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A monoidal model category S\mathbf{S} with extra properties guaranteeing a particularly good homotopy theory of S\mathbf{S}-enriched categories is referred to in (Lurie) as an excellent model category.

(The term excellent model category has also been used by Lumsdaine and Shulman for a slightly different notion; see type-theoretic model category. This article is about Lurie’s sense.)



Let S\mathbf{S} be a symmetric monoidal model category. It is called excellent if

This is (Lurie, def. A.3.2.16), except that he also includes the “invertibility hypothesis” (see below).



The invertibility hypothesis

Lurie originally included the an additional condition in the definition, called the invertibility hypothesis. Fortunately, it was shown later by Lawson that the invertibility hypothesis is redundant: any model category satisfying the other axioms of an excellent model category always satisfies the invertibility hypothesis.

Description of the invertibility hypothesis

The invertibility hypothesis states:

  • for any homotopy equivalence ff in an S\mathbf{S}-enriched category CC, the localization functor CC[f 1]C \to C[f^{-1}] is a weak equivalence of S\mathbf{S}-enriched categories.

The invertibility hypothesis requires some more explanation. Firstly, by a “homotopy equivalence” we mean a morphism that becomes an isomorphism in the homotopy category hCh C, which is defined by applying to the hom-objects of CC the monoidal localization functor ShS\mathbf{S}\to h \mathbf{S} from S\mathbf{S} to its homotopy category (defined in the usual way for a model category by inverting its weak equivalences). This makes hCh C an hSh \mathbf{S}-enriched category, and we ask that ff be invertible in the underlying ordinary category of hCh C.

Secondly, by C[f 1]C[f^{-1}] we mean a homotopy invariant notion of “localization”. One way to define this is to assume that ff is classified by a functor [f]:2 SC[f]:\mathbf{2}_{\mathbf{S}} \to C that is a cofibration in the model structure on enriched categories, where 2 S\mathbf{2}_{\mathbf{S}} denotes the S\mathbf{S}-enriched interval category, and take the ordinary localization which can be defined as a pushout

2 S [f] C 2 S Cf 1.\array{ \mathbf{2}_{\mathbf{S}} & \xrightarrow{[f]} & C \\ \downarrow && \downarrow \\ \mathbf{2}^\cong_{\mathbf{S}} & \to & C\langle f^{-1}\rangle.}

Here 2 S \mathbf{2}^\cong_{\mathbf{S}} denotes the 2-object contractible groupoid regarded as an S\mathbf{S}-category. Another way to define it (without this assumption on ff) is to factor the functor 2 S2 S \mathbf{2}_{\mathbf{S}}\to \mathbf{2}^\cong_{\mathbf{S}} through a cofibration 2 SE\mathbf{2}_{\mathbf{S}}\to E and weak equivalence E2 S E\to \mathbf{2}^\cong_{\mathbf{S}} in the model structure on enriched categories, and take the ordinary pushout

2 S [f] C E C[f 1].\array{ \mathbf{2}_{\mathbf{S}} & \xrightarrow{[f]} & C \\ \downarrow && \downarrow \\ E & \to & C[f^{-1}].}

Finally, by a “weak equivalence of S\mathbf{S}-enriched categories” we mean a weak equivalence in the model structure on enriched categories.

Simplicial sets satisfy the invertibility hypothesis

We Lurie says (A.3.2.18) that this is “one of the main theorems” of DK80, but some further details may be helpful to understand how this comes about. Note that this proof is not subsumed by Lawson’s proof that all excellent model categories satisfy the invertibility hypothesis; instead Lawson uses this theorem to transfer the property from simplicial sets to all others excellent model categories.

The relevant theorem appears to be Proposition 10.4 of DK80 which states

Let VBsOCatV\to B\in \mathrm{s} O-Cat be a strong cofibration. Then the induced map BB[V 1]B\to B[V^{-1}] is a weak equivalence if and only if every map of π 0B\pi_0 B which is in the image of π 0V\pi_0 V is invertible.

Here sOCat\mathrm{s} O-Cat is the category of simplicially enriched categories with a fixed object set OO. Thus this proposition does not apply as written to the map [f]:2 SC[f] : \mathbf{2}_{\mathbf{S}}\to C from the invertibility hypothesis, since it is not a bijection on objects. Instead we have to replace 2 S\mathbf{2}_{\mathbf{S}} by a simplicially enriched category 2 S,O\mathbf{2}_{\mathbf{S},O} with set of objects O=ob(C)O = ob(C) in which the only nonidentity morphism “is” ff. Note that [f][f] being a cofibration implies that it is injective on objects, so that ff has distinct domain and codomain; thus in 2 S,O\mathbf{2}_{\mathbf{S},O} the unique nonidentity morphism is not an endomorphism and hence we don’t have to worry about any nontrivial composites.

Now the construction of B[V 1]B[V^{-1}] in DK80 is essentially the same as the pushout quoted above from Lurie. And π 0B\pi_0 B denotes taking the homwise set of connected components of a simplicially enriched category, which can be factored as first taking the enriched homotopy category (which is a category enriched over h(sSet)h(sSet)) and then its underlying ordinary category; thus the assumption that “every map of π 0B\pi_0 B which is in the image of π 0V\pi_0 V is invertible” is equivalent to the assumption that ff is a homotopy equivalence.

Finally, a “strong cofibration” is defined in DK80 to be a cofibration with cofibrant domain, and 2 S,O\mathbf{2}_{\mathbf{S},O} is easily checked to indeed be cofibrant (essentially because the unit object of sSetsSet is cofibrant).

All excellent model categories satisfy the invertibility hypothesis

Lawson now shows that:

  1. Cubical sets are monoidally Quillen equivalent to simplicial sets, hence also satisfy the invertibility hypothesis.
  2. Any excellent model category admits a monoidal Quillen functor from cubical sets, hence also satisfies the invertibility hypothesis.


Last revised on February 19, 2019 at 11:43:50. See the history of this page for a list of all contributions to it.