nLab compactly generated model category

under construction – warning – currently inconsistent

Definition

Definition

A cofibrantly generated simplicial model category $C$ is compactly generated if

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

• such that

  1. each $K \in S$ is cofibrant;

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

     $$\mathbb{L}\lim_{\to_i} C(K, Y_i) \simeq C(K, \mathbb{L}\lim_{\to_i} Y_i)$$

     (where $\mathbb{L}\lim_\to$ denotes the $Ho_C$ is the homotopy category of $C$) is a weak homotopy equivalence in sSet;

  3. a morphism $X \to Y$ in $C$ is a weak equivalence precisely if for all $K \in S$ the induced morphism

     $$Ho_C(K, X) \to Ho_C(K,Y)$$

     is a bijection.

Something needs to be added/fixed here!!

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

Related concepts

• compactly generated triangulated category

• compactly generated (∞,1)-category

References

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

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society Volume 82, Issue 2, 2000 (pdf, doi:10.1112/S0024611501012692)