Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

# Contents

## Definition

###### Definition

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

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

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

Last revised on September 13, 2023 at 19:38:01. See the history of this page for a list of all contributions to it.