Some new categorical invariants

Viewing $D^b(K(l))$ as a non-commutative curve, we observe that it is reasonable to count non-commutative curves in any category which lies in a small neighborhood (w.r. to our topology) of a given non-commutative curve. 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 general definition, which specializes to the non-commutative curve-counting invariants.