标题： 不可压缩张量范畴 显示英文标题

标题： Incompressible tensor categories

摘要： 一个对称张量范畴$\mathcal D$在代数闭域$k$上是不可压缩的，如果从$\mathcal D$出发的每个张量函子都是嵌入。例如，范畴$Vec$和$sVec$的（超）向量空间是不可压缩的。 此外，根据德利涅定理，如果特征为$(k)=0$，则任何中等增长的张量范畴唯一地在$sVec$上纤维化，因此$Vec$和$sVec$是此类中的唯一不可压缩范畴。 类似地，在特征$p>0$下，我们有不可压缩的韦尔林德范畴$Ver_p$，并且任何中等增长的弗罗贝尼乌斯正合范畴唯一地在$Ver_p$上纤维化。 更一般地，Verlinde范畴$Ver_{p^n}$，$Ver_{p^n}^+$是不可压缩的，一个关键猜想是每个中等增长的张量范畴唯一地纤维化到$Ver_{p^\infty}$。这将使上述成为该类中唯一的不可压缩范畴。我们证明了这个猜想的一部分，表明每个中等增长的张量范畴都纤维化到一个不可压缩的范畴。因此，仍需理解不可压缩范畴。我们说$\mathcal D$是次终的，如果每个张量范畴最多有一个到它的纤维函子，而是一个Bezrukavnikov范畴，如果纤维化到$\mathcal D$的张量范畴类在商上是封闭的。显然，一个次终的Bezrukavnikov范畴是不可压缩的，我们猜想其逆成立。我们证明了$Ver_p$是Bezrukavnikov，推广了Bezrukavnikov对$Vec$的结果。我们还找到了不可压缩性和次终性的内在充分条件。 也就是说，当对称幂的增长率最小时，$\mathcal D$是极大幂零的。 我们证明了一个有限的极大幂零范畴是不可压缩的，并且如果满足额外的几何约化条件，则是次终端的。 然后我们验证了这些条件对于$Ver_{2^n}$成立。 显示英文摘要

摘要： A symmetric tensor category $\mathcal D$ over an algebraically closed field $k$ is incompressible if every tensor functor out of $\mathcal D$ is an embedding. E.g., the categories $Vec$ and $sVec$ of (super)vector spaces are incompressible. Moreover, by Deligne's theorem, if char$(k)=0$ then any tensor category of moderate growth uniquely fibres over $sVec$, so $Vec$ and $sVec$ are the only incompressible categories in this class. Similarly, in characteristic $p>0$, we have the incompressible Verlinde category $Ver_p$, and any Frobenius exact category of moderate growth uniquely fibres over $Ver_p$. More generally, the Verlinde categories $Ver_{p^n}$, $Ver_{p^n}^+$ are incompressible, and a key conjecture is that every tensor category of moderate growth uniquely fibres over $Ver_{p^\infty}$. This would make the above the only incompressible categories in this class. 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. We say that $\mathcal D$ is subterminal if it every tensor category admits at most one fibre functor to it, and a Bezrukavnikov category if the class of tensor categories that fibre over $\mathcal D$ is closed under quotients. Clearly, a subterminal Bezrukavnikov category is incompressible, and we conjecture the converse. We prove that $Ver_p$ is Bezrukavnikov, generalizing the result of Bezrukavnikov for $Vec$. We also find intrinsic sufficient conditions for incompressibility and subterminality. Namely, $\mathcal D$ is maximally nilpotent if the growth rates of symmetric powers are minimal. We show that a finite maximally nilpotent category is incompressible, and also subterminal if it satisfies an additional geometric reductivity condition. Then we verify these conditions for $Ver_{2^n}$.