nLab
adjunction

Definition

An adjunction in a 2-category is a pair of objects $C, D$ together with morphisms $L : C \to D$ , $R : D \to C$ and 2-cells $η : 1_{C} \to R \circ L$ , $ϵ : L \circ R \to 1_{D}$ satisfying the equations

R ϵ . η R = 1_{R} i.e. \begin{matrix} 1_{C} \\ D & \overset{R}{\to} & C & \overset{L}{\to} & D & \overset{R}{\to} & C \\ 1_{D} \end{matrix} = D \overset{R}{\to} C

and

ϵ L . L η = 1_{L} i.e. \begin{matrix} 1_{C} \\ C & \overset{L}{\to} & D & \overset{R}{\to} & C & \overset{L}{\to} & D \\ 1_{D} \end{matrix} = C \overset{L}{\to} D

variously called the triangle identities or the zig-zag identities. We call $L$ the left adjoint (of $R$ ) and $R$ the right adjoint (of $L$ ). We call $η$ the unit? and $ϵ$ the counit? of the adjunction.

When interpreted in the prototypical 2-category Cat, $C$ and $D$ are categories, $L$ and $R$ are functors, and $η$ and $ϵ$ are natural transformations. In this case (which was of course the first to be defined) there are a number of equivalent definitions of an adjunction, which can be found on the page adjoint functor. Conversely, the definition in any 2-category can be obtained by internalization from the definition in $Cat$ .

Remarks

Essentially everything that makes category theory nontrivial and interesting can be derived from the concept of adjunction. In particular universal constructions such as limits and colimits are examples of certain adjunctions.

String Diagrams

The definition of an adjunction may be nicely expressed using string diagrams. The data $L : C \to D$ , $R : D \to C$ and 2-cells $η : 1_{C} \to R \circ L$ , $ϵ : L \circ R \to 1_{D}$ are depicted as

String diagram for a left adjoint (for 'Adjunction') String diagram of a right adjoint (for 'Adjunction') String diagram of an adjunction unit (for 'Adjunction')

(where 1-cells read from right to left and 2-cells from bottom to top), and the zigzag identities are expressed as “pulling zigzags straight” (hence the name):

String diagram of first zigzag identity (for 'Adjunction')

Often, arrows on strings are used to distinguish $L$ and $R$ , and most or all other labels are left implicit; so the zigzag identities, for instance, become:

Minimal string diagram of first zigzag identity (for 'Adjunction') Minimal string diagram of second zigzag identity (for 'Adjunction')

References

Catsters, Adjunctions (YouTube)

Revised on January 14, 2010 09:21:19 by Andrew Stacey (129.241.15.70)

nLab adjunction

Definition

Remarks

String Diagrams

References

nLab
adjunction