(sliced adjoints)
Let
be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) has all pullbacks (homotopy pullbacks).
Then:
For every object there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form
where:
For every object there is induced a pair of adjoint functors between the slice categories of the form
where:
is the evident induced functor (applying to the entire triangle diagrams in which represent the morphisms in );
is the composite
of
the evident functor induced by ;
the composition with the -counit at (i.e. the left base change along ).
(in 1-category theory)
Recall that (this Prop.) the hom-isomorphism that defines an adjunction of functors (this Def.) is equivalently given in terms of composition with
the adjunction unit
as follows:
Using this, consider the following transformations of morphisms in slice categories, for the first case:
(1a)
(2a)
(2b)
(1b)
Here:
(1a) and (1b) are equivalent expressions of the same morphism in , by (at the top of the diagrams) the above expression of adjuncts between and and (at the bottom) by the triangle identity.
(2a) and (2b) are equivalent expression of the same morphism in , by the universal property of the pullback.
Hence:
starting with a morphism as in (1a) and transforming it to and then to (1b) is the identity operation;
starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.
In conclusion, the transformations (1) (2) consitute a hom-isomorphism that witnesses an adjunction of the first claimed form (1).
The second case follows analogously, but a little more directly since no pullback is involved:
(1a)
(2)
(1b)
In conclusion, the transformations (1) (2) consitute a hom-isomorphism that witnesses an adjunction of the second claimed form (2).
(left adjoint of sliced adjunction forms adjuncts)
The sliced adjunction (Prop. ) in the second form (2) is such that the sliced left adjoint sends slicing morphism to their adjuncts , in that (again by this Prop.):
Last revised on July 26, 2021 at 04:26:21. See the history of this page for a list of all contributions to it.