model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
A locally cartesian closed model category is a locally cartesian closed category which is equipped with the structure of a model category in a compatible way, namely such that all right base change-adjunctions along fibrations are Quillen adjunctions, hence such that all their dependent product-functors are right Quillen functors.
(relation to cartesian closed model categories)
Beware that, despite the terminology, the axioms on a locally cartesian closed model category (Def. ) do not imply that the underlying model category (or any of its slice model categories) is a cartesian closed model category – and in most examples this is not the case. Namely, the axioms here (2) only require Quillen functors in one variable (the second variable for internal homs, with the other variable a fixed fibrant object) where those of a cartesian closed model category require Quillen bifunctors.
(locally cartesian closed model category)
A locally cartesian closed model category is
a model category $\mathcal{C}$,
whose underlying category is a locally cartesian closed category
such that for every fibration between fibrant objects
the dependent product/base change adjunction
is a Quillen adjunction between the corresponding slice model structures.
Concretely, this means that both cofibrations and trivial cofibrations are stable under pullback along fibrations between fibrant objects.
Equivalently this means that for all fibrations $A \to B$ the internal hom adjunction in the slice category over $B$
is a Quillen adjunction.
Any right proper model category whose underlying category is locally cartesian closed and in which the cofibrations are the monomorphisms is a locally cartesian closed model category.
The fiber product/pullback functor $g^\ast$
is a left adjoint by local cartesian closure of the underlying category,
preserves cofibrations because these are the monomorphisms and hence are preserved by pullback (by this prop.),
preserves weak equivalences, and hence acyclic cofibrations by the previous item, due to right properness – using here the assumption (1) that $g$ is a fibration.
In summary this means that $g^\ast$ is a left Quillen functor.
Example subsumes the following classes of examples, in increasing generality:
any injective global model structure on simplicial presheaves,
any injective local model structure on simplicial presheaves whose weak equivalences are detected stalk-wise (by the discussion there),
any right proper Cisinski model structure.
It is easy to see that the $(\infty,1)$-category presented by a locally cartesian closed model category is itself locally cartesian closed: With the assumption (1) that $g$ is a fibration between fibrant objects, it follows (by this Prop) that pullback along $g$ models the correct homotopy pullback.
Conversely, any locally presentable locally cartesian closed $(\infty,1)$-category can be presented by some right proper Cisinski model category, which is therefore a locally cartesian closed model category; see there for the proof.
Last revised on May 31, 2023 at 12:22:48. See the history of this page for a list of all contributions to it.