nLab reduced simplicial set



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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].



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.


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.

As a model for \infty-groups

There is a Quillen equivalence

(GW¯):sGrpW¯ΩsSet 0 (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 GG, the simplicial loop space construction, is a concrete model for the loop space construction with values in simplicial groups.

Last revised on June 11, 2022 at 16:46:26. See the history of this page for a list of all contributions to it.