Highest weight categories arising from Khovanov's diagram algebra I: cellularity

This is the first of four articles studying some slight generalisations $H^n_m$ of Khovanov's diagram algebra, as well as quasi-hereditary covers $K^n_m$ of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions $K^\infty_m$, $K^{\pm\infty}_m$ and $K^\infty_\infty$. In this article we prove that $H^n_m$ is a cellular symmetric algebra and that $K^n_m$ is a cellular quasi-hereditary algebra. In subsequent articles, we relate $H^n_m$, $K^n_m$ and $K^\infty_m$ to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category $\mathcal O$ and the general linear supergroup, respectively.