A simplicial set is (sometimes) called reduced if it has a single vertex, .
More generally, for a simplicial set is -reduced if its -skeleton is the point, .
This is a reflective subcategory. The reflector
identifies all vertices of a simplicial set.
Write for the category of pointed simplicial sets. There is also a full inclusion . This has a right adjoint which sends a pointed simplicial set to the subobject all whose -cells have as 0-faces the given point.
There is a Quillen equivalence
between the model structure on simplicial groups and the model structure on reduced simplicial sets (thus exhibiting both of these as models for infinity-groups). Its left adjoint , the simplicial loop space construction, is a concrete model for the loop space construction with values in simplicial groups.