Let be a pointed fibrant simplicial set. The homotopy group , as a set, is defined by , i.e. the set of the pointed homotopy classes of all pointed simplicial maps from to . as sets.
An element is said to be spherical if for all .
Given a spherical element , then its representing map factors through the simplicial quotient set . Conversely, any simplicial map gives a spherical element , where is the nondegenerate element in . This gives a one-to-one correspondence from the set of spherical elements in to the set of simplicial maps .
Path product and fundamental groupoids
Let . A path is a simplicial map .