Commutative exact superalgebras and super-modular tensor categories

We extend the theory of commutative exact algebras in braided finite tensor categories (Shimizu-Yadav) to superalgebras, motivated by extensions of vertex operator (super)algebras. For a braided category $\text{C}$ let $\text{B}={\text{C}}^{\text{gr}}$ be the category of $\mathbb{Z}\mathrm{/}2$-graded objects with the Koszul-signed braiding. It is the underlying category of the supercategory of Creutzig-Kanade-McRae. A supercommutative superalgebra $A=A_0\oplus A_1$ in $\text{C}$ is the same as a commutative algebra $\text{bA}$ in $\text{B}$, and super-local modules are the local $\text{bA}$-modules. Hence rigidity, pivotality and ribbon statements are instances of the ordinary theorems, with the ribbon condition ${\theta }_{A_0}=\text{id}$, ${\theta }_{A_1}=\epsilon \text{id}$, where $\epsilon =\pm 1$ is fixed by the statistics of the odd part. For $\text{C}$ non-degenerate and $\text{bA}$ haploid and exact we prove that the category of super-local modules is a braided finite tensor category whose Müger center is $\mathsf{s}\mathsf{V}\mathsf{e}\mathsf{c}\mathsf{t}$. It is the centralizer of a fermion in a non-degenerate category obtained by condensing in $\text{C}⊠(\text{toric code})$. If $\text{C}$ is ribbon and $\text{bA}$ is symmetric Frobenius it is ribbon. We construct examples from graded simple currents, treat the free fermion and the symplectic fermion, and compare with Creutzig-Kanade-McRae and Creutzig-McRae-Shimizu-Yadav.