Pathmap sheaf
July 6, 2022
Pathmap sheaf #
Introduction #
This chapter introduces the concept of the pathmap sheaf from a graph \( \graph{G} \)$ to a graph \( \graph{H} \)$. This will allow us to formalize the notion of an atlas on a graph \( \graph{G} \)$, and for special choices of \( \graph{H} \)$, define a kind of curvature on \( \graph{G} \)$ as a kind of sheaf cohomology.
The pathmap sheaf is a structure whose elements are partial pathmaps from \( \graph{G} \)$ to \( \graph{H} \)$ -- pathmaps that are not necessarily defined for all paths in \( \graph{G} \)$ -- those for which it is defined are called the support of the partial pathmap. The sheaf structure is what allows us to assemble compatible pathmaps to form new pathmaps with larger support.
We will proceed by defining a partial pathmap, and then elaborate how these form the pathmap sheaf.
Partial pathmap #
A partial pathmap [FSF[\( \pathHomomorphism{ \rho } \), \( \graph{G} \), \( \graph{H} \)]] is equivalent to a partial groupoid homomorphism between group