all overlaps.
+
+In words:  
+**local data that is compatible on overlaps comes from a unique global datum.**
+
+---
+
+### Final Mental Model
+
+- $F(U)$: **global data**
+- $F(U_i)$: **data on each patch**
+- $F(U_i \times_U U_j)$: **data on overlaps**
+
+The sheaf condition says:
+
+> **Global data = families of local data that agree on overlaps.**
+
+This is expressed precisely in the equalizer diagram.
+
+---
+
+## Classical Topology Version (Open Sets)
+
+If $U$ is a topological space and $\{U_i\}$ an open cover, and $F$ assigns
+(e.g.) functions, sections, or other data to open sets:
+
+- $F(U)$ is the data on the whole space,
+- $F(U_i)$ is the data on each open set,
+- $F(U_i \cap U_j)$ is the data on the overlaps.
+
+The sheaf condition becomes:
+
+> If we choose data $s_i \in F(U_i)$ on each $U_i$,  
+> and if for every pair $(i,j)$ the restrictions of $s_i$ and $s_j$ to  
+> $U_i \cap U_j$ are equal,  
+> then there exists a **unique** $s \in F(U)$ whose restriction to each $U_i$ is $s_i$.
+
+This is the **gluing condition** from topology.