Affine and cyclotomic $q$-Schur categories via webs

We formulate two new $\mathbb Z[q,q^{-1}]$-linear diagrammatic monoidal categories, the affine $q$-web category and the affine $q$-Schur category, as well as their respective cyclotomic quotient categories. Diagrammatic integral bases for the Hom-spaces of all these categories are established. In addition, we establish the following isomorphisms, providing diagrammatic presentations of these $q$-Schur algebras for the first time: (i)~ the path algebras of the affine $q$-web category to R.~Green's affine $q$-Schur algebras, (ii)~ the path algebras of the affine $q$-Schur category to Maksimau-Stroppel's higher level affine $q$-Schur algebras, and most significantly, (iii)~ the path algebras of the cyclotomic $q$-Schur categories to Dipper-James-Mathas' cyclotomic $q$-Schur algebras.