An element, cohomology class, representation, or other invariant that does not lie in the image of a specified lifting map or functor; it signals the failure to lift an object from a target category or base to a source (for example from characteristic p to characteristic zero, or from a quotient to a full object).

Principle

Liftability is controlled by obstruction theories and deformation functors: a nonliftable class is detected by nonzero obstruction elements in the obstruction groups that measure the failure to extend a local deformation or to find a preimage under a given map of spaces or moduli.

Demonstration

Concrete scenario: a Galois representation over a finite field whose deformation ring has no points lifting to characteristic zero gives a mod p representation class that is nonliftable to characteristic zero; equivalently, a cohomology class may fail to have a preimage under a natural map due to a nonvanishing obstruction in H^2.

Misapplication

Confusing nonliftability with nonexistence of any related structure (for instance, assuming no deformation whatsoever exists) or failing to check whether liftability might hold after passing to an extension or allowing weakened targets (e.g., allowing twists, enlarging coefficient rings).

Consequence

A nonliftable class restricts possible deformations and global constructions, may force working in positive characteristic only or using stacky/derived enhancements, and often indicates nontrivial obstruction groups that one must study to understand deformation-theoretic limits.

Reversal

Liftable class: the element admits a preimage under the chosen lifting map, enabling passage to the larger category or base and often opening richer structural or arithmetic consequences.

Boundary

Depends on the specified lifting map, the categories involved, and auxiliary choices (level structures, coefficients, topology); a class may be nonliftable with respect to one lifting problem but liftable after base change or refinement, so nonliftability is not absolute without context.

Semantic Tension

Tension between 'nonliftable' and 'obstructed' or 'rigid': nonliftable emphasizes failure to find a preimage for a particular map, while 'obstructed' highlights cohomological reasons; 'rigid' suggests uniqueness of deformations rather than impossibility—these notions overlap but differ in focus.

Synthesis

A nonliftable class records a concrete obstruction in a lifting problem: it pinpoints where deformation- or descent-theoretic techniques fail to produce a preimage, guiding attention to obstruction groups, possible relaxations (twists, base change), and to the precise localization of failure rather than to a vague sense of impossibility.