nLab
cartesian closed model category
Context
Model category theory
model category

Definitions
category with weak equivalences

weak factorization system

homotopy

small object argument

resolution

Morphisms
Quillen adjunction

Universal constructions
homotopy Kan extension

homotopy limit /homotopy colimit

Bousfield-Kan map

Refinements
monoidal model category

enriched model category

simplicial model category

cofibrantly generated model category

algebraic model category

compactly generated model category

proper model category

cartesian closed model category , locally cartesian closed model category

stable model category

Producing new model structures
on functor categories (global)

on overcategories

Bousfield localization

transferred model structure

Grothendieck construction for model categories

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

simplicial localization

(∞,1)-categorical hom-space

presentable (∞,1)-category

Model structures Cisinski model structure for $\infty$ -groupoids for ∞-groupoids

on topological spaces

Thomason model structure

model structure on presheaves over a test category

on simplicial sets , on semi-simplicial sets

model structure on simplicial groupoids

on cubical sets

on strict ∞-groupoids , on groupoids

on chain complexes /model structure on cosimplicial abelian groups

related by the Dold-Kan correspondence

model structure on cosimplicial simplicial sets

for $n$ -groupoids
for n-groupoids /for n-types

for 1-groupoids

for $\infty$ -groups
model structure on simplicial groups

model structure on reduced simplicial sets

for $\infty$ -algebras general
on monoids

on simplicial T-algebras , on homotopy T-algebra s

on algebas over a monad

on algebras over an operad ,

on modules over an algebra over an operad

specific
model structure on differential-graded commutative algebras

model structure on differential graded-commutative superalgebras

on dg-algebras over an operad

model structure on dg-modules

for stable/spectrum objects
model structure on spectra

model structure on ring spectra

model structure on presheaves of spectra

for $(\infty,1)$ -categories
on categories with weak equivalences

Joyal model for quasi-categories

on sSet-categories

for complete Segal spaces

for Cartesian fibrations

for stable $(\infty,1)$ -categories on dg-categories for $(\infty,1)$ -operads
on operads , for Segal operads

on algebras over an operad ,

on modules over an algebra over an operad

on dendroidal sets , for dendroidal complete Segal spaces , for dendroidal Cartesian fibrations

for $(n,r)$ -categories
for (n,r)-categories as ∞-spaces

for weak ∞-categories as weak complicial sets

on cellular sets

on higher categories in general

on strict ∞-categories

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

model structure for (2,1)-sheaves /for stacks

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Contents
Idea
A cartesian closed model category is a cartesian closed category that is equipped with the structure of a monoidal model category in a compatible way, which combines the axioms for a monoidal model category and an enriched model category .

Definition
A cartesian model category (following Rezk (2010) and Simpson (2012)) is a cartesian closed category equipped with a model structure that satisfies the following additional axioms:

(Pushout?product axiom ). If $f : X \to Y$ and $f' : X' \to Y'$ are cofibrations, then the induced morphism $(Y \times X') \cup^{X \times X'} (X \times Y') \to Y \times Y'$ is a cofibration that is trivial if either $f$ or $f'$ is.

(Unit axiom). The terminal object is cofibrant.

Examples
References
Last revised on March 9, 2016 at 16:37:36.
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