Orbit category of a fusion system

Definition

Suppose $P$ is a group of prime power order and ${\mathcal {F}}$ is a fusion system on $P$. The orbit category of ${\mathcal {F}}$ is defined as the following category:

• Its objects are the subgroups of $P$.
• The morphisms between two objects $Q$ and $R$ are the orbits in the set $\operatorname {Hom} (Q,R)$ under the action of the group of inner automorphisms of $R$ (denoted $\operatorname {Aut} _{R}(R)$ by post-composition. In other words, two elements $\alpha ,\beta :Q\to R$ are in the same equivalence class iff there exists $g\in R$ such that for all $x\in Q$, $\beta (x)=g\alpha (x)g^{-1}$.

Note that the orbit category is not a concrete category because its morphisms are equivalence classes of maps rather than actual maps.