Incompressible tensor categories

A symmetric tensor category $D$ over an algebraically closed field k is called incompressible if its objects have finite length ($D$ is pretannakian) and every tensor functor out of $D$ is an embedding of a tensor subcategory. E.g., the categories $\mathrm{Vec}$, $\mathrm{sVec}$ of vector and supervector spaces are incompressible. Moreover, by Deligne's theorem [15], if $\mathrm{char}\left(k\right)=0$ then any tensor category of moderate growth uniquely fibres over $\mathrm{sVec}$. This implies that $\mathrm{Vec}$, $\mathrm{sVec}$ are the only incompressible categories over k in this class, and perhaps altogether, as we expect that all incompressible categories have moderate growth.

Similarly, in characteristic $p>0$, we also have incompressible Verlinde categories ${\mathrm{Ver}}_p,{\mathrm{Ver}}_p^{+}$, and by [10] any Frobenius exact category of moderate growth uniquely fibres over ${\mathrm{Ver}}_p$, meaning that, in this class, the above categories are the only incompressible ones. More generally, the Verlinde categories ${\mathrm{Ver}}_{p^n}$, ${\mathrm{Ver}}_{p^n}^{+}$, $n\le \infty$ introduced in [3], [7] are incompressible, and a key conjecture is that every tensor category of moderate growth uniquely fibres over ${\mathrm{Ver}}_{p^{\infty }}$. This would make the above the only incompressible categories in this class (and perhaps altogether).

We prove a part of this conjecture, showing that every tensor category of moderate growth fibres over an incompressible one. So it remains to understand incompressible categories, and we prove several results in this direction. Namely, let $D$-Tann be the category of tensor categories that fibre over $D$. Then we say that $D$ is subterminal if it is a terminal object of $D$-Tann (i.e., fibre functors to $D$ are unique when exist), and that $D$ is a Bezrukavnikov category if $D$-Tann is closed under taking images of tensor functors (=quotient categories). Clearly, a subterminal Bezrukavnikov category is incompressible, and we conjecture that the converse also holds; for instance, tensor subcategories of ${\mathrm{Ver}}_p$ are known to be subterminal. We prove that furthermore they are Bezrukavnikov categories, generalizing the result of Bezrukavnikov [4] in the case of $\mathrm{Vec}$. Finally, we show that a tensor subcategory of a finite incompressible category is incompressible.

We also find intrinsic sufficient conditions for incompressibility and subterminality. Namely, $D$ is called maximally nilpotent if the growth rate of the length of the symmetric powers of every object is the minimal one theoretically possible. We show that a finite maximally nilpotent category is incompressible, and also subterminal if it satisfies an additional geometric reductivity condition (for every morphism $X\twoheadrightarrow 1$ in $D$, there exists $n>0$ for which ${\mathrm{Sym}}^{n}X\twoheadrightarrow 1$ is split). Then we verify these conditions for the category ${\mathrm{Ver}}_{2^n}$, thereby showing that it is subterminal.