# transfinite composition of weak equivalences in sSet

Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition.

What's a reference for that?

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More specifics would be helpful. Do you mean that you have a transfinite composite of maps which are weak equivalences and want to show that all the objects are equivalent to the colimit? The homotopy colimit? Do you mean that you have two transfinite composites with a map between them that is a levelwise weak equivalence, and you want to show the colimits are weakly equivalent? The homotopy colimits? –  Tyler Lawson Jan 23 '10 at 18:20
Sorry if this was't clear: I want to know if the cocone component on the first object of the colimiting cocone that defines the transfinite composition is a weak equivalence. That's the morphism that is the "transfinite composite" of a transfinite sequence of weak equivalences. –  Urs Schreiber Jan 23 '10 at 20:39

• there is a functor $\mathrm{Ex}^\infty$, which replaces a simplicial set with a weakly equivalent fibrant replacement, and which commutes with filtered colimits. (See ch. 3 of Goerss-Jardine.)