nLab derived adjunction unit

Definition

Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be model categories, and let

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}$$

be a Quillen adjunction. Then

1. a derived adjunction unit at an object $d \in \mathcal{D}$ is a composition of the form

   $$Q(d) \overset{\eta_{Q(d)}}{\longrightarrow} R(L(Q(d))) \overset{R( j_{L(Q(d))} )}{\longrightarrow} R(P(L(Q(d)))$$

   where

   1. $\eta$ is the ordinary adjunction unit;

   2. $\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{Q(d)}}{\longrightarrow} Q(d) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{Q(d)}}{\longrightarrow} d$ is a cofibrant resolution in $\mathcal{D}$;

   3. $L(Q(d)) \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_{L(Q(d))}}{\longrightarrow} P(L(Q(d))) \underoverset{\in Fib_{\mathcal{C}}}{q_{L(Q(d))}}{\longrightarrow} \ast$ is a fibrant resolution in $\mathcal{C}$;

2. a derived adjunction counit at an object $c \in \mathcal{C}$ is a composition of the form

   $$L(Q(R(P(c)))) \overset{ L( p_{R(P(c))} ) }{\longrightarrow} L R(P(c)) \overset{\epsilon_{P(c)}}{\longrightarrow} P(c)$$

   where

   1. $\epsilon$ is the ordinary adjunction counit;

   2. $c \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_c}{\longrightarrow} P c \underoverset{\in Fib_{\mathcal{C}}}{q_c}{\longrightarrow} \ast$ is a fibrant resolution in $\mathcal{C}$;

   3. $\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{R(P(c))}}{\longrightarrow} Q(R(P(c))) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{R(P(c))}}{\longrightarrow} R(P(c))$ is a cofibrant resolution in $\mathcal{D}$.

Properties

• A Quillen adjunction is a Quillen equivalence precisely if both its derived adjunction units and derived adjunction counit are weak equivalences.

• A Quillen adjunction is a Quillen reflection precisely if the components of its derived adjunction counit are weak equivalences.

• A Quillen adjunction is a Quillen co-reflection precisely if the components of its derived adjunction unit are weak equivalences.

Related concepts

• adjunction unit

• derived functor

• Quillen equivalence, Quillen reflection