We study quotients of mapping class groups Γ g,1 of oriented surfaces with one boundary component by terms of their Johnson filtrations, and we show that the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity. We also compute the stable (co)homology with constant rational coefficients for one family of such quotients.Remark 2.3. C -Mod is abelian, with kernels and cokernels computed pointwise. Hence it makes good sense to talk of sub-C -modules, quotient C -modules etc. Now suppose further that the category C is strict monoidal, where the monoidal product ⊕ is given by n ⊕ m = n + m on objects, and that the monoidal unit 0 is initial in C .Definition 2.4. We will call such C a category with objects the naturals, or CON for short.Consider the functor R := 1 ⊕ (−) : C → C . Since 0 is initial, the unique arrow 0 → 1 induces a natural transformation ρ : Id ⇒ R from the identity Id = 0 ⊕ (−) to R.