Morphism of fusion systems

Definition

Suppose $P_{1},P_{2}$ are both finite $p$ -groups (i.e., groups of prime power order for some prime number $p$ ). Suppose ${\mathcal {F}}_{1}$ is a fusion system on $P_{1}$ and ${\mathcal {F}}_{2}$ is a fusion system on $P_{2}$ . A morphism of fusion systems from ${\mathcal {F}}_{1}$ to ${\mathcal {F}}_{2}$ is a pair $(\alpha ,\Phi )$ where:

$\alpha$ is a homomorphism of groups from $P_{1}$ to $P_{2}$ .
$\Phi$ is a covariant functor from the category ${\mathcal {F}}_{1}$ to the category ${\mathcal {F}}_{2}$ .
For every subgroup $Q\leq P$ , $\alpha (Q)=\Phi (Q)$ . Here, $\alpha (Q)$ is the homomorphic image and $\Phi (Q)$ is the image of the object $Q\in {\mathcal {F}}_{1}$ under the covariant functor $\Phi$ .
If $\varphi :Q\to R$ is a morphism in ${\mathcal {F}}_{1}$ , then $\Phi (\varphi )\circ \alpha =\alpha \circ \varphi$ .

References

Expository references

Introduction to Fusion Systems by Markus Linckelmann^Weblink^{More info}