It sais here that the canonical model structure on $Cat$ is cofibrantly generated. I found out that a generating trivial cofibration is the functor $I:*\rightarrow E $, where $E$ is the category with 2 isomorphic objects (and $*$ is mapped into one of them), but i can't find any information about the set of generating cofibrations (and I don't know how to work it out on my own). Any suggestion?

More in general does anyone know a book/article in which this argument is treated extensively?

  • 2
    $\begingroup$ A reference for this is Rezk's article “A model structure on categories”. $\endgroup$
    – Dmitri P.
    Jan 18, 2016 at 10:00
  • $\begingroup$ Are you sure the title is correct? I cannot find it anywhere. $\endgroup$
    – LK512
    Jan 18, 2016 at 14:26
  • 1
    $\begingroup$ "A model category for categories". It's available here. $\endgroup$
    – Zhen Lin
    Jan 18, 2016 at 14:51
  • $\begingroup$ @LK512: What Zhen Lin said. $\endgroup$
    – Dmitri P.
    Jan 18, 2016 at 22:25

1 Answer 1


The generating cofibrations are the inclusions $\emptyset \hookrightarrow \{ \ast \}$ and $\{ 0, 1 \} \hookrightarrow \{ 0 \to 1 \}$ plus the projection $\{ 0 \rightrightarrows 1 \} \to \{ 0 \rightarrow 1 \}$. It is easy to see that a functor has the right lifting property with respect to these if and only if it is fully faithful and surjective on objects.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .