Let h(C) be the homotopy category of a stable infinity category C. Then the homotopy category h(C-delta 1) of morphisms in the stable infinity category C is also triangulated. Hence, the space Stab h(C-delta 1) of stability conditions on h(C-delta 1) is well-defined though the nonemptiness of Stab h(C-delta 1) is not obvious. Our basic motivation is a comparison of the homotopy type of Stab h(C) and that of Stab h(C-delta 1). Under the motivation, we show that functors d(0) and d(1) : C-delta 1 -> C induce continuous maps from Stab h(C) to Stab h(C-delta 1) contravariantly where d(0) (resp., d(1)) takes a morphism to the target (resp., source) of the morphism. As a consequence, if Stab h(C) is nonempty, then so is Stab h(C-delta 1). Assuming C is the derived infinity category of the projective line over a field, we further study basic properties of d(0)* and d(1)*. In addition, we give an example of a derived category which does not have any stability condition.
展开▼
