# nLab compactly generated model category

Contents

under construction – warning – currently inconsistent

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

• 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.