nLab
reduced simplicial set

Contents

Definition

A simplicial set XX is (sometimes) called reduced if it has a single vertex, X 0*X_0 \simeq *.

More generally, for nn \in \mathbb{N} a simplicial set is nn-reduced if its nn-skeleton is the point, sk nX=Δ[0]sk_n X = \Delta[0].

Properties

Reflection

Write sSet 0sSet_0 \hookrightarrow sSet for the full subcategory inclusion of the reduced simplicial sets into all of them.

This is a reflective subcategory. The reflector

red:sSetsSet 0 red : sSet \to sSet_0

identifies all vertices of a simplicial set.

Write sSet */sSet^{*/} for the category of pointed simplicial sets. There is also a full inclusion sSet 0sSet */sSet_0 \hookrightarrow sSet^{*/}. This has a right adjoint red:sSet */sSet 0red : sSet^{*/} \to sSet_0 which sends a pointed simplicial set to the subobject all whose nn-cells have as 0-faces the given point.

Coreflection

The inclusion sSet 0sSet */sSet_0 \hookrightarrow sSet^{*/} into 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 by Urs Schreiber (82.169.65.155)