# Contents

## Definition

A simplicial set $X$ is (sometimes) called reduced if it has a single vertex, ${X}_{0}\simeq *$.

More generally, for $n\in ℕ$ a simplicial set is $n$-reduced if its $n$-skeleton is the point, ${\mathrm{sk}}_{n}X=\Delta \left[0\right]$.

## Properties

### Reflection

Write ${\mathrm{sSet}}_{0}↪$ sSet for the full subcategory inclusion of the reduced simplicial sets into all of them.

This is a reflective subcategory. The reflector

$\mathrm{red}:\mathrm{sSet}\to {\mathrm{sSet}}_{0}$red : sSet \to sSet_0

identifies all vertices of a simplicial set.

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

### Coreflection

The inclusion ${\mathrm{sSet}}_{0}↪{\mathrm{sSet}}^{*/}$ 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)