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? 
 A: I don't have a complete reference (and like Tyler, I don't know exactly what result you want).  But here are some observations:


*

*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.)

*if you have a transfinite composition(s) in which all the simplicial sets are fibrant, it is straightforward to understand their behaviour with respect to weak equivalences, using the formula for simplicial homotopy groups, which gives the right homotopy groups for Kan complexes; (in particular, simplicial homotopy groups commute with filtered colimits in this setting.)
[Added later:] in particular, in any transfinite composition of weak equivalences between Kan complexes, the cocone componenent of the first object (i.e., the map from the first object to the colimit of the trasnfinite sequence) has to be a weak equivalence.
[Added:] These two facts taken together imply that a "transfinite composition" of weak equivalences is a weak equivalence.  
A: This answer serves to record two explicit proofs of this fact in the literature:
Corollary 5.1 in Raptis and Rosický, “The accessibility rank of weak equivalences”, arXiv:1403.3042v2.
Theory and Applications of Categories 30:19 (2015), 687—703.
Theorem 4.6 in Barnea and Schlank, “Model structures on Ind categories and the accessibility rank of weak equivalences”, arXiv:1407.1817v6.
