Homotopy, homology and GL2

We define weak 2-categories of finite-dimensional algebras with bimodules, along with collections of operators ¿(c, x) on these 2-categories. We prove that special examples ¿p of these operators control all homological aspects of the rational representation theory of the algebraic group GL2, over a field of positive characteristic. We prove that when x is a Rickard tilting complex, the operators ¿(c, x) honour derived equivalences in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight 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.