It was meant to be the other way around, of course.

]]>Fixed an error in the phrasing of the property that Ex^\infty reflects weak equivalences.

]]>Please feel invited to edit the entry to clarify/fix.

Recently I made “simplicial weak equivalence” redirect to *classical model structure on simplicial sets*, but it would deserve it’s own entry.

Ex^∞(f) is a simplicial weak equivalences if and only if f is a simplicial homotopy equivalence.

Isn't that incorrect? It's clear that the statement

Ex^∞(f) is a weak equivalence in the classical model structure if and only if f is a weak equivalence in the classical model structure

is true. But, AFAIK, the class of homotopy equivalences expressed using the cylinder X+X -> X×Δ[1] does not coincide with the class of weak equivalences. (if these do coincide, it's surprisingly hard to find this fact explicitly stated...)

Was the statement meant the other way around? i.e.

Ex^∞(f) is a simplicial homotopy equivalences if and only if f is a simplicial weak equivalence.

is also true, since Ex^∞(f) is a morphism between fibrant objects, so it is a homotopy equivalence iff it is a weak equivalence.

I'm assuming "simplicial homotopy equivalence" means you have simplicial homotopies fg=>1 and 1=>gf.

I'm assuming the term "simplicial weak equivalence" means a weak equivalence in the classical model structure, but AFAICT that term is barely used on the ncatlab and is never defined. ]]>

Added redirects “Ex^∞” and “Ex-infinity”.

]]>Added more properties and applications.

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