  We define weak 2-categories of finite dimensional algebras with bimodules,
along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories.
We prove that special examples $\mathbb{O}_p$ of these operators control all
homological aspects of the rational representation theory of the algebraic
group $GL_2$, over a field of positive characteristic. We prove that when $x$
is a Rickard tilting complex, the operators $\mathbb{O}_{(c,x)}$ honour derived
equivalences, in a differential graded setting. We give a number of
representation theoretic corollaries, such as the existence of tight
$\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid
group actions on the derived categories of blocks of these Schur algebras.
