examples of adjoint functors

This entry lists examples for pairs of adjoint functors.

For examples of the other universal constructions see

Free/forgetful functors

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

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

For instance

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

Nerves and realization

For CC a category equipped with cosimplicial objects Δ C:ΔC\Delta_C : \Delta \to C and tensored over SetSet;

N D:CSet Δ opN_D : C \to Set^{\Delta^{op}}


|| C:Set Δ opC |-|_C : Set^{\Delta^{op}} \to C
N D(c):Δ opΔ C opC opC(,C)Set N_D(c) : \Delta^{op} \stackrel{\Delta_C^{op}}{\to} C^{op} \stackrel{C(-,C)}{\to} Set


|| C:Set Δ opC |-|_C : Set^{\Delta^{op}} \to C
|S | C= [n]ΔS nΔ C[n] |S_\bullet|_C = \int^{[n] \in \Delta} S_n \cdot \Delta_C[n]

adjunction || CN D|-|_C \dashv N_D

Last revised on October 12, 2010 at 15:36:40. See the history of this page for a list of all contributions to it.