Adjoints of fully faithful functors #
A left adjoint is fully faithful, if and only if the unit is an isomorphism (and similarly for right adjoints and the counit).
Adjunction.restrictFullyFaithful
shows that an adjunction can be restricted along fully faithful
inclusions.
Future work #
The statements from Riehl 4.5.13 for adjoints which are either full, or faithful.
If the left adjoint is fully faithful, then the unit is an isomorphism.
See
- Lemma 4.5.13 from [Riehl][riehl2017]
- https://math.stackexchange.com/a/2727177
- https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)
If the right adjoint is fully faithful, then the counit is an isomorphism.
See https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)
If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given by L whiskered with the counit.
If the unit is an isomorphism, bundle one has an isomorphism L ⋙ R ⋙ L ≅ L
.
Instances For
If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given by R whiskered with the unit.
If the counit of an is an isomorphism, one has an isomorphism (R ⋙ L ⋙ R) ≅ R
.
Instances For
If the unit is an isomorphism, then the left adjoint is full
Instances For
If the unit is an isomorphism, then the left adjoint is faithful
If the counit is an isomorphism, then the right adjoint is full
Instances For
If the counit is an isomorphism, then the right adjoint is faithful
If C
is a full subcategory of C'
and D
is a full subcategory of D'
, then we can restrict
an adjunction L' ⊣ R'
where L' : C' ⥤ D'
and R' : D' ⥤ C'
to C
and D
.
The construction here is slightly more general, in that C
is required only to have a full and
faithful "inclusion" functor iC : C ⥤ C'
(and similarly iD : D ⥤ D'
) which commute (up to
natural isomorphism) with the proposed restrictions.