nLab
embedding

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higher category theory

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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 CC, we not only need a good notion of image. Note that it is not enough to have the image of f:XYf\colon X \to Y as a subobject imf\im f of YY; we also need to be able to interpret ff as a morphism from XX to imf\im f, because it is this morphism that we are asking to be an isomorphism.

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:XYf : X \to Y in CC the definition of image as an equalizer says that the image of ff is

imf:=lim (YY XY). im f := \lim_\leftarrow ( Y \stackrel{\to}{\to} Y \coprod_X Y) \,.

In particular we have a factorization of ff as

f:Xf˜imfY, f : X \stackrel{\tilde f}{\to} im f \hookrightarrow Y \,,

where the morphism on the right is a monomorphism.

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

Examples

In TopTop

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

In SmothMfdSmothMfd

Revised on November 10, 2013 22:34:42 by Urs Schreiber (89.204.137.233)