Let be a space. We can have a fibrant simplicial set, namely the singular simplicial set , where is the set of all continuous maps from the -simplex to . However seems too large as there are uncountably many elements in each . On the other hand, we need the fibrant assumption to have simplicial homotopy groups. This means that the simplicial model for a given space cannot be too small. We wish to have a smallest fibrant simplicial set which will be the idea behind minimal simplicial sets.
Let be a fibrant simplicial set. For we say that if the representing maps and are homotopic relative to .
A fibrant simplicial set is said to be minimal if it has the property that implies .
Let be a fibrant simplicial set. is minimal iff for any , such that for all implies .
In other words, it means that a fibrant simplicial set is minimal iff for any two elements with all faces but one the same, then the missed face must be the same.