Linearized topologies and deformation theory

In this paper, for an underlying small category UU endowed with a Grothendieck topology τ  , and a linear category aa which is graded over UU in the sense of [13], we define a natural linear topology TτTτ on aa, which we call the linearized topology  . Grothendieck categories in (non-commutative) algebraic geometry can often be realized as linear sheaf categories over linearized topologies. With the eye on deformation theory, it is important to obtain such realizations in which the linear category contains a restricted amount of algebraic information. We prove several results on the relation between refinement (eliminating both objects, and, more surprisingly, morphisms) of the non-linear underlying site (U,τ)(U,τ), and refinement of the linearized site (a,Tτ)(a,Tτ). These results apply to several incarnations of (quasi-coherent) sheaf categories, leading to a description of the infinitesimal deformation theory of these categories in the sense of [17] which is entirely controlled by the Gerstenhaber deformation theory of the small linear category aa, and the Grothendieck topology τ   on UU. Our findings extend results from [17], [12] and [7] and recover the examples from [21] and [20].