View source for Adjoint functor

A concept expressing the universality and naturalness of many important mathematical constructions, such as a free universal algebra, various completions, and direct and inverse limits.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108201.png" /> be a covariant functor in one argument from a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108202.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108203.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108204.png" /> induces a functor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108205.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108206.png" /> is the category dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108208.png" /> is the category of sets, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108209.png" /> is the basic set-valued functor. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082010.png" /> is contravariant in the first argument and covariant in the second. Similarly, any covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082011.png" /> induces a functor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082012.png" /></td> </tr></table>

which is also contravariant in the first argument and covariant in the second. The functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082014.png" /> are adjoint, or form an adjoint pair, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082016.png" /> are isomorphic, that is, if there is a natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082017.png" /> that establishes a one-to-one correspondence between the sets of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082019.png" /> for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082021.png" />. The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082022.png" /> is called the adjunction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082025.png" /> is called the left adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082027.png" /> the right adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082028.png" /> (this is written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082029.png" />, or simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082030.png" />). The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082031.png" /> is called the co-adjunction.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082032.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082034.png" />, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082035.png" /></td> </tr></table>

The families of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082037.png" /> define natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082039.png" />, called the unit and co-unit of the adjunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082040.png" />. They satisfy the following equations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082041.png" /></td> </tr></table>

In general, a pair of natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082043.png" /> leads to an adjoint pair (or adjunction) if the following equations hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082044.png" /></td> </tr></table>

for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082046.png" />. A natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082047.png" /> is the unit of some adjunction if and only if for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082049.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082052.png" />. This property expresses the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082053.png" /> is a free object over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082054.png" /> with respect to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082055.png" /> in the sense of the following definition. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082056.png" /> together with a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082057.png" /> is free over an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082058.png" /> if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082059.png" /> can be written uniquely in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082060.png" /> for some morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082061.png" />. A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082062.png" /> has a left adjoint if and only if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082063.png" /> there is an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082064.png" /> that is free over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082065.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082066.png" />.

===Examples of adjoint functors.===


1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082068.png" /> is the category of sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082069.png" /> has a left adjoint only if it is representable. A representable functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082070.png" /> has a left adjoint if and only if all co-products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082071.png" /> exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082072.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082074.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082075.png" />.

2) In the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082076.png" /> of sets, for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082077.png" /> the basic functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082078.png" /> is the right adjoint of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082079.png" />.

3) In the category of Abelian groups, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082080.png" /> is the right adjoint of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082081.png" /> of tensor multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082082.png" />, and the imbedding functor of the full subcategory of torsion groups is the left adjoint of the functor of taking the torsion part of any Abelian group.

4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082083.png" /> be the forgetful functor from an arbitrary variety of universal algebras into the category of sets. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082084.png" /> has a left adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082085.png" />, which assigns to every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082086.png" /> the free algebra of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082088.png" /> as set of free generators.

5) The imbedding functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082089.png" /> of an arbitrary reflective subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082090.png" /> of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082091.png" /> is the right adjoint of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082092.png" />-reflector (cf. also [[Reflexive subcategory|Reflexive subcategor]]). In particular, the imbedding functor of the category of Abelian groups in the category of groups has a left adjoint, which assigns to every group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082093.png" /> its quotient group by the commutator subgroup.

===Properties of adjoint functors.===
The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors. Left adjoints commute with co-limits (e.g. co-products) and send null objects and null morphism into null objects and null morphisms, respectively.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082095.png" /> be categories that are complete on the left and locally small on the left. A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082096.png" /> has a left adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082097.png" /> if and only if the following conditions hold: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082098.png" /> commutes with limits; b) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082099.png" />, at least one of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820101.png" />, is non-empty; and c) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820102.png" />, there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820103.png" /> such that every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820104.png" /> is representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820108.png" />.

By passing to dual categories, one may establish a duality between the concepts of a  "left adjoint functor"  and a  "right adjoint functor" ; this enables one to deduce the properties of right adjoints from those of left adjoints.

The concept of an adjoint functor is directly connected with the concept of a [[Triple|triple]] (or monad) in a category.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,   E.G. Shul'geifer,   "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Maclane,   "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>



====Comments====
A category is called complete on the left if small diagrams have limits. A category is called locally small on the left if it has small hom-sets. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.