The $q$-Schur category and polynomial tilting modules for quantum $GL_n$

The $q$-Schur category is a $\mathbb{Z}[q,q^{-1}]$-linear monoidal category closely related to the $q$-Schur algebra. We explain how to construct it from coordinate algebras of quantum $GL_n$ for all $n \geq 0$. Then we use Donkin's work on Ringel duality for $q$-Schur algebras to make precise the relationship between the $q$-Schur category and an integral form for the $U_q\mathfrak{gl}_n$-web category of Cautis, Kamnitzer and Morrison. We construct explicit integral bases for morphism spaces in the latter category, and extend the Cautis-Kamnitzer-Morrison theorem to polynomial representations of quantum $GL_n$ at a root of unity over a field of any characteristic.