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We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(bf k)$ by the $n$-th Steinberg module, and ${rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(bf k)$-modules. We show that ${rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $mathbb{Z}[2cos(2pi/p^n)]$, and that ${rm Ver}_{p^n}$ embeds into ${rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $bf k$ admits a fiber functor to the union ${rm Ver}_{p^infty}$ of the nested sequence ${rm Ver}_{p}subset {rm Ver}_{p^2}subsetcdots$. This would provide an analog of Delignes theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${rm Ver}_p$.
We develop a theory of Frobenius functors for symmetric tensor categories (STC) $mathcal{C}$ over a field $bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoi
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine
We describe graded commutative Gorenstein algebras ${mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)cong{mathcal E}_{n}(p)$, where $mathsf{Ver}_{p^{n+1}}$ are the new symmet
This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of the curre
It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,lambda)$, where $G$ is a compact Lie group, and $lambda in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones subfactor