nLab modelizer

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\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 MM and a subcategory WW satisfying these conditions: * (M,W)(M, W) is a saturated homotopical category, meaning WW is precisely the class of morphisms in MM that become invertible in the localization M[W 1]M [W^{-1}]. * 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 MM equipped with a functor π:MHo(Top)\pi : M \to Ho(Top) such that, for WW the class of morphisms inverted by π\pi, the induced functor M[W 1]Ho(Top)M [W^{-1}] \to Ho(Top) is an equivalence of categories.

A morphism of modelizers (M,W)(M,W)(M, W) \to (M', W') is a functor F:MMF : M \to M' such that: * FF sends morphisms in WW to morphisms in WW'. * The functor M[W 1]M[W 1]M [W^{-1}] \to M' [W'^{-1}] so induced is an equivalence of categories. * The composite MFMπHo(Top)M \overset{F}{\to} M' \overset{\pi}{\to} Ho(Top) is isomorphic to MπHo(Top)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 AA is a test category, then there exists a model structure on [A op,Set][A^{op}, Set] that is Quillen-equivalent to the standard model structure on sSetsSet.

References

Last revised on June 19, 2022 at 10:31:08. See the history of this page for a list of all contributions to it.