nLab
examples of adjoint functors

This entry lists examples for pairs of adjoint functors.

For examples of the other universal constructions see

• limits and colimits by example

• examples of Kan extensions.

Free/forgetful functors

The classical examples of pairs of adjoint functors are $L⊣R$ where the right adjoint $R:C\prime \to C$ forgets structure in that it is a faithful functor. In these case the left adjoint $L:C\to C\prime$ usually is the free functor that “adds structure freely”.

In fact, one usually turns this around and defines the free $C\prime$-structure on an object $c$ of $C$ as the image of that object under the left adjoint (if it exists) to the functor $R:C\prime \to C$ that forgets this structure.

For instance

• forgetful right adjoint $R:$ Grp $\to$ Set forgets the group structure on a group and just remembers the underlying set – the left adjoint $L:\mathrm{Set}\to \mathrm{Grp}$ sends each set to the free group over it.

Nerves and realization

For $C$ a category equipped with cosimplicial objects ${\Delta }_C:\Delta \to C$ and tensored over $\mathrm{Set}$;

$N_D:C\to {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$

nerve

$$\mid -{\mid }_C:{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to C$$|-|_C : Set^{\Delta^{op}} \to C

$$N_D(c):{\Delta }^{\mathrm{op}}\stackrel{{\Delta }_C^{\mathrm{op}}}{\to }{C}^{\mathrm{op}}\stackrel{C(-,C)}{\to }\mathrm{Set}$$N_D(c) : \Delta^{op} \stackrel{\Delta_C^{op}}{\to} C^{op} \stackrel{C(-,C)}{\to} Set

realization

$$\mid -{\mid }_C:{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to C$$|-|_C : Set^{\Delta^{op}} \to C

$$\mid S_{•}{\mid }_C={\int }^{[n]\in \Delta }{S}_n\cdot {\Delta }_C[n]$$|S_\bullet|_C = \int^{[n] \in \Delta} S_n \cdot \Delta_C[n]

adjunction $\mid -{\mid }_C⊣N_D$