Holmstrom Anodyne extension

Consider the category of simplicial sets. We define the set $I$ as the set of inclusions $\partial \Delta [n] \to \Delta [n]$ for $n \geq 0$. Define $J$ to be the set of inclusions $\Lambda^r[n] \to \Delta[n]$ for $n>0, \ 0 \leq r \leq n$. A map $f$ is a cofibration iff it is in $I-cof$. A map is a (Kan) fibration iff it is in $J-inj$. A map $f$ is a weak equivalence iff its geometric realization is a weak equivalence of topological spaces. The maps in $J-cof$ are called anodyne extensions. See Cofibrantly generated for the notation.

Every anodyne extension is a trivial cofibration of simplicial sets.

I think the anodyne extensions include the maps $\Lambda_k^n \to \Delta^n$. Possibly these suffice for certain “testing”.

Another formulation: A class of simplicial set monomorphisms is called saturated if contains all IMs, is closed under pushouts, retracts, countable compositions and arbitrary disjoint unions. An anodyne extension is a member of the smallest saturated class which contains the standard inclusions of horns.

Kan fibrations have the RLP wrt all standard inclusions of horns, and hence wrt all anodyne extensions.

nLab page on Anodyne extension