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I have a question concerning Remark 3.1.1.3 in Lurie's "Higher Topos Theory" about the definition of the class of marked anodyne morphisms. There, it is mentioned that "it suffices to allow $K$ to range over a set of representatives for all isomorphism classes of Kan complexes with only countably many simplices". Perhaps I'm wrong, but wouldn't it even be possible to replace the class (4) by the class (4') containing only the inclusion $K^{\flat} \hookrightarrow K^{\sharp}$, where $K$ is the simplicial set having two 0-simplices ( - $x$ and $y$ - ), two 1-simplices ( - one going from $x$ to $y$ and the other going from $y$ to $x$ - ), and two 2-simplices witnessing that the two 1-simplices shall be equivalences, and besides that only degenerate simplices.

This is intimately related to the proof of Proposition 3.1.1.6, specifically line 7 on page 150. There, $K$ is defined to denote the largest Kan complex contained in $X_{p(x)}$; but wouldn't it also be possible to define $K$ to be the simplicial set described above, and, after choosing an extension of $d_2(\sigma) : \Delta^1 \to X$ to $K$, argue that $d_2(\sigma)$ must be marked because $p$ has the right lifting property with respect to the inclusion $K^{\flat} \hookrightarrow K^{\sharp}$?

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    $\begingroup$ When you refer to the groupoid freely generated by $\Delta^1$, do you actually mean the groupoid in the sense of ordinary category theory (where the map of $\Delta^1$ has an actual inverse)? Or do you have something less structured in mind? $\endgroup$ Oct 6, 2015 at 2:50
  • $\begingroup$ Oh yes, you're right. I confused the groupoid freely generated by $\Delta^1$ with the "walking equivalence", i.e. the simplicial set having two 0-simplices ( - $x$ and $y$ - ), two 1-simplices ( - one going from $x$ to $y$ and the other going from $y$ to $x$ - ), and two 2-simplices witnessing the fact that the two 1-simplices shall be equivalences, and besides that only degenerate simplices. I will correct my post. Thank you for the hint. $\endgroup$ Oct 6, 2015 at 3:49
  • $\begingroup$ I'm hesitating because I'm not sure yet whether the inclusion $W^{\flat} \hookrightarrow W^{\sharp}$, where $W$ denotes the "walking equivalence", is marked anodyne at all. Let me sleep on that ... $\endgroup$ Oct 6, 2015 at 5:43
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    $\begingroup$ @RooibosTee It's not clear to me that the simplicial set you describe is genuinely the "walking equivalence". I think there are some coherence issues. On the other hand, the nerve of the ordinary groupoid generated by $\Delta^1$ is a model for "fully coherent" equivalences. $\endgroup$
    – Zhen Lin
    Oct 6, 2015 at 13:39
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    $\begingroup$ @RooibosTee: yes, always :-) I notice now, though, that Zhen Lin had already mentioned E^1 by a different name, since it is exactly the nerve of the ordinary groupoid generated by Δ[1], which he suggests. $\endgroup$ Oct 6, 2015 at 21:34

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