under construction – warning – currently inconsistent
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 cofibrantly generated simplicial model category $C$ is compactly generated if
such that
each $K \in S$ is cofibrant;
each $K \in S$ is a homotopy compact object: for all filtered colimit diagram $Y : D \to C$ the morphism
(where $\mathbb{L}\lim_\to$ denotes the $Ho_C$ is the homotopy category of $C$) is a weak homotopy equivalence in sSet;
a morphism $X \to Y$ in $C$ is a weak equivalence precisely if for all $K \in S$ the induced morphism
is a bijection.
Something needs to be added/fixed here!!
See (Jardine11, page 14), (Marty, def 1.7).
Page 88 (14) of
215 (2011) (pdf)
Def. 1.7 of
A different meaning of “compactly generated model category” is used in Definition 5.9 of
Last revised on November 15, 2021 at 17:30:34. See the history of this page for a list of all contributions to it.