Fusion systems and localities -- a dictionary
Linking systems were introduced to provide algebraic models for $p$-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system $\mathcal{F}$ corresponds to a group-like structure called a locality. Given such a locality $\mathcal{L}$, we prove that there is a one-to-one correspondence between the partial normal subgroups of $\mathcal{L}$ and the normal subsystems of the fusion system $\mathcal{F}$. This is then used to obtain a kind of dictionary, which makes it possible to translate between various concepts in localities and corresponding concepts in fusion systems. As a byproduct, we obtain new proofs of many known theorems about fusion systems and also some new results. For example, we show in this paper that, in any saturated fusion system, there is a sensible notion of a product of normal subsystems.