Blocks of Mackey categories

For a suitable small category F of homomorphisms between finite groups, we introduce two subcategories of the biset category, namely, the deflation Mackey category MF← and the inflation Mackey category MF→. Let G be the subcategory of F consisting of the injective homomorphisms. We shall show that, for a field K of characteristic zero, the K-linear category KMG=KMG←=KMG→ has a semisimplicity property and, in particular, every block of KMG owns a unique simple functor up to isomorphism. On the other hand, we shall show that, when F is equivalent to the category of finite groups, the K-linear categories KMF← and KMF→ each have a unique block.