nLab concrete subobject

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 $|-| : \mathcal{C} \to Set$, for an object $X$, when an injection $i : A \to | X |$ naturally induces (this is deliberately vague)

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

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

Compare with Grothendieck fibration

Examples

Algebra

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

Topology

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