fiberrelative

Fiberrelative is a term used in some mathematical discussions to denote properties or invariants that are defined relative to a base space for a family of objects parameterized by that base. In this usage, one studies a morphism f: E -> B and regards the fibers E_b = f^{-1}(b) as a family indexed by points b in B. A fiberrelative invariant assigns to each fiber a quantity that may depend on b only through the fiber, and one analyzes how these invariants vary as b varies over B.

In the formal setting of algebraic geometry or topology, a fiberrelative invariant is a rule that assigns to each fiber E_b an invariant I(E_b), such that the collection {I(E_b) | b in B} captures how the fibers behave in the family. Examples include the dimension of E_b, the cohomology groups H^i(E_b), or Hilbert polynomials in algebraic geometry. Monodromy and local triviality considerations describe how these invariants can change when one moves along paths in B.

Key properties of fiberrelative invariants include their behavior under base change, semicontinuity phenomena, and stratifications of the base according to the constancy of the invariant. In many contexts, invariants are locally constant on strata where the family is well-behaved, leading to moduli or deformation-theoretic conclusions.

Because “fiberrelative” is not a widely standardized term, its precise meaning can vary by context. It serves as a descriptive label for invariants that are natural to study in a family over a base, highlighting the emphasis on base-dependent behavior rather than on absolute properties of the total space. See also fiber bundle, base change, and relative cohomology.