(sliced adjoints)
Let
be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).
Then:
For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form
where:
$L_{/b}$ is the evident induced functor (applying $L$ to the entire triangle diagrams in $\mathcal{C}$ which represent the morphisms in $\mathcal{C}_{/b}$);
$R_{/b}$ is the composite
of
the evident functor induced by $R$;
the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).
For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form
where:
$R_{/b}$ is the evident induced functor (applying $R$ to the entire triangle diagrams in $\mathcal{D}$ which represent the morphisms in $\mathcal{D}_{/b}$);
$L_{/b}$ is the composite
of
the evident functor induced by $L$;
the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).
(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 $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$
the adjunction counit $\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d$
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 $f$ in $\mathcal{D}_{/L(b)}$, by (at the top of the diagrams) the above expression of adjuncts between $\mathcal{C}$ and $\mathcal{D}$ and (at the bottom) by the triangle identity.
(2a) and (2b) are equivalent expression of the same morphism $\tilde f$ in $\mathcal{C}_{/b}$, by the universal property of the pullback.
Hence:
starting with a morphism as in (1a) and transforming it to $(2)$ 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) $\leftrightarrow$ (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) $\leftrightarrow$ (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 $\tau$ to their adjuncts $\widetilde{\tau}$, 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.