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, ${\mathrm{sk}}_{n}X=\Delta [0]$.

Properties

Reflection

Write ${\mathrm{sSet}}_0\hookrightarrow$ sSet for the full subcategory inclusion of the reduced simplicial sets into all of them.

This is a reflective subcategory. The reflector

$$\mathrm{red}:\mathrm{sSet}\to {\mathrm{sSet}}_0$$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 ${\mathrm{sSet}}_0\hookrightarrow {\mathrm{sSet}}^{*/}$. 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 ${\mathrm{sSet}}_0\hookrightarrow {\mathrm{sSet}}^{*/}$ 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.

Related concepts

• Eilenberg subcomplex

• simplicial group