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	~~In [[category theory]]~~, ~~a '''faithful functor''' (resp~~. ~~a '''full functor''')~~ is ~~a [[functor]] that is [[injective]]~~ (~~resp~~. ~~[[surjective]]) when restricted~~ to ~~each set~~ of ~~[[morphism]]s that have a given source and target~~.		To sure extend, the exceptionally low interest rate atmosphere which was brought on by Sub prime crisis might be well one of many fueling factors to spur up the property market growth because property buyers can simply gear as much as leverage extra. But if we go deeper, is it obtained to do extra with the advertising gimmicks by the developer?<br><br>HDB Flats are flats constructed and maintained by the Housing Growth Board (HDB). More than eighty% of Singaporeans reside in HDB housing estates. HDB housing estates are usually self-contained cities with clinics, faculties, supermarkets, food centres, in addition to sports activities and leisure facilities. For the classification of HDB flats, the living room is counted as one room. 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	~~Explicitly~~, ~~let ''C''~~ and ~~''D'' be ([[locally small category\|locally small]]) [[category (mathematics)\|categories]] and let ''F'' : ''C'' → ''D'' be~~ a ~~functor from ''C'' to ''D''~~. ~~The functor ''F'' induces~~ a ~~function~~
	:<~~math~~>~~F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(F(X),F(Y))~~<~~/math~~>
	~~for every pair of objects ''X'~~' and ~~''Y'' in ''C''~~. The ~~functor ''F''~~ is ~~said to be~~
	*'~~''faithful''' if ''F''<sub>''X'',''Y''</sub> is [[injective]]<ref>Mac Lane (1971), p~~. ~~15</ref><ref name="Jacobson-09">Jacobson (2009)~~, p. 22<~~/ref~~>
	*'''full''' if ''F''<~~sub~~>''~~X''~~,~~''Y''</sub>~~ is ~~[[surjective]]<ref name="Jacobson~~-~~09"/><ref>Mac Lane (1971)~~, p. 14<~~/ref~~>
	*'''fully faithful''' if ''F''<~~sub~~>'~~'X''~~,~~''Y''</sub> is [[bijective]]~~
	~~for each ''X''~~ and ~~''Y'' in ''C''.~~

	~~A faithful functor need not be injective~~ on ~~objects or morphisms~~. ~~That~~ is, two ~~objects ''X~~'' and ~~''X''′ may map~~ to the ~~same object~~ in ~~''D'' (which is why~~ the ~~range of~~ a ~~full and faithful functor~~ is ~~not necessarily isomorphic to ''C''), and two morphisms ''f'' : ''X'' → ''Y'' and ''f''′ : ''X''′ → ''Y''′ (with different domains/codomains) may map~~ to the ~~same morphism in ''D''~~. ~~Likewise, a full functor need not~~ be ~~surjective~~ on ~~objects or morphisms~~. ~~There may be objects in~~ '~~'D'' not~~ of the ~~form ''FX'' for some ''X'' in ''C'~~'. ~~Morphisms between such objects clearly cannot come from morphisms~~ in ~~''C''.~~

	~~A full~~ and ~~faithful functor is necessarily injective on objects up~~ to ~~isomorphism~~. ~~That is~~, ~~if ''F'' : ''C'' → ''D'' is~~ a ~~full and faithful functor and <math>F~~(~~x)\cong F(y~~)<~~/math~~> ~~then~~ <~~math>x \cong y</math~~>.

	~~==Examples==~~
	* The [[forgetful functor]] ''U'' : '''Grp''' → '''Set''' is faithful as each group maps to ~~a unique set~~ and the ~~group homomorphism are a subset~~ of ~~the functions~~. ~~This functor is not full as there are functions between [[group (mathematics)\|group~~]~~]s which are not [[group homomorphism]]s~~. ~~A category with~~ a ~~faithful functor to '''Set''' is (by definition)~~ a ~~[[concrete category]];~~ in ~~general, that forgetful functor is~~ not ~~full.~~

	* Let ''F'' : '''Set''' → '''Set''' be the ~~functor which maps every set to~~ the ~~[[empty set]]~~ and ~~every function~~ to ~~the [[empty function]]. Then ''F'' is full~~, ~~but is neither injective on objects nor on morphisms~~.

	* The ~~inclusion functor '''Ab''' → '''Grp'''~~ is ~~fully faithful~~.

	~~==See also==~~
	*[[full ~~subcategory]]~~
	*[[equivalence of ~~categories]]~~

	~~== Notes ==~~
	~~<references/>~~

	~~==References==~~
	* {{Cite book
	~~\|first=Saunders~~
	~~\|last=Mac Lane~~
	~~\|authorlink=Saunders Mac Lane~~
	~~\|title=[[Categories~~ for ~~the Working Mathematician]]~~
	~~\|edition=second~~
	~~\|date=September 1998~~
	~~\|publisher=Springer~~
	~~\|isbn=0-387-98403-8}}~~
	* {{Cite book\| last=Jacobson\| first=Nathan\| author-link=Nathan Jacobson\| year=2009\| title=Basic algebra\| edition=2nd\| volume = 2 \| series= \| publisher=Dover\| isbn = 978-0-486-47187-7}}

	~~{{Functors}}~~

	~~{{DEFAULTSORT:Full And Faithful Functors}}~~
	~~[[Category:Functors]]~~

en>Jamesx12345: Reverted edits by Aleksa is cool (talk) to last version by 210.77.5.190

2013-09-28T16:54:17Z

Reverted edits by Aleksa is cool (talk) to last version by 210.77.5.190

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In [[category theory]], a '''faithful functor''' (resp. a '''full functor''') is a [[functor]] that is [[injective]] (resp. [[surjective]]) when restricted to each set of [[morphism]]s that have a given source and target.

Explicitly, let ''C'' and ''D'' be ([[locally small category|locally small]]) [[category (mathematics)|categories]] and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function
:<math>F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(F(X),F(Y))</math>
for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be
*'''faithful''' if ''F''''X'',''Y'' is [[injective]]<ref>Mac Lane (1971), p. 15</ref><ref name="Jacobson-09">Jacobson (2009), p. 22</ref>
*'''full''' if ''F''''X'',''Y'' is [[surjective]]<ref name="Jacobson-09"/><ref>Mac Lane (1971), p. 14</ref>
*'''fully faithful''' if ''F''''X'',''Y'' is [[bijective]]
for each ''X'' and ''Y'' in ''C''.

A faithful functor need not be injective on objects or morphisms. That is, two objects ''X'' and ''X''′ may map to the same object in ''D'' (which is why the range of a full and faithful functor is not necessarily isomorphic to ''C''), and two morphisms ''f'' : ''X'' → ''Y'' and ''f''′ : ''X''′ → ''Y''′ (with different domains/codomains) may map to the same morphism in ''D''. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in ''D'' not of the form ''FX'' for some ''X'' in ''C''. Morphisms between such objects clearly cannot come from morphisms in ''C''.

A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if ''F'' : ''C'' → ''D'' is a full and faithful functor and <math>F(x)\cong F(y)</math> then <math>x \cong y</math>.

==Examples==
* The [[forgetful functor]] ''U'' : '''Grp''' → '''Set''' is faithful as each group maps to a unique set and the group homomorphism are a subset of the functions. This functor is not full as there are functions between [[group (mathematics)|group]]s which are not [[group homomorphism]]s. A category with a faithful functor to '''Set''' is (by definition) a [[concrete category]]; in general, that forgetful functor is not full.

* Let ''F'' : '''Set''' → '''Set''' be the functor which maps every set to the [[empty set]] and every function to the [[empty function]]. Then ''F'' is full, but is neither injective on objects nor on morphisms.

* The inclusion functor '''Ab''' → '''Grp''' is fully faithful.

==See also==
*[[full subcategory]]
*[[equivalence of categories]]

== Notes ==
<references/>

==References==
* {{Cite book
|first=Saunders
|last=Mac Lane
|authorlink=Saunders Mac Lane
|title=[[Categories for the Working Mathematician]]
|edition=second
|date=September 1998
|publisher=Springer
|isbn=0-387-98403-8}}
* {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | series= | publisher=Dover| isbn = 978-0-486-47187-7}}

{{Functors}}

{{DEFAULTSORT:Full And Faithful Functors}}
[[Category:Functors]]

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117.192.105.14 at 21:46, 13 May 2014

en>Jamesx12345: Reverted edits by Aleksa is cool (talk) to last version by 210.77.5.190