Editing Subcategory (section)

== Embeddings ==
Given a subcategory ''S'' of ''C'', the inclusion functor {{math|''I'' : ''S'' → ''C''}} is both a faithful functor and [[injective]] on objects. It is [[full functor|full]] if and only if ''S'' is a full subcategory.

Some authors define an '''embedding''' to be a [[full and faithful functor]]. Such a functor is necessarily injective on objects up to [[isomorphism]]. For instance, the [[Yoneda embedding]] is an embedding in this sense.

Some authors define an '''embedding''' to be a full and faithful functor that is injective on objects.<ref>{{cite web|author=Jaap van Oosten|title=Basic category theory|url=http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf}}</ref>

Other authors define a functor to be an '''embedding''' if it is
faithful and
injective on objects.
Equivalently, ''F'' is an embedding if it is injective on morphisms. A functor ''F'' is then called a '''full embedding''' if it is a full functor and an embedding.

With the definitions of the previous paragraph, for any (full) embedding ''F'' : ''B'' → ''C'' the [[Image (mathematics)|image]] of ''F'' is a (full) subcategory ''S'' of ''C'', and ''F'' induces an [[isomorphism of categories]] between ''B'' and ''S''. If ''F'' is not injective on objects then the image of ''F'' is [[equivalence of categories|equivalent]] to ''B''.

In some categories, one can also speak of morphisms of the category being [[embedding#Category theory|embeddings]].