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We show that the subgroup of the Picard group of a $p$-block of a finitegroup given by bimodules with endopermutation sources modulo the automorphismgroup of a source algebra is determined locally in terms of the fusion systemon a defect group. We show that the Picard group of a block over the a completediscrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraicclosure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picardgroups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the resultsto blocks with an abelian defect group and Frobenius inertial quotient, andspecialise this further to blocks with cyclic or Klein four defect groups.

We show that the subgroup of the Picard group of a $p$-block of a finitegroup given by bimodules with endopermutation sources modulo the automorphismgroup of a source algebra is determined locally in terms of the fusion systemon a defect group. We show that the Picard group of a block over the a completediscrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraicclosure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picardgroups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the resultsto blocks with an abelian defect group and Frobenius inertial quotient, andspecialise this further to blocks with cyclic or Klein four defect groups.

In this article we define the $-_+$-construction and the $-^+$-construction,that was crucial in the theory of canonical induction formulas (see\cite{Boltje1998b}), in the setting of biset functors, thus providing thenecessary framework to define and construct canonical induction formulas forrepresentation rings that are most naturally viewed as biset functors.Additionally, this provides a unified approach to the study of a class offunctors including the Burnside ring, the monomial Burnside ring and globalrepresentation ring.

In this article we define the $-_+$-construction and the $-^+$-construction,that was crucial in the theory of canonical induction formulas (see\cite{Boltje1998b}), in the setting of biset functors, thus providing thenecessary framework to define and construct canonical induction formulas forrepresentation rings that are most naturally viewed as biset functors.Additionally, this provides a unified approach to the study of a class offunctors including the Burnside ring, the monomial Burnside ring and globalrepresentation ring.

The theory of biset functors, introduced by Serge Bouc, gives a unifiedtreatment of operations in representation theory that are induced bypermutation bimodules. In this paper, by considering fibered bisets, weintroduce and describe the basic theory of fibered biset functors which is anatural framework for operations induced by monomial bimodules. The main resultof this paper is the classification of simple fibered biset functors.

The theory of biset functors, introduced by Serge Bouc, gives a unifiedtreatment of operations in representation theory that are induced bypermutation bimodules. In this paper, by considering fibered bisets, weintroduce and describe the basic theory of fibered biset functors which is anatural framework for operations induced by monomial bimodules. The main resultof this paper is the classification of simple fibered biset functors.