  We introduce several notions and give examples.
  We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for
$l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD,
where ${\rm Stab}( \langle \mathcal T_1,\mathcal T_2\rangle )\not \cong{\rm
Stab}(\mathcal T_1)\times {\rm Stab}(\mathcal T_2)$.
  This example suggest a new notion of a norm, strictly increasing on
$\{D^b(K(l))\}_{l\geq 2}$. To a triangulated category $\mathcal T$ which has
property of a phase gap we attach a non-negative number $\Vert \mathcal T
\Vert_{\varepsilon}$. Natural assumptions on a SOD imply $ \Vert \langle
\mathcal T_1,\mathcal T_2\rangle \Vert_{\varepsilon}\geq {\rm max}\{ \Vert
\mathcal T_1 \Vert_{\varepsilon}, \Vert\mathcal T_2 \Vert_{\varepsilon}\}$.
Using this we define a topology on the set of equivalence classes of
triangulated categories with a phase gap, where the set of discrete derived
categories is a discrete subset and the rationality of a smooth surface $S$
ensures that $[D^b(point)] \in {\rm Cl}([D^b(S)])$.
  Viewing $D^b(K(l))$ as a non-commutative curve, we observe that it is
reasonable to count non-commutative curves in any category in a small
neighborhood of $D^b(K(l))$. Examples show that this idea (non-commutative
curve-counting) opens directions to new categorical structures and connections
to number theory and classical geometry. We give a definition, which
specializes to the non-commutative curve-counting invariants. In an example
arising on the A side we specialize our definition to non-commutative
Calabi-Yau curve-counting, where the entities we count are a Calabi-Yau
modification of $D^b(K(l))$.
  Finally we speculate that one might consider a holomorphic family of
categories, introduced by Kontsevich, as a non-commutative extension with the
norm playing a role similar to the classical notion of degree of an extension
in Galois theory.
