nLab
model structure on reduced simplicial sets

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

The model structure on reduced simplicial sets is a presentation of the full sub-(∞,1)-category

∞Grpd 1 */{}^{*/}_{\geq 1} \hookrightarrow ∞Grpd */{}^{*/} \simeq Top */{}^{*/}

of pointed ∞-groupoids on those that are connected.

By the looping and delooping-equivalence, this is equivalent to the (∞,1)-category of ∞-groups and this equivalence is presented by a Quillen equivalence to the model structure on simplicial groups.

Definition

Definition

A reduced simplicial set is a simplicial set SS with a single vertex:

S 0=*. S_0 = * \,.

Write sSet 0sSet_0 \subset sSet for the full subcategory of the category of simplicial sets on those that are reduced.

Proposition

There is a model category structure on sSet 0sSet_0 whose

  • weak equivalences

  • and cofibrations

are those in the classical model structure on simplicial sets.

This appears as (Goerss-Jardine, ch V, prop. 6.2).

Properties

Proposition

The simplicial loop space functor GG and the delooping functor W¯()\bar W(-) (discussed at simplicial group) constitute a Quillen equivalence

(GW¯):sGrW¯GsSet 0 (G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0

with the model structure on simplicial groups.

This appears as (Goerss-Jardine, ch. V prop. 6.3).

Proposition

Under the forgetful functor U:sSet 0sSetU : sSet_0 \hookrightarrow sSet

  • a fibration f:XYf : X \to Y maps to a fibration precisely if it has the right lifting property against *S 1:=Δ[1]/Δ[1]* \to S^1 := \Delta[1]/ \partial \Delta[1];

In particular

  • every fibrant object maps to a fibrant object.

The first statment appears as (Goerss-Jardine, ch. V, lemma 6.6.). The second (an immediate consequence) appears as (Goerss-Jardine, ch. V, corollary 6.8).

References

A standard textbook reference is chapter V of

Revised on February 20, 2017 07:15:41 by Urs Schreiber (94.220.94.100)