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 standard model structure on simplicial sets.

This appears as (GoerssJardine, 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 (GoerssJardine, 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 (GoerssJardine, ch. V, lemma 6.6.). The second (an immediate consequence) appears as (GoerssJardine, ch. V, corollary 6.8).

References

A standard textbook reference is chapter V of

Revised on April 14, 2012 10:23:31 by Urs Schreiber (82.113.106.163)