nLab simplicial weak equivalence

Idea

A simplicial weak equivalence is a weak equivalence in the standard homotopy theory of simplicial sets, hence with respect to the classical model structure on simplicial sets.

Definition

A morphism $f$ between simplicial sets is a simplicial weak equivalence if any of the following equivalent conditions is satisfied (where $Ex^\infty$ denotes Kan fibrant replacement):

• $Ex^\infty(f)$ is a simplicial homotopy equivalence.

• $Ex^\infty(f)$ has the right homotopy lifting property with respect to $\partial\Delta^n\to\Delta^n$.

• $Ex^\infty(f)$ induces isomorphisms on simplicial homotopy groups, i.e., $Ex^\infty(f)$ induces an isomorphism on $\pi_0$ and all homotopy groups for any choice of basepoints.

• $Hom(f, A)$ is a simplicial homotopy equivalence for every Kan complex $A$ (where $Hom$ denotes the hom-functor).

• $f$ has the right homotopy lifting property with respect to $Sd^i \partial\Delta^n \to Sd^i \Delta^n$ (allowing subdivisions for homotopies also).

• $f$ belongs to the class of weak equivalences in the classical model structure on simplicial sets, whose cofibrations are monomorphisms and fibrant objects are Kan complexes.

• The morphism $f$ is a composition of a trivial cofibration and a trivial fibration on the classical model structure on simplicial sets, both of which are defined using lifting properties.

• Applying the category of elements functor produces a Thomason weak equivalence of categories. The class of Thomason weak equivalences forms the smallest basic localizer, i.e., the smallest class of functors between small categories that contains identities, is closed under retracts and the 2-out-of-3 property, contains all functors $A \to 1$ for which the category A has a terminal object, and is locally determined: if $u\colon A\to B$ and $w\colon B\to C$ are functors, with $v=w\circ u\colon A\to C$, and for any $c\in C$ the induced functor of comma categories $v/c\to w/c$ is a Thomason weak equivalence, then so is $u$.

Related concepts

• simplicial homotopy equivalence

• weak equivalence

• weak homotopy equivalence