Generalizing the polynomial web category, we introduce a diagrammatic -linear monoidal category, the affine web category, for any commutative ring . Integral bases consisting of elementary diagrams are obtained for the affine web category and its cyclotomic quotient categories. Connections between cyclotomic web categories and finite W-algebras are established, leading to a diagrammatic presentation of idempotent subalgebras of W-Schur algebras introduced by Brundan-Kleshchev. The affine web category will be used as a basic building block of another -linear monoidal category, the affine Schur category, formulated in a sequel.
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Song et al. (2024) studied this question.