Let be a simplicial set. Then is fibrant if and only if every simplicial map has an extension for each .
Assume that is fibrant. Let . The elements are matching faces with respect to . This is because for and ,
Thus, since is fibrant, there exists an element such that for . Then, the representing map , , is an extension of .
Conversely let be any elements that are matching faces with respect to . Then the representing maps for defines a simplicial map such that the diagram
commutes for each .
By the assumption, there exists an extension such that . Let . Then for . Thus is fibrant.