nLab reduced simplicial set

Definition

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

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

Properties

Reflection

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

This is a reflective subcategory. The reflector

$$red : sSet \to sSet_0$$

identifies all vertices of a simplicial set.

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

Coreflection

The inclusion $sSet_0 \hookrightarrow sSet^{*/}$ 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.

As a model for $\infty$-groups

There is a Quillen equivalence

$$(G \dashv \bar W) \;\colon\; sGrp \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0$$

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 $G$, the simplicial loop space construction, is a concrete model for the loop space construction with values in simplicial groups.

Related concepts

• Eilenberg subcomplex

• simplicial group