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), the l-Kronecker quiver, is biholomorphic to $${{\mathbb {C}}} \times {\mathcal {H}}$$ for $$l\ge 3$$ . In the present paper we define a new notion of norm, which distinguishes $$\{D^b(K(l)) \}_{l\ge 2}$$ . More precisely, to a triangulated category $${\mathcal {T}}$$ which has property of a phase gap we attach a non-negative real number $$\left\| {\mathcal {T}}\right\| ^{\varepsilon }$$ . Natural assumptions on a SOD $${\mathcal {T}} =\langle {\mathcal {T}}_1,{\mathcal {T}}_2\rangle $$ imply $$\left\| \langle {\mathcal {T}}_1,{\mathcal {T}}_2\rangle \right\| ^{\varepsilon }\le {\mathrm{min}}\{\left\| {\mathcal {T}}_1 \right\| ^{\varepsilon },\left\| {\mathcal {T}}_2\right\| ^{\varepsilon } \}$$ . 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 $$[D^b(point)] \in \mathrm{Cl}([D^b(S)])$$ . Categories in a neighborhood of $$D^b(K(l))$$ have the property that $$D^b(K(l))$$ 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 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 $$D^b(K(l))$$ . 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.