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 $f:X\to Y$ as a subobject $imf$ of $Y$; we also need to be able to interpret $f$ as a morphism from $X$ to $imf$, because it is this morphism that we are asking to be an isomorphism.

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

• effective monomorphism $\Rightarrow$ regular monomorphism $\iff$ 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

$$\mathrm{im}f:=\underleftarrow{\lim }(Y\overrightarrow{\to }Y\coprod _{X}Y).$$im f := \lim_\leftarrow ( Y \stackrel{\to}{\to} Y \coprod_X Y) \,.

In particular we have a factorization of $f$ as

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

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 $\mathrm{SmothMfd}$

• embedding of smooth manifolds