# nLab modelizer

Contents

### 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

# Contents

## Idea

A modelizer is a presentation of the (∞,1)-category of ∞-groupoids, or at least, the homotopy category thereof.

## Definition

A modelizer is a category $M$ and a subcategory $W$ satisfying these conditions:

• $(M, W)$ is a saturated homotopical category, meaning $W$ is precisely the class of morphisms in $M$ that become invertible in the localization $M [W^{-1}]$.
• $M [W^{-1}]$ is equivalent to the category of weak homotopy types, i.e. Ho(Top) (with respect to weak homotopy equivalences).

More precisely, it is a category $M$ equipped with a functor $\pi : M \to Ho(Top)$ such that, for $W$ the class of morphisms inverted by $\pi$, the induced functor $M [W^{-1}] \to Ho(Top)$ is an equivalence of categories.

A morphism of modelizers $(M, W) \to (M', W')$ is a functor $F : M \to M'$ such that:

• $F$ sends morphisms in $W$ to morphisms in $W'$.
• The functor $M [W^{-1}] \to M' [W'^{-1}]$ so induced is an equivalence of categories.
• The composite $M \overset{F}{\to} M' \overset{\pi}{\to} Ho(Top)$ is isomorphic to $M \overset{\pi}{\to} Ho(Top)$.

An elementary modelizer is a modelizer whose underlying category is the category of presheaves on a test category, with the weak equivalences the ones described at the linked page.

## Examples

The main examples turn out to be model categories:

## Cisinski’s theorem

###### Theorem (Cisinski)

If $A$ is a test category, then there exists a model structure on $[A^{op}, Set]$ that is Quillen-equivalent to the standard model structure on $sSet$.