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derived adjunction unit

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Definition

Definition

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

𝒞 QuRAALAA𝒟 \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 d𝒟d \in \mathcal{D} is a composition of the form

    Q(d)η Q(d)R(L(Q(d)))R(j L(Q(d)))R(P(L(Q(d))) 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. η\eta is the ordinary adjunction unit;

    2. Cof 𝒟i Q(d)Q(d)W 𝒟Fib 𝒟p Q(d)d\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 𝒟\mathcal{D};

    3. L(Q(d))W 𝒞Cof 𝒞j L(Q(d))P(L(Q(d)))Fib 𝒞q L(Q(d))*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 𝒞\mathcal{C};

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

    L(Q(R(P(c))))p R(P(c))LR(P(c))ϵ P(c)P(c) L(Q(R(P(c)))) \overset{ p_{R(P(c))} }{\longrightarrow} L R(P(c)) \overset{\epsilon_{P(c)}}{\longrightarrow} P(c)

    where

    1. ϵ\epsilon is the ordinary adjunction counit;

    2. cW 𝒞Cof 𝒞j cPcFib 𝒞q c*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 𝒞\mathcal{C};

    3. Cof 𝒟i R(P(c))Q(R(P(c)))W 𝒟Fib 𝒟p R(P(c))R(P(c))\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 𝒟\mathcal{D}.

Properties

Last revised on July 12, 2018 at 04:03:55. See the history of this page for a list of all contributions to it.