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 cartesian model category (alias 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.
A cartesian closed model category (following Rezk 09, 2.2 and Simpson 12) is
equipped with a model category-structure
that satisfies
the following equivalent axioms:
For $f \colon X \to Y$ and $f' \colon X' \to Y'$ cofibrations, the induced morphism
is a cofibration that is a weak equivalence if at least one of $f$ or $f'$ is;
For $f \colon X \to Y$ a cofibration and $f' \colon A \to B$ a fibration, the induced morphism
is a fibration, and a weak equivalence if at least one of $f$ or $f'$ is.
the unit axiom:
The terminal object is cofibrant.
Cartesian monoidal model categories include:
the standard Quillen model structure on topological spaces on compactly generated weakly Hausdorff topological spaces
(see there)
the fine model structure on topological G-spaces
(see there)
the canonical model structure on categories (and that on groupoids)
(see there)
the model structure for complete Segal spaces
(see there)
the model structure for Theta-spaces
(see there)
cartesian closed category, locally cartesian closed category
cartesian closed (∞,1)-categorylocally cartesian closed (∞,1)-category
Charles Rezk, A cartesian presentation of weak $n$-categories, Geom. Topol. 14(1): 521-571 (2010). (arXiv:0901.3602, doi:10.2140/gt.2010.14.521)
Carlos Simpson, Chapter 10 of: Homotopy Theory of Higher Categories, Cambridge University Press 2011 (hal:00449826, ISBN:9780521516952)
Last revised on November 1, 2023 at 16:56:50. See the history of this page for a list of all contributions to it.