Modular Hecke algebras over Möbius categories

We extend the modular Hecke operators of Connes and Moscovici by taking values in the modular incidence algebra M[C] over a Möbius category C. Here M is the 'modular tower' consisting of all modular forms of all weights across all levels. We construct multiple product structures on the collection A(Γ)[C] of modular Hecke operators of level Γ over C, where Γ is a principal congruence subgroup. These product structures are then shown to be well behaved with respect to Hopf actions on A(Γ)[C]. While A(Γ)[C] already carries an action of the Hopf algebra H1 of codimension 1-foliations, the noncommutativity of the modular incidence algebra M[C] allows us to construct additional operators on A(Γ)[C]. The use of Möbius categories provides a single framework for describing modular Hecke operators taking values in various rings: from formal power series rings over M to arithmetic functions over M and algebras of upper triangular matrices with entries in M. Moreover, we use functors between Möbius categories to study relations between various modular Hecke algebras.