Linearised Categories

In this note we study several ways of turning a small category into a ringoid. We refer to such processes as linearisation, for lack of a better name. Although this material is very abstract, our main application is concrete: the category Mod(X) of sheaves of modules over a scheme X can be viewed as a localisation of a category of modules over a ringoid, very naturally constructed from the structure sheaf of the scheme. We freely use the notation and concepts introduced in our notes on Algebra in a Category. We allow noncommutative rings and noncommutative sheaves of rings. Throughout all additive topologies are right additive topologies (notation of our Localisation of Ringoids notes).