Editing Adjoint functors (section)

==Formal definitions==

There are various equivalent definitions for adjoint functors:

* The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint.  They are also the most analogous to our intuition involving optimizations.
* The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word ''adjoint''.
* The definition via counit–unit adjunction is convenient for proofs about functors that are known to be adjoint, because they provide formulas that can be directly manipulated.

The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics.  Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be repeated separately in every subject area.

===Conventions===

The theory of adjoints has the terms ''left'' and ''right'' at its foundation, and there are many components that live in one of two categories ''C'' and ''D'' that are under consideration.  Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category ''C'' or the "righthand" category ''D'', and also to write them down in this order whenever possible.

In this article for example, the letters ''X'', ''F'', ''f'', ε will consistently denote things that live in the category ''C'', the letters ''Y'', ''G'', ''g'', η will consistently denote things that live in the category ''D'', and whenever possible such things will be referred to in order from left to right (a functor ''F'' : ''D'' → ''C'' can be thought of as "living" where its outputs are, in ''C''). If the arrows for the left adjoint functor F were drawn they would be pointing to the left; if the arrows for the right adjoint functor G were drawn they would be pointing to the right.

===Definition via universal morphisms===

By definition, a functor 
<math>F: \mathcal{D} \to \mathcal{C}</math> is a '''left adjoint functor''' if for each object <math>X</math> in <math>\mathcal{C}</math> there exists a [[universal morphism]] 
from <math>F</math> to <math>X</math>. Spelled out, this means that for each object <math>X</math> in <math>\mathcal{C}</math> there exists an object 
<math>G(X)</math> in <math>\mathcal{D}</math> and a morphism <math>\epsilon_X: F(G(X)) \to X</math> such that for every object 
<math>Y</math> in <math>\mathcal{D}</math> and every morphism <math>f: F(Y) \to X</math> there exists a unique morphism 
<math>g: Y \to G(X)</math> with <math>\epsilon_X \circ F(g) = f</math>.

The latter equation is expressed by the following [[commutative diagram]]:
[[File:Definition of the counit of an adjunction.svg|center|Here the counit is a universal morphism.|190px]]

In this situation, one can show that <math>G</math> can be turned into a functor <math>G :\mathcal{C} \to \mathcal{D}</math> in a unique way such that 
<math>\varepsilon_X \circ F(G(f)) = f \circ \varepsilon_{X'}</math> 
for all morphisms <math>f: X' \to X</math> in <math>\mathcal{C}</math>; <math>F</math> is then called a '''left adjoint''' to <math>G</math>.

Similarly, we may define right-adjoint functors. A functor <math>G: \mathcal{C} \to \mathcal{D}</math> is a '''right adjoint functor''' if for each object <math>Y</math> in <math>\mathcal{D}</math>, 
there exists a [[universal morphism]] from <math>Y</math> to <math>G</math>. Spelled out, this means that for each object <math>Y</math> in <math>\mathcal{D}</math>, 
there exists an object <math>F(Y)</math> in <math>C</math> and a morphism <math>\eta_Y: Y \to G(F(Y))</math> such that for every object <math>X</math> in <math>\mathcal{C}</math> 
and every morphism <math>g: Y \to G(X)</math> there exists a unique morphism <math>f: F(Y) \to X</math> with <math>G(f) \circ \eta_Y = g</math>.
[[File:Definition of the unit of an adjunction 1.svg|center|The existence of the unit, a universal morphism, can prove the existence of an adjunction.|190px]]

Again, this <math>F</math> can be uniquely turned into a functor <math>F: \mathcal{D} \to \mathcal{C}</math> such that <math>G(F(g)) \circ \eta_Y = \eta_{Y'} \circ g</math> for <math>g: Y \to Y'</math> a morphism in <math>\mathcal{D}</math>; <math>G</math> is then called a '''right adjoint''' to <math>F</math>.

It is true, as the terminology implies, that <math>F</math> is left adjoint to <math>G</math> if and only if <math>G</math> is right adjoint to <math>F</math>.

These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.

===Definition via Hom-sets===

Using [[hom-set]]s, an adjunction between two categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math> can be defined as consisting of two [[functor]]s <math>F : \mathcal{D} \to \mathcal{C}</math> and <math>G : \mathcal{C} \to \mathcal{D}</math> and a [[natural isomorphism]]
:<math>\Phi:\mathrm{hom}_\mathcal{C}(F-,-) \to \mathrm{hom}_\mathcal{D}(-,G-)</math>.
This specifies a family of bijections
:<math>\Phi_{Y,X}:\mathrm{hom}_\mathcal{C}(FY,X) \to \mathrm{hom}_\mathcal{D}(Y,GX)</math>
for all objects <math>X\in\mathcal{C}</math> and <math>Y\in\mathcal{D}</math>.

In this situation, <math>F</math> is left adjoint to <math>G</math> and '''<math>G</math> is right adjoint to <math>F</math>'''.

This definition is a logical compromise in that it is more difficult to establish its satisfaction than the universal morphism definitions, and has fewer immediate implications than the counit–unit definition.  It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.

In order to interpret <math>\Phi</math> as a ''natural isomorphism'', one must recognize <math>\text{hom}_\mathcal{C}(F-,-)</math> and <math>\text{hom}_\mathcal{D}(-,G-)</math> as functors. In fact, they are both [[bifunctor]]s from <math>\mathcal{D}^\text{op} \times \mathcal{C}</math> to <math>\mathbf{Set}</math> (the [[category of sets]]). For details, see the article on [[hom functor]]s. Spelled out, the naturality of <math>\Phi</math> means that for all [[morphism]]s <math>f : X \to X'</math> in <math>\mathcal{C}</math> and all morphisms <math>g : Y' \to Y</math> in <math>\mathcal{D}</math> the following diagram [[commutative diagram|commutes]]:

[[File:Natural phi.svg|center|Naturality of Φ|350px]]

The vertical arrows in this diagram are those induced by composition. Formally, <math>\text{Hom}(Fg, f) : \text{Hom}_\mathcal{C}(FY, X) \to \text{Hom}_\mathcal{C}(FY', X')</math> is given by <math>h \mapsto f \circ h \circ Fg</math> for each <math>h \in \text{Hom}_\mathcal{C}(FY, X).</math> <math>\text{Hom}(g, Gf)</math> is similar.

===Definition via counit–unit===

A third way of defining an adjunction between two categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math> consists of two [[functor]]s <math>F : \mathcal{D} \to \mathcal{C}</math> and <math>G : \mathcal{C} \to \mathcal{D}</math> and two [[natural transformation]]s
:<math>\begin{align}
\varepsilon &: FG \to 1_\mathcal{C} \\
\eta &: 1_\mathcal{D} \to GF\end{align}</math>
respectively called the '''counit''' and the '''unit''' of the adjunction (terminology from [[universal algebra]]), such that the compositions
:<math>F\xrightarrow\overset{}{\;F\eta\;}FGF\xrightarrow\overset{}{\;\varepsilon F\,}F</math>
:<math>G\xrightarrow\overset{}{\;\eta G\;}GFG\xrightarrow\overset{}{\;G \varepsilon\,}G</math>
are the identity morphisms <math>1_F</math> and <math>1_G</math> on {{mvar|F}} and {{mvar|G}} respectively.

In this situation we say that {{mvar|F}} '''is left adjoint to''' {{mvar|G}} and {{mvar|G}} '''is right adjoint to''' {{mvar|F}}, and may indicate this relationship by writing <math>(\varepsilon,\eta):F\dashv G</math>&nbsp;, or, simply <math>F\dashv G</math>&nbsp;.

In equational form, the above conditions on <math>(\varepsilon,\eta)</math> are the '''counit–unit equations'''
:<math>\begin{align}
1_F &= \varepsilon F\circ F\eta\\
1_G &= G\varepsilon \circ \eta G
\end{align}</math>
which imply that for each <math>X \in \mathcal{C}</math> and each <math>Y \in \mathcal{D},</math>
:<math>\begin{align}
1_{FY} &= \varepsilon_{FY}\circ F(\eta_Y) \\
1_{GX} &= G(\varepsilon_X)\circ\eta_{GX}
\end{align}</math>.

Note that <math>1_{\mathcal C}</math> denotes the identify functor on the category <math>\mathcal C</math>, <math>1_F</math> denotes the identity natural transformation from the functor {{mvar|F}} to itself, and <math>1_{FY}</math> denotes the identity morphism of the object <math>FY.</math>
[[File:String diagram adjunction.svg|thumb|String diagram for adjunction.]]
These equations are useful in reducing proofs about adjoint functors to algebraic manipulations.  They are sometimes called the ''triangle identities'', or sometimes the ''zig-zag equations'' because of the appearance of the corresponding [[string diagram]]s.  A way to remember them is to first write down the nonsensical equation <math>1=\varepsilon\circ\eta</math> and then fill in either {{mvar|F}} or {{mvar|G}} in one of the two simple ways that make the compositions defined.

Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an ''initial'' property whereas the counit morphisms  satisfy ''terminal'' properties, and dually for limit versus unit.  The term ''unit'' here is borrowed from the theory of [[Monad (category theory)|monads]], where it looks like the insertion of the identity {{math|1}} into a [[monoid]].