# Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

## 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.

• 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

The fact that in $\operatorname{Set}_{\Delta}$ the class of inner anodyne maps has the right cancellation property (within the class of monomorphisms) was proven by Danny Stevenson in his 2016 paper Stability for inner fibrations revisited, which was also published in TAC.