3
$\begingroup$

It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration.

Now, I see that there is a section, by using lifting property with $\emptyset \to Y$ on the left (all objects are cofibrant!). I see from 2-out-of-3 that this section is an equivalence. How can I see that it is a cofibration?

Thank you, Sasha

  • 5
    OK, I now see how to show it. It works only in our specific model category of simplicial sets, though. Indeed, I build a section as above; and it is a weak equivalence. That it is a cofibration just follows from it being an inverse (from the correct direction), since it implies that it is a monomorphism, hence a cofibration (since in our model structure cofibrations are exactly monomorphisms). Why I have not succedded: Because I wanted to much generality.2012-04-01

0 Answers 0