Let be simplicial maps. We say that is homotopic to (denoted by ) if there exists a simplicial map such that and for all . If is a simplicial subset of and are simplicial maps such that , we say that if there is a homotopy such that , and for all , , .
Let be a simplicial set. The elements are said to be matching faces with respect to if for and .
A simplicial set is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each :
Let be any elements that are matching faces with respect to . Then there exists an element such that for .
We define as the simplicial subset of generated by all for , where is the nondegenerate element.
Proposition: Let be a simplicial set. Then is fibrant if and only if every simplicial map has an extension for each .