fiberkategori

The fiber category, or fiberkategori, is a construction in category theory associated with a functor F: E -> B and a chosen object b in B. The fiber E_b is the comma category (F ↓ b). Its objects are pairs (e, φ) where e is an object of E and φ: F(e) -> b is a morphism in B. A morphism from (e, φ) to (e', φ') is a morphism f: e -> e' in E such that φ = φ' ∘ F(f). Intuitively, E_b collects all objects of E together with their structure maps to the base object b.

If F is a fibration, the fiber construction interacts with arrows in B. For a morphism u: b' -> b in B and an object (e, φ) in E_b, there exists a cartesian lift over u to an object (e', φ') in E_{b'} and a morphism in E that projects to u. This induces a reindexing functor u^*: E_b -> E_{b'} encoding base change along u. In many contexts, fibers vary functorially with the base, and E is assembled from its fibers via the Grothendieck construction.

Relation to other concepts: The fiber E_b is the over-category (id_B ↓ b) when F is the identity functor on B, i.e., it generalizes familiar constructions such as B/b. Fibers are central to the study of fibrations, descent theory, stacks, and the Grothendieck construction, where a family of categories indexed by B is encoded by a single total category together with its fiber functors.

See also: comma category, over category, fibration, Grothendieck construction, descent.