Categorifying non‐commutative deformations

Abstract We define the functor of non‐commutative deformations of an ‐tuple of objects in an arbitrary ‐linear abelian category . In our categorified approach, we view the underlying spaces of infinitesimal flat deformations as Deligne finite categories, that is, finite length abelian categories admitting projective generators, with isomorphism classes of simple objects. More generally, we define the functor of non‐commutative deformations of an exact functor of abelian categories. Here the role of an infinitesimal non‐commutative thickening of is played by an abelian category containing and such that generates by extensions. The functor assigns to such the set of equivalence classes of exact functors which extend . We prove that an exact functor on an infinitesimal extension is fully faithful if and only if it is fully faithful on the first infinitesimal neighborhood. We show that if is fully faithful, then the functor is ind‐represented by the extension closure of the essential image of . We prove that for a flopping contraction with the fiber over a closed point , where are irreducible curves, is the set of simple objects in the null‐category for . We conclude that the null‐category ind‐represents the functor .