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 $L \dashv R$ where the right adjoint $R : C' \to C$ forgets structure in that it is a faithful functor. In these case the left adjoint $L : C \to C'$ usually is the free functor that “adds structure freely”.

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

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

nerve

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

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

realization

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

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

adjunction $|-|_C \dashv N_D$