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 $\mathbf{Web}^{A,a}_I$. 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 $\mathbf{Web}^{A,a}_I$ to the monoidal supercategory of $\mathfrak{gl}_n(A)$-modules generated by symmetric powers of the natural module. When this functor is full, the single diagrammatic supercategory $\mathbf{Web}^{A,a}_I$ provides a combinatorial description of this module category for all $n \geq 1$. We also use these results to establish Howe dualities between $\mathfrak{gl}_{m}(A)$ and $\mathfrak{gl}_{n}(A)$ when $A$ is semisimple.