We define the functor $\textrm{ ncDef}_{ (Z_1,\ldots,Z_n)}$ of non-commutative deformations of an $n$-tuple of objects in an arbitrary $k$-linear abelian category $\mathcal{ Z}$. In our categorified approach, we view the underlying spaces of infinitesimal flat deformations as Deligne finite categories, i.e. finite length abelian categories admitting projective generators, with $n$ isomorphism classes of simple objects. More generally, we define the functor $\textrm{ ncDef}_{ \zeta}$ of non-commutative deformations of an exact functor $\zeta \colon \mathcal{ A} \to \mathcal{ Z}$ of abelian categories. Here the role of an infinitesimal non-commutative thickening of $\mathcal{ A}$ is played by an abelian category $\mathcal{ B}$ containing $\mathcal{ A}$ and such that $\mathcal{ A}$ generates $\mathcal{ B}$ by extensions. The functor $\textrm{ ncDef}_{ \zeta}$ assigns to such $\mathcal{ B}$ the set of equivalence classes of exact functors $\mathcal{ B} \to \mathcal{ Z}$ which extend $\zeta$. 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 neighbourhood. We show that if $\zeta$ is fully faithful, then the functor $\textrm{ ncDef}_{ \zeta}$ is ind-represented by the extension closure of the essential image of $\zeta$. We prove that for a flopping contraction $f\colon X\to Y$ with the fiber over a closed point $C = \bigcup_{ i=1}^n C_i$, where $C_i$'s are irreducible curves, $\{ \mathcal{ O}_{ C_i}(-1)\}$ is the set of simple objects in the null-category for $f$. We conclude that the null-category ind-represents the functor $\textrm{ ncDef}_{ (\mathcal{ O}_{ C_1}(-1),\ldots,\mathcal{ O}_{ C_n}(-1))}$.
Categories: math.AG