nLab compactly generated model category


under construction – warning – currently inconsistent


Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Compact objects




A cofibrantly generated simplicial model category CC is compactly generated if

  • there exists a small set SObj(C)S \subset Obj(C) of objects

  • such that

    1. each KSK \in S is cofibrant;

    2. each KSK \in S is a homotopy compact object: for all filtered colimit diagram Y:DCY : D \to C the morphism

      𝕃lim iC(K,Y i)C(K,𝕃lim iY i) \mathbb{L}\lim_{\to_i} C(K, Y_i) \simeq C(K, \mathbb{L}\lim_{\to_i} Y_i)

      (where 𝕃lim \mathbb{L}\lim_\to denotes the Ho CHo_C is the homotopy category of CC) is a weak homotopy equivalence in sSet;

    3. a morphism XYX \to Y in CC is a weak equivalence precisely if for all KSK \in S the induced morphism

      Ho C(K,X)Ho C(K,Y) Ho_C(K, X) \to Ho_C(K,Y)

      is a bijection.

Something needs to be added/fixed here!!

See (Jardine11, page 14), (Marty, def 1.7).


Page 88 (14) of

  • J. F. Jardine, Representability theorems for presheaves of spectra J. Pure Appl. Algebra

    215 (2011) (pdf)

Def. 1.7 of

  • Florian Marty, Smoothness in relative geometry (2009) (pdf)

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.