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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.