Pullbackkiinnittyjät

Pullbackkiinnittyjät is a concept in category theory describing a class of subobjects that remains stable under pullback. The idea is to formalize when a collection of subobjects behaves well when changing the base object along a morphism.

Formally, let C be a category with pullbacks. A class P of monomorphisms (subobjects) in C is called pullbackkiinnittyjät if for every morphism f from an object X to an object Y and every monomorphism m: A → Y belonging to P, the pullback of m along f yields a morphism m′: A′ → X that also lies in P. Intuitively, pulling back along any map does not take you out of the class P.

Examples help illustrate the notion. In the category of sets, subobjects are just subsets, and the pullback of an inclusion along any map is the preimage of the subset, which is again a subset; thus subobjects are pullbackkiinnittyjät. In topological spaces, the pullback of a closed embedding along any continuous map is a closed embedding, so closed subspaces form a pullbackkiinnittyjät class. In the category of modules over a ring, the preimage of a submodule under a homomorphism is again a submodule, making submodules pullbackkiinnittyjät as well.

Applications of pullbackkiinnittyjät appear in descent theory and sheaf theory, where stability under base change (pullback) is essential for defining coherent structures across different contexts. The notion also aligns with broader studies of fibred categories and Grothendieck topologies, where stable classes of subobjects support robust constructions.