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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?

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    $\begingroup$ 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
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    $\begingroup$ "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

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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.

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