nLab embedding

This page consider the very general concept of embeddings. For the special cases traditionally considered see at embedding of topological spaces or embedding of smooth manifolds.

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”.

In dependent type theory

Assume that the dependent type theory has identity types, dependent identity types, and uniqueness quantifiers. Then a family of elements $x:A \vdash f(x):B$ is an embedding if there is a family of elements

$$x:A, y:A, q:f(x) =_B f(y) \vdash \eta(x, y, q):\exists! p:x =_A y.ap_f(x, y, p) =_{f(x) =_B f(y)} q$$

which states that for all identities $q:f(x) =_B f(y)$ there is a unique identity $p:x =_A y$ up to identity such that $ap_f(x, y, p) =_{f(x) =_B f(y)} q$.

Equivalently, if the dependent type theory also has existential quantifiers, then $x:A \vdash f(x):B$ is an embedding if there is a family of elements

$$y:B, z:\exists x:A.f(x) =_B y \vdash p(y, z):\exists! x:A.f(x) =_B y$$

which states that for all $y:B$, if there exists an element $x:A$ such that $f(x) =_B y$, then that element is unique up to identity.

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 $SmoothMfd$

• embedding of smooth manifolds

 Related concepts

• embedding type