ScalingStacks

1.4.3 The fiber functor[000C]

The fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) on which Theorem B is built, can be viewed as a categorified and graded analog of the forgetful functor from a category of quantum group representations to \(\mathrm{Vec}\). Since its target \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is braided monoidal (in fact symmetric monoidal), it is tempting to apply the machinery of subsection 1.4.2 to obtain a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\).

However, this does not work because the analogy with the classical situation breaks down in an important way. In the classical setting, the fiber functor is faithful and monoidal, but in general not braided. By contrast, our categorified fiber functor will be braided monoidal but not faithful (in the sense of Subsection 1.4.2). This lack of faithfulness prevents us from directly using the reduction results from Subsection 1.4.2. In Subsection 1.4.4, we will discuss a restricted version of the fiber functor that is faithful, which allows us to leverage the results indicated in Subsection 1.4.2.

In the end, the non-faithfulness turns out to be an essential feature of our fiber functor. This feature is what allows \(H_{\mathrm{loc}}\) and its source category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to be equipped with a non-symmetrically braided monoidal structure, even though the target category \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is symmetric monoidal. A faithful braided fiber functor would force the source category to be symmetric!

In the classical situation, the existence of a fiber functor arises from the fact that quantum group representations are ultimately categories of modules for a quasitriangular Hopf algebra, which can be recovered from the fiber functor via Tannakian reconstruction. We would be very interested to see an application of Tannakian reconstruction to our categorified fiber functor.

We expect our fiber functor to be important for future applications and computations. For instance, the fact that it is braided may give a means of recursively computing all the algebraic data implicit in the braided monoidal structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) on a cell-by-cell basis.