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Recall:

Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we have the implication $gf\in S \Rightarrow g\in S$.

Recall that the class of inner-anodyne morphisms in the category of simplicial sets is defined to be the class $\operatorname{llp}(\operatorname{rlp}(E))$, where $E$ denotes the set of inner-horn inclusions $\iota^n_k:\Lambda^n_k \hookrightarrow \Delta^n$ for $0<k<n.$

Question:

Is the class of inner-anodynes right-cancellative with respect to the class of monomorphisms? This is certainly the case for the Joyal-trivial cofibrations, but being inner-anodyne is a rather stronger condition.

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Incidentally, do you know if right plus left anodyne implies inner anodyne? –  Akhil Mathew Dec 3 '11 at 7:32
    
@Akhil: The map $\Delta[1] \to \Delta[2]$ induced by $\delta_1 : [1] \to [2]$ is both left and right anodyne, but it's not inner anodyne since it's not bijective on $0$-simplices. –  Karol Szumiło Dec 3 '11 at 8:41
    
@Karol: thanks! –  Akhil Mathew Dec 3 '11 at 14:24
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