nLab
embedding

Idea

An embedding is, generally, a morphism which in some sense is an isomorphism onto its image

For this to make sense in a given category $C$, we not only need a good notion of image. Note that it is not enough to have the image of $f\colon X \to Y$ as a subobject $\im f$ of $Y$; we also need to be able to interpret $f$ as a morphism from $X$ to $\im f$, because it is this morphism that we are asking to be an isomorphism.

• effective epimorphism $\Rightarrow$ regular epimorphism $\Leftrightarrow$ covering

• effective monomorphism $\Rightarrow$ regular monomorphism $\Leftrightarrow$ embedding .

As regular or effective monomorphisms

Definition

One general abstract way to define an embedding morphism is to say that this is equivalently a regular monomorphism.

If the ambient category has finite limits and colimits, then this is equivalently an effective monomorphism. In terms of this we recover a formalization of the above idea, that an embedding is an iso onto its image :

For a morphism $f : X \to Y$ in $C$ the definition of image as an equalizer says that the image of $f$ is

$$im f := \lim_\leftarrow ( Y \stackrel{\to}{\to} Y \coprod_X Y) \,.$$

In particular we have a factorization of $f$ as

$$f : X \stackrel{\tilde f}{\to} im f \hookrightarrow Y \,,$$

where the morphism on the right is a monomorphism.

The morphism $f$ being an effective monomorphism means that $\tilde f$ is an isomorphism, hence that $f$ is an “isomomorphism onto its image”.

Examples

In $Top$

A morphism $U \to X$ of topological spaces is a regular monomorphism precisely if this is an injection such that the topology on $U$ is the induced topology. This is an embedding of topological spaces.

In $SmothMfd$

• embedding of smooth manifolds