  We introduce the notions of categorical systoles and categorical volumes of
Bridgeland stability conditions on triangulated categories. We prove that for
any projective K3 surface, there exists a constant C depending only on the rank
and discriminant of its Picard group, such that $$\mathrm{sys}(\sigma)^2\leq
C\cdot\mathrm{vol}(\sigma)$$ holds for any stability condition on the derived
category of coherent sheaves on the K3 surface. This is an algebro-geometric
generalization of a classical systolic inequality on two-tori. We also discuss
applications of this inequality in symplectic geometry.
