I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are generated by a (small) set of weak equivalences under filtered colimits. It seems so doable, yet I am at a loss. The only references I have found on the matter, state it as a consequence of the fact that sSet is a combinatorial model category so that's no help.

Maybe his alternative characterisation of accessibility (Proposition A.2.6.5) can help?

Any help and references would be much appreciated.

Edit for clarification: I need to prove that sSet is a combinatorial model category via Smith's theorem, meaning that I can only use that sSet is locally presentable and that I have to prove a few conditions for the classes of weak equivalences and cofibrations. One of these conditions is that weak equivalences are generated by filtered colimits from a set. Therefore I cannot use that sSet is a model category, and I have to prove this statement in some more direct way.

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are generated by a (small) set of weak equivalences under filtered colimits. It seems so doable, yet I am at a loss. The only references I have found on the matter, state it as a consequence of the fact that sSet is a combinatorial model category so that's no help.

Maybe his alternative characterisation of accessibility (Proposition A.2.6.5) can help?

Any help and references would be much appreciated.

Edit for clarification: I need to prove that sSet is a combinatorial model category via Smith's theorem, meaning that I can only use that sSet is locally presentable and that I have to prove a few conditions for the classes of weak equivalences and cofibrations. One of these conditions is that weak equivalences are generated by filtered colimits from a set. Therefore I cannot use that sSet is a model category, and I have to prove this statement in some more direct way.

Also, weak equivalences are the morphisms that become weak homotopy equivalences in the realisation.

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are generated by a (small) set of weak equivalences under filtered colimits. It seems so doable, yet I am at a loss. The only references I have found on the matter, state it as a consequence of the fact that sSet is a combinatorial model category so that's no help.

Maybe his alternative characterisation of accessibility (Proposition A.2.6.5) can help?

Any help and references would be much appreciated.

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are generated by a (small) set of weak equivalences under filtered colimits. It seems so doable, yet I am at a loss. The only references I have found on the matter, state it as a consequence of the fact that sSet is a combinatorial model category so that's no help.

Maybe his alternative characterisation of accessibility (Proposition A.2.6.5) can help?

Any help and references would be much appreciated.

Edit for clarification: I need to prove that sSet is a combinatorial model category via Smith's theorem, meaning that I can only use that sSet is locally presentable and that I have to prove a few conditions for the classes of weak equivalences and cofibrations. One of these conditions is that weak equivalences are generated by filtered colimits from a set. Therefore I cannot use that sSet is a model category, and I have to prove this statement in some more direct way.