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Fiber functors and reconstruction of Hopf algebras
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 03 June 2024, pp. 1-44
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The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if
$F:{\mathcal B}\to {\mathcal C}$ is an exact faithful monoidal functor of tensor categories, one would like to realize 
${\mathcal B}$ as category of representations of a braided Hopf algebra 
$H(F)$ in 
${\mathcal C}$. We prove that this is the case iff 
${\mathcal B}$ has the additional structure of a monoidal 
${\mathcal C}$-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 fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.
