(01/22/2021, 10:49 PM)JmsNxn Wrote: Just so you know, branching goes brrrrrrrr! for me because I think of it as local solutions. And then, we paste local solutions together to get a **somewhat** global solution--minus the branch cuts.
This pasting reminds me alot the definition of manifold and of sheaves: a basic example of sheaf \( \mathcal F \) is a procedure that instead of associating to a geometric object \( X \) an algebraic information, because in some cases you can't, you consider the full lattice of the "parts" of \( X \) (e.g. subspaces) and you associate to every one of them an algebraic gadget in a functorial way, i.e. if a part \( U \) sits inside a part \( V \) then the gadgets \( {\mathcal F}(U) \) and \( {\mathcal F}(V) \) must be related or there must be a transoformation between them.
In this way instead of defining a unique map \( X\to ... \) we define something piecewise, over the pieces of X but with some regularity: this is just a functor. It becomes a sheaf when you also ask that if two parts of X overlaps you have some rule to glue the maps in those intersections, like we do with an atlas of charts for a top. space.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)