Affine and cyclotomic webs

Generalizing the polynomial web category, we introduce a diagrammatic $k$-linear monoidal category, the affine web category, for any commutative ring $k$. 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 $k$-linear monoidal category, the affine Schur category, formulated in a sequel.