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

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \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 August 7, 2013 at 17:02:10. See the history of this page for a list of all contributions to it.