On a Conjecture by J.h.smith

We show that the class of weak equivalences of a combinatorial model category can be detected by an accessible functor into simplicial sets. The purpose of this short note is to prove the following result that was conjectured by J.H. Smith in [[2],p. 460]. 0.1. Theorem. For every combinatorial model categoryM, there is an accessible functor F :M→ SSet that detects the weak equivalences, i.e., a morphism f in M is a weak equivalence if and only if F (f) is a weak homotopy equivalence. Proof. By [[3], Theorem 1.1], there is a small category C, a set of morphisms S in SSet and a Quillen equivalence F : LSSSet M : G, where LSSSet denotes the left Bousfield localisation of SSet with the projective model structure at the set of morphisms S (see [4]). By [[3], Proposition 7.1], there is a fibrant replacement functor R :M→M that is accessible. Let u : ObC → C denote the inclusion of the objects of C (as a discrete category) into C. Then let F :M→ SSet be the following composition of functors M R →M G → SSet u ∗ → SSet ∏ → SSet. F is accessible because it is a composition of accessible functors. The functors G, u∗ and ∏ are accessible because they are right adjoints between locally presentable categories (see [[1], 1.66]). Since G is a right Quillen equivalence, a morphism f in M is a weak equivalence if and only if GR(f) is a weak equivalence in LSSSet . The functor GR maps into the category of fibrant (or S-local) objects, therefore GR(f) is a weak equivalence in LSSSet (i.e. S-local equivalence) if and only if it is a weak equivalence in SSet , i.e., a pointwise weak equivalence (see [[4], Theorem 3.2.13]). The morphism GR(f) is a pointwise weak equivalence if and only if u∗GR(f) is a pointwise weak equivalence in SSet . By the combinatorial definition of homotopy groups, the product functor ∏ detects the pointwise weak equivalences between pointwise fibrant objects. Since u∗GR takes values in pointwise fibrant objects, it follows that F detects the weak equivalences. As an immediate corollary, we have that the class of weak equivalences of a combinatorial model category is accessible and accessibly embedded. This was proved by different methods in [[5], Corollary A.2.6.6] and [[6], Theorem 4.1]. Received by the editors 2010-02-08 and, in revised form, 2010-04-13. Transmitted by J. Rosicky. Published on 2010-04-05. 2000 Mathematics Subject Classification: 55U35, 18C35.