  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 $Rep(\underline{P})$,
possessing nice universal properties among tensor categories over the category
$\mathtt{sVect}$ of finite-dimensional complex vector superspaces.
  First, it is the "abelian envelope" of the Deligne category corresponding to
the periplectic Lie superalgebra, in the sense of arXiv:1511.07699.
  Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact
tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs
$(X, \omega)$ in $\mathcal{C}$ where $\omega: X \otimes X \to \Pi \mathbf{1}$
is a non-degenerate symmetric form and $X$ not annihilated by any Schur
functor.
  The category $Rep(\underline{P})$ is constructed in two ways. The first
construction is through an explicit limit of the tensor categories
$Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second
construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the
category of representations of a periplectic Lie supergroup in the Deligne
category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$.
  An upcoming paper by the authors will give results on the abelian and tensor
structure of $Rep(\underline{P})$.
