Editing Embedding (section)

==Category theory==

In [[category theory]], there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any [[monomorphism#Related concepts|extremal monomorphism]] is an embedding and embeddings are stable under [[Pullback (category theory)|pullback]]s.

Ideally the class of all embedded [[subobject]]s of a given object, up to isomorphism, should also be [[small class|small]], and thus an [[ordered set]]. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a [[closure operator]]).

In a [[concrete category]], an '''embedding''' is a morphism <math>f:A\rightarrow B</math> that is an injective function from the underlying set of <math>A</math> to the underlying set of <math>B</math> and is also an '''initial morphism''' in the following sense:
If <math>g</math> is a function from the underlying set of an object <math>C</math> to the underlying set of <math>A</math>, and if its composition with <math>f</math> is a morphism <math>fg:C\rightarrow B</math>, then <math>g</math> itself is a morphism.

A [[factorization system]] for a category also gives rise to a notion of embedding. If <math>(E,M)</math> is a factorization system, then the morphisms in <math>M</math> may be regarded as the embeddings, especially when the category is well powered with respect to <math>M</math>. Concrete theories often have a factorization system in which <math>M</math> consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.

As usual in category theory, there is a [[dual (category theory)|dual]] concept, known as quotient. All the preceding properties can be dualized.

An embedding can also refer to an [[Subcategory#Embeddings|embedding functor]].