A bunch of stuff I always forget!

  • 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:

Link to Diagram

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 \(X\).