Morphism of fusion systems

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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 \le 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