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 M-F(<-) and the inflation Mackey category M-F(->). 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. (C) 2015 Elsevier Inc. All rights reserved.