under construction – warning – currently inconsistent
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
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
Def. 1.7 of
A different meaning of “compactly generated model category” is used in Definition 5.9 of