Superalgebra deformations of web categories: finite webs

Let $\mathbb{k}$ be a characteristic zero domain. For a locally unital $\mathbb{k}$-superalgebra $A$ with distinguished idempotents $I$and even subalgebra $a\subseteq A_{\bar{0}}$, we define and study an associated diagrammatic monoidal $\mathbb{k}$-linear supercategory ${\mathbf{W}\mathbf{e}\mathbf{b}}_I^{A,a}$. This supercategory yields a diagrammatic description of the generalized Schur algebras $T_a^A(n,d)$. We also show there is an asymptotically faithful functor from ${\mathbf{W}\mathbf{e}\mathbf{b}}_I^{A,a}$ to the monoidal supercategory of ${\mathfrak{g}\mathfrak{l}}_n(A)$-modules generated by symmetric powers of the natural module. When this functor is full, the single diagrammatic supercategory ${\mathbf{W}\mathbf{e}\mathbf{b}}_I^{A,a}$ provides a combinatorial description of this module category for all $n\ge 1$. We also use these results to establish Howe dualities between ${\mathfrak{g}\mathfrak{l}}_m(A)$ and ${\mathfrak{g}\mathfrak{l}}_n(A)$ when $A$ is semisimple.