  Many of the conjectures of current interest in the representation theory of
finite groups in characteristic $p$ are local-to-global statements, in that
they predict consequences for the representations of a finite group $G$ given
data about the representations of the $p$-local subgroups of $G$. The local
structure of a block of a group algebra is encoded in the fusion system of the
block together with a compatible family of K\"ulshammer-Puig cohomology
classes. Motivated by conjectures in block theory, we state and initiate
investigation of a number of seemingly local conjectures for arbitrary triples
$(S,\mathcal{F},\alpha)$ consisting of a saturated fusion system $\mathcal{F} $
on a finite $p$-group $S$ and a compatible family $\alpha$.
