### Article

## Some new categorical invariants

In this paper we introduce new categorical notions and give many examples. In an

earlier paper we proved that the Bridgeland stability space on the derived category of

representations of K(l), thel-Kronecker quiver, is biholomorphic toC×Hfor l ≥ 3. In

the present paper we define a new notion of norm, which distinguishes {Db(K(l))}l≥2.

More precisely, to a triangulated category T which has property of a phase gap we

attach a non-negative real number T ε. Natural assumptions on a SOD T = T1, T2

imply T1, T2ε ≤ min{T1ε , T2ε}. Using the norm we define a topology on

the set of equivalence classes of proper triangulated categories with a phase gap, such

that the set of discrete derived categories is a discrete subset, whereas the rationality

of a smooth surface S ensures that [Db(point)] ∈ Cl([Db(S)]). Categories in a

neighborhood of Db(K(l)) have the property that Db(K(l)) is embedded in each of

them. We view such embeddings as non-commutative curves in the ambient category

and introduce categorical invariants based on counting them. Examples show that the

idea of non-commutative curve-counting opens directions to newcategorical 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 curvecounting,

where the entities we count are a Calabi–Yau modification of Db(K(l)).

In the end we speculate that one might consider a holomorphic family of categories,

introduced by Kontsevich, as a non-commutative extension with the norm, introduced

here, playing a role similar to the classical notion of degree of an extension in Galois

theory.