nLab modelizer

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:

• Tautologically, Top (with weak homotopy equivalences) is a modelizer. This extends to a model structure on topological spaces.
• Quillen showed that there is a model structure on simplicial sets that is Quillen-equivalent to the previous one; in particular, $sSet$ (with weak homotopy equivaleces) is a modelizer.
• Thomason showed that Cat (with nerve equivalences) is a modelizer by exhibiting a model structure Quillen-equivalent to the on $sSet$.

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 $sSet$.

Related concepts

• category with weak equivalences

• localizer

References

• Grothendieck, A. (1983) Pursuing stacks, §28.
• Cisinski, D.-C. (2002, 2006) Les préfaisceaux comme modèles des types d’homotopie.