reduced simplicial set
Paths and cylinders
simplicial set is (sometimes) called X reduced if it has a single vertex, . X 0 ≃ *
More generally, for
a simplicial set is n ∈ ℕ if its -reduced n - n skeleton is the point, . sk n X = Δ [ 0 ] Properties
sSet 0 ↪ sSet for the full subcategory inclusion of the reduced simplicial sets into all of them.
This is a
reflective subcategory. The reflector
red : sSet → sSet 0
red : sSet \to sSet_0
identifies all vertices of a simplicial set.
for the category of sSet * / pointed simplicial sets. There is also a full inclusion . This has a right adjoint sSet 0 ↪ sSet * / which sends a pointed simplicial set to the subobject all whose red : sSet * / → sSet 0 -cells have as 0-faces the given point. n Coreflection
into sSet 0 ↪ sSet * / pointed simplicial sets is coreflective. The coreflector is the Eilenberg subcomplex construction in degree 1. Model structure
There is a
model structure on reduced simplicial sets (see there) which serves as a presentation of the (∞,1)-category of pointed connected ∞-groupoids.
Revised on April 19, 2012 07:43:37