nLab
concrete subobject

Contents

Contents

This entry is meant as a complement to subobject. In particular, the initial author knows that it is there, and that it really doesn’t mean what he intends by “concrete subobject”.

Idea

In a concrete category ||:𝒞Set|-| : \mathcal{C} \to Set , for an object XX, when an injection i:A|X|i : A \to | X | naturally induces (this is deliberately vague)

  • an object A A^\vee with A=|A | A = | A ^ \vee |
  • a monomorphism i :A X i^\vee : A^\vee \to X with |i |=i | i^\vee | = i

then we say A A^\vee is a (concrete) subobject of XX.

Compare with Grothendieck fibration

Examples

Algebra

  • In the category GpGp of groups, a concrete subgroup is exactly a monomorphism in GpGp, which is exactly a map in GpGp whose underlying function is an injection; a map of sets f:A|G|f: A \to |G| underlies a subgroup inclusion ?G ? \to G iff there is exactly one group structure on AA such that ff underlies a homomorphism.

Topology

  • For every topological space XX and every set AA and every map f:A|X|f : A \to |X|, there is a terminal (coarsest) topology on AA with a continuous map to |X||X| over the function ff; a subspace (“embedded” subspace) is exactly an injection of underlying sets where the domain is given this terminal compatible topology.
  • A sub-β\beta-module is a closed subset; that is, an embedding of a compact space.

Geometry

  • An injection of a topological manifold to a smooth manifold may be a topological embedding, but there may be many (or no) compatible smooth structures on the domain; a submanifold is a smooth embedding with injective differential, which is exactly an embedding of a topological manifold such that smooth functions on the codomain restrict to give a smooth atlas on the domain.

Created on September 16, 2013 at 03:27:37. See the history of this page for a list of all contributions to it.