# nLab compactly generated model category

under construction – warning – currently inconsistent

model category

## Definitions

• category with weak equivalences

• weak factorization system

• homotopy

• small object argument

• resolution

• ## Universal constructions

• homotopy Kan extension

• Bousfield-Kan map

• ## Refinements

• monoidal model category

• enriched model category

• simplicial model category

• cofibrantly generated model category

• algebraic model category

• compactly generated model category

• proper model category

• stable model category

• ## Producing new model structures

• on functor categories (global)

• on overcategories

• Bousfield localization

• transferred model structure

• Grothendieck construction for model categories

• ## Presentation of $(\infty,1)$-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

• ## Model structures

• Cisinski model structure
• ### for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

• Strom model structure?
• Thomason model structure

• model structure on presheaves over a test category

• model structure on simplicial groupoids

• on cubical sets

• related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

• ### for $n$-groupoids

• for 1-groupoids

• ### for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

• ### for $\infty$-algebras

#### general

• on monoids

• on algebas over a monad

• on modules over an algebra over an operad

• #### specific

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

• model structure on dg-modules

• ### for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on presheaves of spectra

• ### for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

• ### for stable $(\infty,1)$-categories

• on dg-categories
• ### for $(\infty,1)$-operads

• on modules over an algebra over an operad

• ### for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

• ### for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

• model structure for (2,1)-sheaves/for stacks

• #### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## 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).

## 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

Last revised on February 10, 2014 at 06:51:02. See the history of this page for a list of all contributions to it.