nLab sliced adjoint functors -- section

Slicing of adjoint functors

Proposition

(sliced adjoints)
Let

$$\mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}$$

be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).

Then:

1. For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form

   (1)$$\mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,}$$

   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

     $$R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b}$$

     of

     1. the evident functor induced by $R$;

     2. the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).

2. For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form

   (2)$$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,}$$

   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

     $$L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b}$$

     of

     1. the evident functor induced by $L$;

     2. the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).

The first statement appears, in the generality of (∞,1)-category theory, as HTT, prop. 5.2.5.1. For discussion in model category theory see at sliced Quillen adjunctions.

Proof

(in 1-category theory; see the proof of Proposition in the case of (∞,1)-categories.)

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).

Remark

(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.):

$$L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b}$$

The two adjunctions in admit the following joint generalisation, which is proven HTT, lem. 5.2.5.2. (Note that the statement there is even more general and here we only use the case where $K = \Delta^0$.)

Proposition

(sliced adjoints)
Let

$$\mathcal{C} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}$$

be a pair of adjoint ∞-functors, where the ∞-category $\mathcal{C}$ has all homotopy pullbacks. Suppose further we are given objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ together with a morphism $\alpha: c \to R(d)$ and its adjunct $\beta:L(c) \to d$.

Then there is an induced a pair of adjoint ∞-functors between the slice ∞-categories of the form

(3)$$\mathcal{C}_{/c} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}_{/d} \mathrlap{\,,}$$

where:

• $L_{/c}$ is the composite

  $$L_{/c} \;\colon\; \mathcal{C}_{/{c}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{D}_{/{L(c)}} \overset{\;\;\beta_!\;\;}{\longrightarrow} \mathcal{D}_{/d}$$

  of

  1. the evident functor induced by $L$;

  2. the composition with $\beta:L(c) \to d$ (i.e. the left base change along $\beta$).

• $R_{/d}$ is the composite

  $$R_{/d} \;\colon\; \mathcal{D}_{/{d}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{R(d)}} \overset{\;\;(\alpha^*\;\;}{\longrightarrow} \mathcal{C}_{/c}$$

  of

  1. the evident functor induced by $R$;

  2. the homotopy along $\alpha:c \to R(d)$ (i.e. the base change along $\alpha$).

Proposition (as well as HTT, lem. 5.2.5.2) can be deduced from the following general result (applied to the case where $\mathcal{X}$ and $\mathcal{Y}$ are the arrow categories of $\mathcal{C}$ and $\mathcal{D}$):

Proposition

Let be a (strictly) commutative diagram of quasicategories (a model of (∞,1)-categories), where $U$ is a cartesian fibration and $V$ is a cocartesian fibration. Suppose that $L$ and $F$ have right adjoints $R$ and $G$, and that the canonical map $U\circ R\to G\circ V$ is the identity natural transformation. Let $\alpha \colon c\to G d$ be a morphism with transpose $\beta \colon F c \to d$. Then the composite

$$\mathcal{X}_c\overset{L_c}{\to} \mathcal{Y}_{F c} \overset{\beta_!}{\to} \mathcal{Y}_d$$

admits a right adjoint, given by the composite

$$\mathcal{Y}_d \overset{R_d}{\to} \mathcal{X}_{G d} \overset{\alpha^*}{\to} \mathcal{X}_{c}.$$

Here the subscripts indicate taking fibers, and $\alpha^*$ and $\beta_!$ are co/cartesian transports for $U$ and $V$.

Proof

We show that $\beta_!\circ L_c$ has a right adjoint; the description for the right adjoint follows by inspecting the proof. By HTT, Proposition 5.2.4.1, it suffices to show that for each $y\in \mathcal{Y}_d$, the $\infty$-category $\mathcal{X}_c \times_{\mathcal{Y}_d} \times (\mathcal{Y}_d)_/y$ has a terminal object. For this, set $\mathcal{Y}_\beta =\Delta^1 \times _{\mathcal{D}} \mathcal{Y}$. The retraction $\mathcal{Y}_\beta \to \mathcal{Y}_d$ induces an equivalence $\mathcal{X}_c \times_{\mathcal{Y}_\beta} (\mathcal{Y}_\beta)_{/y} \overset{\simeq}{\to} \mathcal{X}_c \times_{\mathcal{Y}_d} (\mathcal{Y}_d)_{/y}$, so it suffices to show that $\mathcal{X}_c \times_{\mathcal{Y}_\beta} (\mathcal{Y}_\beta)_{/y}$ has a terminal object. This $\infty$-category can be written as $(\mathcal{X}\times_{\mathcal{Y}}\mathcal{Y}_{/y})\times _{\mathcal{C}\times_{\mathcal{D}}\mathcal{D}_{/V y}}\{(c,\beta)\}$. Using the fact that $U$ is a cartesian fibration and $\mathcal{Y}_{/y} \to \mathcal{Y}$ is a right fibration, we find that the projection $\mathcal{X}\times _{\mathcal{Y}}\mathcal{Y}_{/y} \to \mathcal{C}\times _{\mathcal{D}}\mathcal{D}_{/V y}$ is a cartesian fibration (with cartesian edges given by those whose images in $\mathcal{X}$ are cartesian). Moreover, $p$ preserves terminal objects by hypothesis. It follows that the fiber $p^{-1}(c,\alpha)$ has a terminal object, namely, the cartesian transport of the terminal object of $\mathcal{X}\times_{\mathcal{Y}}\mathcal{Y}_{/y}$. The claim follows.