Fibre functors and reconstruction of Hopf algebras

The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful monoidal functor of tensor categories, one would like to realize $B$ as category of representations of a braided Hopf algebra $H(F)$ in $C$. We prove that this is the case iff $B$ has the additional structure of a monoidal $C$-module category compatible with $F$, which equivalently means that $F$ admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fibre functors, and we give some applications. One particular motivation was the logarithmic Kazhdan-Lusztig conjecture.