Fibred Category - Examples

Examples

1. Categories of arrows: For any category E the category of arrows A(E) in E has as objects the morphisms in E, and as morphisms the commutative squares in E (more precisely, a morphism from (f: X → T) to (g: Y → S) consists of morphisms (a: X → Y) and (b: T → S) such that bf = ga). The functor which takes an arrow to its target makes A(E) into an E-category; for an object S of E the fibre E_S is the category E_/S of S-objects in E, i.e., arrows in E with target S. Cartesian morphisms in A(E) are precisely the cartesian squares in E, and thus A(E) is fibred over E precisely when fibre products exist in E.
2. Fibre bundles: Fibre products exist in the category Top of topological spaces and thus by the previous example A(Top) is fibred over Top. If Fib is the full subcategory of A(Top) consisting of arrows that are projection maps of fibre bundles, then Fib_S is the category of fibre bundles on S and Fib is fibred over Top. A a choice of a cleavage amounts to a choice of ordinary inverse image (or pull-back) functors for fibre bundles.
3. Vector bundles: In a manner similar to the previous examples the projections (p: V → S) of real (complex) vector bundles to their base spaces form a category Vect_R (Vect_C) over Top (morphisms of vector bundles respecting the vector space structure of the fibres). This Top-category is also fibred, and the inverse image functors (are the ordinary pull-back functors for vector bundles. These fibred categories are (non-full) subcategories of Fib.
4. Sheaves on topological spaces: The inverse image functors of sheaves make the categories Sh(S) of sheaves on topological spaces S into a (cleaved) fibred category Sh over Top. This fibred category can be described as the full sub-category of A(Top) consisting of etale spaces of sheaves. As with vector bundles, the sheaves of groups and rings also form fibred categories of Top.
5. Sheaves on topoi: If E is a topos and S is an object in E, the category E_S of S-objects is also a topos, interpreted as the category of sheaves on S. If f: T → S is a morphism in E, the inverse image functor f* can be described as follows: for a sheaf F on E_S and an object p: U → T in E_T one has f*F(U) = Hom_T(U, f*F) equals Hom_S(f o p, F) = F(U). These inverse image make the categories E_S into a split fibred category on E. This can be applied in particular to the "large" topos TOP of topological spaces.
6. Quasi-coherent sheaves on schemes: Quasi-coherent sheaves form a fibred category over the category of schemes. This is one of the motivating examples for the definition of fibred categories.
7. Fibred category admitting no splitting: A group G can be considered as a category with one object and the elements of G as the morphisms, composition of morphisms being given by the group law. A group homomorphism f: G → H can then be considered as a functor, which makes G into a H-category. It can be checked that in this set-up all morphisms in G are cartesian; hence G is fibred over H precisely when f is surjective. A splitting in this setup is a (set-theoretic) section of f which commutes strictly with composition, or in other words a section of f which is also a homomorphism. But as is well known in group theory, this is not always possible (one can take the projection in a non-split group extension).
8. Co-fibred category of sheaves: The direct image functor of sheaves makes the categories of sheaves on topological spaces into a co-fibred category. The transitivity of the direct image shows that this is even naturally co-split.