nLab
compactly generated model category

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under construction – warning – currently inconsistent

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Compact objects

objects d∈C such that C(d,−) commutes with certain colimits

Models

Relative version

Contents

Definition

Definition

A cofibrantly generated simplicial model category C is compactly generated if

  • there exists a small set S⊂Obj(C) of objects

  • such that

    1. each K∈S is cofibrant;

    2. each K∈S is a homotopy compact object: for all filtered colimit diagram Y:D→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 → denotes the Ho C is the homotopy category of C) is a weak homotopy equivalence in sSet;

    3. a morphism X→Y in C is a weak equivalence precisely if for all K∈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).

References

Page 88 (14) of

  • Rick 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

Revised on August 28, 2012 14:49:24 by Urs Schreiber (89.204.138.213)
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