Deligne categories and the periplectic Lie superalgebra

We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$.
The paper gives a construction of the tensor category $\mathrm{Rep}(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathrm{sVect}$ of finite-dimensional complex vector superspaces.
First, it is the “abelian envelope” of the Deligne category corresponding to the periplectic Lie superalgebra.
Secondly, given a tensor category $\mathcal{C}$ over $\mathrm{sVect}$, exact tensor functors $\mathrm{Rep}(\underline{P})\rightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$, where $\omega\colon X \otimes X \to \Pi1$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor.
The category $\mathrm{Rep}(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $\mathrm{Rep}(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo–Serganova functors. The second construction (inspired by P. Etingof) describes $\mathrm{Rep}(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathrm{sVect} \boxtimes \mathrm{Rep}(\underline{\mathrm{GL}}_t)$.