nLab
reduced simplicial set
Context
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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents
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 .

Revised on February 20, 2017 07:28:54
by

Urs Schreiber
(94.220.94.100)