Superalgebra deformations of web categories: Affine and cyclotomic webs

Let $\mathbb{k}$ be a characteristic zero domain. We define and study a diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{aff}_{A}$ associated to any locally unital Frobenius $\mathbb{k}$-superalgebra $A$. This category can be viewed variously as an affinization of the finite web category $\mathbf{Web}_{A}$ previously defined by the authors and Zhu, as a thickening of the degenerate affine wreath product algebras defined by Savage, or as a Frobenius deformation of affine web categories defined by Song and Wang. We show that there is an asymptotically faithful family of functors from $\mathbf{Web}^{aff}_{A}$ to the monoidal supercategory of endofunctors of $\mathfrak{gl}_n(A)$-modules for every $n \geq 1$, and use this to establish a basis of `decorated double coset diagrams' for morphism spaces in $\mathbf{Web}^{aff}_{A}$. We also define and establish basis results for the cyclotomic quotient category $\mathbf{Web}^{\Lambda}_{A}$ associated with a cyclotomic datum $\Lambda$.