> This page is about the general concept of embeddings in [[category theory]]. For the special cases see embeddings _[[embedding of topological spaces|of topological spaces]]_, _[[embedding of smooth manifolds|of smooth manifolds]]_, or _[[embedding of types|of types]]_. For the notion in [[model theory]] see instead at *[[elementary embedding]]*. +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Category theory +--{: .hide} [[!include category theory - contents]] =-- #### Higher category theory +--{: .hide} [[!include higher category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea An embedding is 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 images. 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 [[finite 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 isomorphism onto its _image_ : For a [[morphism]] $f : X \to Y$ in $C$, the definition of [an image as an equalizer](http://ncatlab.org/nlab/show/image#AsEqualizer) says that the [[image]] of $f$ is $$ im f \coloneqq \lim_\leftarrow ( Y \stackrel{\to}{\to} Y \coprod_X Y) \,. $$ In particular, we have a factorization of $f$ as $$ f \colon X \stackrel{\tilde f}{\to} im f \hookrightarrow Y \,, $$ where the morphism on the right is a [[monomorphism]]. The morphism $f$ being an [[effective monomorphism]] means that $\tilde f$ is an [[isomorphism]], and hence that $f$ is an "isomomorphism onto its image". ## Examples ### In $Top$ A morphism $U \to X$ of [[topological space]]s is a regular monomorphism precisely if it 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]] ### In $Sch$ * [[Plücker embedding]] ## Related concepts * [[embedding type]] * [[embedding in type theory]] [[!redirects embedding]] [[!redirects embeddings]] [[!redirects imbedding]] [[!redirects imbeddings]] [[!redirects closed embedding]] [[!redirects open embedding]] [[!redirects open embeddings]] [[!redirects imbedding]] [[!redirects imbeddings]]