Fundamental gerbes

For a class of affine algebraic groups C over a field κ, we define the notion of C-fundamental gerbe of a fibered category, generalizing what we did for finite group schemes in a 2015 paper. We give necessary and sufficient conditions on C implying that a fibered category X over κ satisfying mild hypotheses admits a Nori C-fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as C the class of virtually unipotent group schemes, under a properness condition on X. Finally, we prove a general duality result, generalizing the duality between group schemes of multiplicative type and Galois modules, that yields a construction of the multiplicative gerbe of multiplicative type which is independent of the previous theory, and requires weaker hypotheses. This gives a conceptual interpretation of the universal torsor of Colliot-Thélène and Sansuc.