fibrats

Fibrats are a class of mathematical objects that arise in the study of algebraic geometry and topology, serving as a generalization of fiber bundles. They were introduced by the mathematician Alexander Grothendieck in the context of schemes and algebraic stacks, providing a framework to unify various geometric constructions.

A fibrat, or fibration in the categorical sense, is a morphism of mathematical structures (such as schemes, manifolds, or topological spaces) that exhibits a "local triviality" property. Specifically, a fibrat consists of a total space *E*, a base space *B*, and a projection map *π: E → B*, where for every point *b* in *B*, there exists an open neighborhood *U* of *b* such that the restriction of *π* to *π⁻¹(U)* is isomorphic to a product *U × F* for some fixed fiber *F*. This local triviality ensures that the fibrat behaves like a fiber bundle in a generalized setting, though the fibers may not be isomorphic globally.

In algebraic geometry, fibrats are particularly useful in studying families of objects, such as curves or varieties, parameterized by a base scheme. For instance, a family of elliptic curves over a base scheme *S* can be viewed as a fibrat where the total space is a scheme representing the universal deformation of elliptic curves, and the fibers correspond to specific elliptic curves over points of *S*. This perspective allows for the application of homotopical and categorical techniques to problems in algebraic geometry.

Fibrats also appear in the study of moduli spaces, where they help describe the deformation theory of geometric objects. For example, the moduli space of stable bundles over a curve can be constructed using fibrats to capture the infinitesimal deformations of such bundles. The theory of fibrats extends beyond classical algebraic geometry, influencing areas such as derived algebraic geometry and higher category theory, where they provide tools for understanding more complex geometric phenomena.