# Appendix

A bunch of stuff I always forget!

\envlist

• A section is just an element $$s\in {\mathcal{F}}(U)$$.

• A stalk of a (pre)sheaf $${\mathcal{F}}$$ at a point $$p$$ is defined as \begin{align*} {\mathcal{F}}_p \coloneqq\colim_{p\ni U_i} ({\mathcal{F}}(U_i), \operatorname{res}_{ij}) .\end{align*}

• A germ $$\tilde f_p$$ at a point $$p$$ is an element in a stalk $${\mathcal{F}}_p$$. It can concretely be described as \begin{align*} \tilde f_p = [(U\ni p, s\in {\mathcal{F}}(U))]/\sim && (U, s)\sim (V, t) \iff \exists W \subseteq U \cap V,\, { \left.{{s}} \right|_{{W}} } = { \left.{{t}} \right|_{{W}} } .\end{align*}

Given a diagram $$J$$ in a category $$\mathsf{C}$$, regard it as a functor $$F: \mathsf{J}\to \mathsf{C}$$ where $$\mathsf{J}$$ is the diagram category of $$J$$. Then the colimit of $$J$$ is defined as the initial object in the category of co-cones over $$F$$.

• A co-cone of $$F$$ is an $$N\in {\operatorname{Ob}}(\mathsf{C})$$ and a family of morphisms $$\left\{{ \psi_X: F(X)\to N{~\mathrel{\Big\vert}~}X\in {\operatorname{Ob}}(\mathsf{J})}\right\}$$.

• The category of co-cones over $$F$$ is the comma category $$F \downarrow \Delta$$, where $$\Delta: \mathsf{C} \to {\mathsf{Fun}}(\mathsf{J}, \mathsf{C})$$ is the diagonal functor sending $$N\in {\operatorname{Ob}}(\mathsf{C})$$ to the constant functor to $$N$$: \begin{align*} \Delta(N):\mathsf{J} &\to \mathsf{C} \\ X &\mapsto N .\end{align*}

• The comma category generalizes slice categories: given categories and functors \begin{align*} \mathsf{A} \mapsto{S} \mathsf{C} \mapsfrom{T} \mathsf{B} ,\end{align*} the comma category $$S\downarrow T$$ is given by triples $$(A, B, h: S(A)\to T(B))$$ making the obvious diagrams commute:

Taking $$\mathsf{C} = A$$, $$S = \operatorname{id}_{\mathsf{A}}$$, and $$\mathsf{B} \coloneqq{\operatorname{pt}}$$ to be a 1-object category with only the identity morphism forces $$X\coloneqq T({\operatorname{pt}}) \in {\operatorname{Ob}}(\mathsf{A})$$ to be a single object and $$(\mathsf{A} \downarrow X)$$ is the usual slice category over $$X$$.